This video is a masterclass in exam-oriented pragmatism that distills complex physics into a tactical checklist for high scores. It prioritizes the efficiency of standardized testing over deep scientific inquiry, making it a perfect survival guide for the COMEDK system.
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Top 100 Physics MCQs for COMEDK 2026 🔥 Score 45+ Guaranteed | Most Expected Questions本站添加:
So hello and welcome to crash course.
Today is time for me to discuss the top 100qs. Uh so instead of discussing 100 questions, I will be discussing 101 questions. So one question is extra for uh good luck, right? And obviously we have discussed some of the questions and most of the questions throughout our live lectures but maybe you have missed some of them, maybe you have not understood some of them. So let us uh discuss some of the questions again so that your conceptual clarity remains very very high. Okay. Remember in comet K the conceptual clarity is very very very important. So when you are answering a question you need know need to know exactly what you are answering.
Okay. So without any delay let us start with the question number one. And the questions that that I have personally chosen is based on the feedback during the lectures whatever we have received.
Some of you have had difficulty understanding some of the concepts. So those questions are given priorities. So difficult questions obviously we are going to discuss. So don't go anywhere.
Please watch till the end and whatever doubts you have you can ask me in our next class. Uh so don't forget to me uh note down the question numbers where you face difficulty. Okay. So in the doubt class mostly we'll be discussing doubts related to this uh lecture only. So before you come into the doubt class this lecture is must watch. So do uh do not forget to check the lecture and if you do not have time watch it on a 1.5x time 1.5x speed so that you can finish the lecture quickly. So let's start with the first question. Shown in the figure are two point charges plus q and minus q inside a cavity of spherical shell. The charges are kept near the surface of the cavity on opposite sides of the center of the shell. If sigma 1 is the surface charge on the inner surface and Q1 net charge on it and sigma 2 is the surface charge on the outer surface and Q2 is net charge on it. Then so first of all let us understand if you see that one is a positive Q charge another one is a minus Q charge. So two charges are there inside a cavity. So if you if you are ask me to apply Goss's law from Goss' law the flux coming out should be zero.
Okay. So flux coming out should be zero.
That that much is clear. But why then they have given this question. This question is not about flux. This question is mainly about the surface charge density. So let's check. So here this positive Q charge is there.
So this positive charge Q will induce a negative charge on this uh nearest surface right? Uh it will induce a near negative charge here.
Okay. Induction electro electric induction. Okay. Now and this negative charge will induce a positive charge on this nearby surface.
And these charges are bound charges. So when you brought a charge uh near a surface okay there are two charges one charge is bound charge another charge is free charge. For example if you have a simple uh spherical object you have an external charged object brought near this spherical object it will induce same uh opposite charge in the near surface and same charge on the opposite surface. This charge is called bound charge.
Bound charge means as long as the source charge is present. This is source charge obviously. So as long as the source is present, these charges are not allowed to move. These charge can move if you ground it. If you ground it, this charge can move. Okay? Or if you connect it to another object, these charge can move to that object. But these negative charges will are are not allowed to move at all.
So that means these are bound these are called free charges.
So if you consider this surface the negative charge induced here they are bound charges and the opposite charge which is induced on the outer surface these are the free charges. Okay.
Similarly this there these negative charges are bound charges and the negative charge induced here they are free charges. Okay. Now what what does that mean and what is the amount of charge that will be induced? Amount of charge induced will be same as the source charge. So whatever value of this plus Q here we will on the inner surface we'll get a minus Q and the outer surface we'll get a plus Q. Similarly, this minus Q will induce a plus Q on the inner surface and a minus Q on the outer surface. Okay. Now, these negative and these positive charges are free charges.
Free charges means they will actually can move. So, these electrons will move and neutralize the entire positive Q.
So, this negative Q will completely neutralize the positive Q. So the outer surface will not have any charge. The outer surface will not have any charge.
Uh so total charge on the outer surface is zero. The charge density is zero.
Okay. So the entire charge will be gone.
Uh so outer surface will no longer have any charges. But what about the inner surface? Inner surface will still hold onto the charges.
Inner surface will still hold onto the charges. So uh this minus q will induce a plus q over here and this plus q will induce a minus q over there. So these charges are bound charges. They are not allowed to move. So if you if you try to find the surface charge density surface charge density of the inner surface is not zero. Even though the total charge if you try to calculate total charge plus and minus total charge will be zero but surface charge density will not be zero. Why? So total char surface charge density is not zero because this is a bound charge. Bound charge are not allowed to move. So correct answer is sigma 1 is not zero. Everything else is zero. Okay. So sigma 1 is not zero.
Everything else is zero. So correct answer is option C. Okay.
Let's move on to the next question. A charge capital Q is placed on the center of the line joining two point charges plus Q and plus Q as shown in the figure. If the system is in equilibrium, the ratio of charges Q, capital Q and small Q is so here we have a plus Q charge. Here we have a capital Q charge.
They are asking what is the ratio if the system is in equilibrium. Now how can a system be in equilibrium? The system be in equilibrium if you consider each and every charge and net force acting on each and every charge is zero. Now there is a trick to solve this question. You should not take any central charge. You should take any one of the side charges.
Any one of the side charge you should take and find its uh equilibrium condition. So what are the force acting on it? The force acting on it on this plus q charge is due to this plus q which is given by f_sub_1 say okay f_sub_1 is given by k + q into plus q divided by the distance between them who squared. What is the distance? x + x. So total distance between them is 2x.
Right? This is the total distance. This is the first force f_sub_1. What else?
What else force acting on this? The other force that is acting on it is due to this plus Q char. This capital Q charge you do not know the nature of this charge. We will find that out don't worry. So plus this capital Q and this plus Q they are also having some electrostatic force between them and that is given by K small Q capital Q divided by X².
Now net force is zero.
What is the meaning of net force? Net force is basically f_sub_1 plus f_sub_2, right? Sum of the forces on this charge.
We are trying to find the net net force on this molecule charge. Okay? So this is our point. So net force on this charge is zero. That means you have to add k q ² by 4x² + k q capital q by x² should be equals to z. So basically we have put the values here right? Now simplify k q² by 4x² equ= to minus k q capital q by x² x² x² cancels uh k cancels one of the q is also cancelled and we get capital q is equal to minus qx4 okay capital q is minus qx4. So the ratio of capital Q to small Q is - 1 by4 right the ratio of capital Q by small Q is minus 1 by4 clear do not overink when you are solving a question just follow the process you should find the answer plain square sheet of side.5 m has uniform surface charge density an electron at 1 cm from the center of the sheet experiences a force of 1.6 6 into 10 ^ of -12 Newton directed away from the sheet the total charge on the plain square sheet is. So we know that uh when it there is a plain charged sheet plain charged sheet if the distance of the uh object from the sheet is small we can consider the sheet to be an infinite sheet and we can say that the electric field is given by sigma by 2 epsylon KN. This is the formula for electric field near a charge sheet. Right? Now this this electric field is basically dependent on surface charge density. Now if you keep an electron if you keep an electron near the surface what will happen? It will experience as a force because of this electric field. Now it is mentioned that it is experiencing a force away from. Away from means what is the charge of an electron? Electron is negatively charged. So if an electron is experiencing a force away from then what is the nature of charge on the sheet the sheet must be negatively charged. The sheet must be negatively charged because negative negative charge repel each other. That's why it is experiencing a force away from. If it is getting attracted then we could say that the sheet is positive electron is negative.
But because the electron is experiencing a repulsive force that means uh the sheet must be negatively charged. We have to find the total charge. Total charge we will find that find that. But first let us find the force. Force is given by charge into E. Okay. Which is charge of an electron into electric field. So from there we can say that it is nothing but E into sigma by 2 epsylon KN or sigma equals to 2 epsylon KN F upon E right so we just need to multiply with the area to calculate the total charge total charge will be negative no doubt about it okay so let's find that out uh sigma the surface charge density is basically given by 2 into 8.85 4 into 10 ^ of -12 into uh force is 1.6 into 10 ^ of -12 again divided by the charge of an electron which is 1.6 into 10 ^ of -19. So 1.6 1.6 6 cancels and you get u 16 17 uh uh 17.6 17.7 approximately uh right 16 uh um 16.6 17.6 17 Yes. 17.7 into 10 ^ of uh -4 + 19 - 24 + 19 means uh - 24 + 19 means uh - 5. Okay. So this is the surface charge density. Now you have to find the total charge on the surface. Obviously as I mentioned this charge is negative. So total charge on the surface is minus sigma into area. Why it is negative?
because of the repulsive force. So the surface is negative. Okay, surface is negative because of the negative uh repulsive force. Here it is mentioned that it is away from directed away from.
So the force must be negative, right? So minus 17.7 into 10 ^ of -5 into area. What is the area? It's a a square sheet and side are 5 m. Uh.5 m. That means it will be 5 into 5 into 10 ^ of -2.5 means 5 into 10 ^ of -1. Okay. So it will be 25 into 17. 25 into 17 into 10 ^ of - 7. Um so that will be around uh 25 into uh 10 is uh 250. 250 uh plus 140.
Okay, let me do it. 25 into 17.
Okay. 5 140 um uh 75 into 25 5 2 4. So it is approximately uh approximately 40 425 into 10 ^ of -7 or approximately - 425 microum approximately 42 420.5 microum so if you see the options clearly the answer is option C minus 44.27 27 microcooler.
Okay.
Anyway, you are have you get an access to simple online calculator. You can use it. It generally available on the top corner. Top right corner. You can use that. Now, let's move on. The figure below shows not below. It is showing on the left hand side. The figure below shows the electric field lines due to two positive charges. The magnitude of magnitudes EA, EB and EC of the electric fields at points A, B and C respectively are related as. So here electric fields are given A, B and C. So if you see that how the electric fields are related. So electric fields so this is positive.
This is positive. So at the center of this combination what will happen? So basically if you put a positive charge it will experience force on both sides.
So the net force on this charge will be zero. So electric field at the center of two positive charges will always be zero. If the charges are positive electric field at the center will be zero. Now what about B and C? So if you see that the electric fields uh the point A is located between two electric field lines. Okay. So point A is located between two electric field lines. So EA should be stronger than EB. Why? EB is located outside the electric field line.
So and correspondingly if you measure the distance from the center, this will be slightly closer. This will be slightly further. Okay. So at this point because it is located inside the electric field lines and we know the property of electric field lines which states that closer the electric field lines, stronger the electric field, further they are, weaker they are. So as you are moving away from this uh this uh closer electric field lines electric field actually decreases. That means electric field at B is less than that of A only because the point A is located between field lines. If it was given like this suppose here another point is given say D. So in that case A and D should be same. Similarly another point is given say F here. So in that case also a and f should have been same. Okay. Now what about along the axis? Along the axis is not super clear. You have to find the distances corresponding distance wise it will change. Okay. So they have not given a complicated point. They have given easier points. Okay. A B and C and along the line you need to know the distances the values based on that you need to calculate and find the values.
But if the they are same distance suppose it is suppose it is a distance X from this side X from this side the electric field lines again should be same. Okay. So obviously electric field at C is zero and uh EA and EB EA is greater than EB. Okay. So EA is greater than EB and EB both of them are greater than EC but obviously EC is zero. So that means correct answer is option A. Analyze picture based questions are common in comet K. Analyze the picture and then answer it. Okay. Analyzing the picture is very important. Even though EC equals to0 is not given in the question. It is given that who is bigger and who is smaller. So zero is smaller obviously and EA and EB and electric both the both the charges are positive that means no sign problem here. Okay. So magnitude wise EA should be greater than EB. Now next question. A non-conducting solid sphere of radius capital R is charged uniformly. The electrostatic potential V plotted as a function of the uh distance from the center of the sphere. Which of the following represents the resulting curve? Now in this case it is very common to make mistake because it is not a conducting sphere. It is a non-conducting sphere. So very very important to note it is a non-conducting sphere. If the sphere is conducting the answer would have been this. Okay.
Electric field is constant inside the up to radius R then it decreases. But for non-conducting sphere the equation is slightly tricky which is given by 1x 4 pi epsilon Q by 2R. R is radius 3 - R² by R². Okay. So for distances less than uh r you will get a value distance greater than r you will get another value and the eventual uh graph plot is a nonlinear graph from the beginning and nonlinear graph means no straight line portion is involved. So only one graph which is completely nonlinear is this one. So this should be your answer.
Okay. Uh just remember if it is a non-conducting relationships are not linear. Okay. Uh relationships are not linear.
Now next uh four identical capacitors are connected in series with a 10V battery as shown in the figure.
Potentials at A and B are. So here we have a um combination of capacitors. So if you see that all capacitors are connected in series. So for now initially ignore this uh ignore this uh grounding portion.
Just imagine four cells are four four capacitors are connected in series. The total voltage applied is 10 volt. That means this 10 volt should equally distribute among all the capacitors because the capacitors are same. Capacitances are same. So the 10 volt will equally distribute among the capacitors. So here what will be the voltage across each capacitors? 2.5. So 10 divided by 10 divided by 4 that is 2.5. So each of them have a voltage of 2.5 across them.
Okay. So each of them will have equal voltage across them because entire voltage will be distributed. You can do that another way as well. You can say that charge is constant Q is equal to CV. Uh so and then you can add 4V that will be 10 volt. that is also possible but not required here. It is very uh evident from the diagram itself because all the capacitors are same. So the problem is not complicated. So don't try to make the problem complicated by using the the equations which are not required. It is very evident from the diagram itself that all the capacitors are same voltage is 10 volt. So that means the 10 volt should be equally distributed among all the capacitors. Now let us uh uh add all the voltages. So uh now we will see that uh the voltage of A suppose voltage at A is VA.
Voltage at N is VN and voltage at at B is VB. Now obviously the end point is grounded. So VN must be zero.
Right? Now let us uh consider what is the voltage difference between VA and VN. Okay. So VA minus VN.
Obviously this is a positive. So current will flow from positive to negative right. So, VA minus VN must be sum of all the voltages across the capacitors that is 2.5 + 2.5 + 2.5 that is 2.5 into 3 that is 7.5.
So, VA equals to 7.5 because VN equals to zero.
Right? Now what about B? Now let us check the same thing for N and B. Okay, V N minus VB. Obviously again the same logic. The current should flow or the charge should flow from positive to negative. That means this must be at high potential. This must be at low potential. So high potential minus low potential. Now how many capacitors are there? Only one capacitor is there. So it will be 2.5.
Okay. So 0 minus VB = to 2.5 or VV = -2.5.
So correct answer is uh 7.5 and minus 2.5. Option B.
Next. A parallel plate capacitor with air between the plates has a capacitance of 10 poparad. The capacitors if the distance between the plate is reduced by half and the space between them is filled with the substance of dialectric constant four is so initially the capacitor is a epsylon KN by D which is given by 10 poparad.
Now what you have done you have added a dialectric medium and the distance is now reduced to half.
Okay. So the equivalent capacitance is 2k a epsylon kn by d. So a epsylon by d is 10. So it is 2k into 10. Now 2 into k is 4 into 10. So it will be 80 poparad.
So answer is 80 poparad.
Let's move on. The capacitance of parall plate capacitor with air as dialectric C. Uh if the slab of dialectric constant K and the same thickness as the separation between the plate is introduced. So as it filled 1/4 of the capacitor shown in the figure when the new capacitors is. So first of all here we have a normal air capacitor. Normal air capacitor the this distance is D.
What is this distance? Now this is 1/4 of the entire distance say capital D.
Okay. So 1/4 of capital D is small D. So capital D is actually 4D. Okay. because it's a it's mentioned in the question that 1/4 of the uh distance is filled with the electric constant. Okay. So, a total distance must be 4D. Okay. So, what is the area? Area must be so this is a this is not a two one-dimensional straight line. It is actually a plate.
Okay. Of length capital D and it has also some uh width as well. So, say the width is X. Okay. Basically, it's a plate. This is a capacitor plate. plate means it has some inner depth as well.
Okay, it has some inner depth as well which is obviously not shown in the diagram. So area is this distance capital D into the distance X. Okay, the width X that is 4 DX. So what is the initial capacitance? Initial capacitance is a epsylon kn by small d which is 4 dx by uh epsylon kn by small d small d small d cancelled. So you get x epsylon equals to c upon 4. So this one we will require for any further calculation. x epsylon equals to c by 4.
Okay. Now let us go back and check this diagram again.
Now let us go back and check this diagram again. So this distance is small D. This is 1/4. Total distance is 4D. So how much is this? How much is this? Or how long is this? This must be 3D.
Okay. And the width remains same. Okay.
The width remains same. So this distance must be 3D because it is given in the question. This is D. This is 1/4. From there we can say 4D minus D should be the remaining distance that is 3D.
Okay. Now this kind of combination can be broken down into two parallel capacitors.
Basically it is nothing but two parallel capacitors. Okay. One is with this dialectric medium, another one is with air and they are connected like this.
They are basically parallel capacitor.
This one has a capacitor C1. This one has a capacitor C2. So we have to calculate C1 and C2 separately. Okay.
So let me go and add a page.
So this one is C1 with dialectric medium, right? C1. So what is C1? C1 is uh K A1 epsylon KN by D. Now what is A here?
This distance is small D. The width we already know width is X. So area A1 for this plate will be small D into X.
Right? So c1 is basically k a uh that is uh k dx epsylon kn by small d small d small d cancels and you get k x epsylon kn okay so this is c1 now this is not over we have another capacitor on this side as well so this capacitor is having uh a width 3D D and this width is X. The distance between the plate is same small D but here only air is present. So what is the capacitance of this one? C2. Capacitance of this one C2 is basically uh A2 epsylon by D. Now what is A2 here? Same logic we can say A2 is 3D into X into epsylon KN by small D small D small D cancels it will be 3 X epsylon KN. Now because the capacitors are connected in parallel the equivalent capacitors will be C1 + C2 which is Kxepsylon KN + 3 X epsylon KN. So k + 3 x epsylon kn.
Now we have already seen that x epsylon kn is cx4. So x epsylon kn is cx4. So the equivalent capacitance is cx4 k + 3.
So we get here uh so the answer for this question is uh k + 3 cx4 option b clear.
Now next two batteries of emf 4 volt and 8 volt with internal resistance 1 ohm and 2 ohm are connected in a circuit with resistance 9 ohm as shown in the figure.
the current and potential difference between the points P and Q are respectively. So here we have a battery and they ask you what is the potential difference between these two point. So potential difference between these two points is same as potential difference between uh these two point as well because there is nothing there right. So basically we find the current in the circuit and once we find the current in the circuit if we multiply with 9 ohm we get the potential difference uh across P and Q P and Q RNS same problem same thing okay uh so now how to find the current now if you see that the batteries are connected in reverse polarity so this uh battery is trying to send current in this direction this battery is trying to send current in the opposite direction.
So who will dominate? So if you see that uh this battery is higher value, this battery is smaller value. So this battery will dominate. So equivalent emf of these two batteries will be equivalent emf of these two battery will be 8 minus 4. So whenever you see a circuit diagram based questions, make sure you check the question two times.
Okay. So equivalent emf is uh 4 volt.
Now what is the total resistance? Total resistance basically simply add all the resistances that is 1 + 2 + 9 that is 12 ohm. So what is the current in the circuit? Current in the circuit is 4 by 12. Okay. Now what is the voltage drop between P and Q? I told you voltage drop between PQ is basically the current into 9 ohm resistance that is 4 by 12 into 9 uh 3 uh 9 3 4 are 12 4 cancel so it will be 3 volt okay so 3 volt is the uh potential difference between the points P and Q and current uh is 1/3 current is 1/3 ampere so current is 1/3 ampere and uh voltage drop is 3 volt. So even if you calculate the current you can get the correct answer for this particular question because current is unique. So you may not need to solve the voltage but because we have done it you can do it by one more step the circuit shown in the figure select the current option. So there are few uh uh circuit. Now in this kind of circuit remember I told you in the class as well that uh uh the voltage of this point and voltage at this point should be same because there is no resistances present at at this junction. That means this resistance can be considered out of the circuit. It is not contributing to anything in the circuit. So this resistance can be cancelled off. Okay, this resistance can be canceled off. If you see this 20 volt uh cell, there is a 4 ohm resistance on this side. So if you consider this loop, this loop apply kop's law. Suppose the current here is I1. Okay. So 4 I1 equals to 20 volt.
That is I1 equals to 5 aere. Now if you take the other loop for this loop suppose the current through this is I2 then again 5 I2 = to 20 volt. So I2 equals to 4 aere.
Now let's see the option. Current passing through 2 ohm resistance is zero. So no current is passing through this. So current is zero.
Current passing through 4 ohm resistance is 5 ampere. Current passing through 4 ohm resistance is 5 ampere. Current passing through 5 ohm resistance is 4 ampere. Current passing through 5 ohm resistance is 4 ampere. Okay. Uh so all of the above is correct answer. All of the above is correct answer.
Now let's move on to the next question.
The I versus V graphs for two different electrical appliances P and Q are shown in the diagram. RP and RQ be the resistances of the devices. Then so uh here we have V versus Igraph.
Okay, V versus Igraph or I versus sorry here we have I versus V graph. Okay, I versus V graph. Now here first of all they are asking you what is the resistances uh what is the relationship between resistance. So there are two ways either you can see that the slope find the slope or uh calculate the slope and then you can do this or what you can do you know that r equals to v upon i. R= to v upon i. So if you can somehow find a same point where the uh current is same. Okay, if you find a point where current is same and check who has the higher voltage that will have higher resistance. So basically we can say that R is proportional to voltage when I is constant.
Right? Now how can I make I constant in this graph? I is varying every point. So to make I constant we have to draw a straight line.
We have to draw a straight line like this.
Now if you draw the straight line what will happen is that uh so obviously all these point have same current. So that means the current for this point say A and current at this point say B both currents are same because it's a uh line which is parallel to Vaxis. So that means current uh for each and every point are same along this line. So basically that means we you get that the voltage here VP and voltage here VQ. Now who is greater? Obviously VP is greater.
So VP is greater than VQ. So VP is greater than VQ. So what we can conclude? RP must be greater than RQ.
Okay. So RP must be greater than RQ. So correct answer is RP greater than RQ.
Okay. Or you can take the slope as well.
But in that case in case of slope you need to be careful that slope is actually opposite that of resistance. So that is one way. This is another easier way.
This is a circuit simplification problem in the part of the circuit shown in the figure. The potential difference between points G and H. So you need to find potential difference between point G and H. So how to solve uh these question very simple current of 2 ampere passing through this resistance and once it reaches over here the same current there is no other path there is no branching.
So same 2 ampere current through uh will pass through the battery and through this 2 ohm resistance. Now here we have a branch. Here we have a branch. So we are not sure if the current is going in this direction or some current is coming from other direction.
So nothing is clear after this point.
So for that we will go slightly ahead in the circuit and we will figure out what is happening at this junction. Okay. So this is called uh uh little bit thinking ahead. Okay. You have to think a little ahead right. So if you see here 1 ampere and 3 ampere are outgoing current. 1 ampere and 3 ampere are outgoing current. So whatever current that is incoming from this side this current is branching into two currents. One is one, one is three. So from Kirchoff's current law, what should be the incoming current? Incoming current should be obviously sum of these two that is 1 + 3 that is 4 ampere. So 4 ampere current must be coming in from this side. So 4 amp current is coming from this side.
The same 4 amp current should come from here as well. Right? Because there is no branching in the battery. Battery is not adding any extra current. current law will always be valid even if there is a battery involved. Okay. So uh so branching is not there that means the current same 4 ampere should come from here as well and uh up to this point we figured out 2 ampere is coming from here. So how can this 2 ampere become a 4 ampere. So 2 aere was coming suddenly it jumps into 4 ampere. How it is possible? It is only possible if another 2 ampere is added by this uh branch.
Okay. So 2 ampere from this side, 2 ampere from this side together they are forming 4 ampere. Right? So this is only possible when this 2 amp. So this is basically we are back calculating. This is called back calculation. Right? So we first figured out that the current at this junction should be 4 ampere and current at this junction should be 4 ampere means who is providing the remaining 2 ampere? The remaining 2 amp is provided by this. Now obviously they ask you find VG minus VH. If it is a simple battery based question it could have been solved. Now because it is given VG by VH you can imagine there is a imaginary battery imaginary battery of potential difference VG minus VH which is sending current in this direction along the uh direction of uh G. Okay. So this is an imaginary cell which is sending a current along G towards H. Okay. So that is what we can imagine.
Now we have to apply Kchov's laws.
Kirchoff's voltage law. So if you see the here we we have four 4 ohm current is 2 amp. So four uh let me here light here. 4 into 2 ampere.
Okay. Ignore the cell. Go to this one.
So 2 into 2.
Okay. Now we started our journey at G.
We have to reach at H. Right. So if you are trying to reach at H. So when you are going up to this point, you are moving in the direction of current. But when you are going in this direction, okay, the current is coming in the upward direction. But to complete the uh point, you have to reach at H. So to reach at H you are growing in the opposite direction of current. So it must be 1 into two. Okay. Resistance into current. Resistance into current.
Resistance into current. Okay. If direction we are going in the direction positive. If we are going in the opposite direction that will be negative. And that should be equals to first is VG minus VH because we imagined an imaginary uh resistance imaginary uh cell over here and that cell is sending current from G towards H. So that is why it is positive. Now this cell can be tricky. Just uh remember that if the if you are going in the direction of the current sent by the cell, if we are going in the direction of the uh current sent by the cell, then on the right hand side the emf of the cell will be positive.
EMF of the cell will be positive. Okay?
Because the cell is also sending current from positive to negative that is in this direction. Okay? So that is why on right hand side it will be positive. Now let's calculate 8 + 4 - 2 = VG - VH + 3.
Okay. So it will be 12 - 2 that is 10 equals to VG minus VH + 3 or VG minus VH = 7 volt. So VG minus VH equals to 7 volt. Correct answer option C. Now let's move on to the next question. The torque required to hold a small circular coil of 10 turns area 1 mm² and carrying a current of 22x 44 ampere in the middle of a long solenoid is 10 ^ of 3 turns per meter carrying a current of 2.5 ampere. what is the perpendicular uh what is the with this axis perpendicular to the axis of the solenoid um okay so basically they try to make a question it's easy question complicated okay so you have to find the magnetic field due to this uh circular solenoid okay solenoid has a magnetic uh field the current carrying coil has a magnetic moment okay and the Torque required is basically torque is equal to MB sin theta. But here sin theta is 1. The angle is 90°. So the magnetic field and the magnetic moment they are making an angle 90°. Okay. Magnetic field is like this. Magnetic moment is like this. So angle between them is 90°. So MB sin 90 is 1. Okay. So that basically we need to find torque is equal to MB. It is almost same formula as dipole moment. In case of dipole moment you have used P. Okay.
Uh dipole moment into electric field.
But here instead of dipole moment you have magnetic moment into magnetic field. Okay. Now what is magnetic moment? Magnetic moment due to a current carrying coil is N I A. Magnetic moment due to a small current carrying coil is NIA. National Investigative Agency.
Okay. What about magnetic field due to a circular solenoid? Magnetic field due to a circular solenoid is mu not n i. Okay.
So mu not ni. So basically you have to multiply everything that the currents are different. Currents are different.
Number of turns are will also be different. Number of turns for the first coil is given as 10 turns. So first coil is 10 turns. So it will be 10 into I is 21 by 44. Don't uh don't surprise actually it is given so that you don't need to calculate the value for pi. Okay. So n is 10 i is 21 by 44.
Okay. A area is given as 1 mm². So that is little you need to be a little careful 1 mm² is equal to 1 into 10 ^ of -6 m²ared okay 1 mm is 10 ^ of - 3 so 1 mm² is 10 ^ of - 6 m² 10 ^ of - 6 and magnetic field is mu not nia a mu not is what is mu not mu not is 4<unk>i into 10 ^ of -7 7 this is mu not right now what is n is 10 ^ of 3 turns per meter. So remember these this n this n is number of turns per unit length whereas capital n is total number of turns. So the these two are different okay into 10 ^ of 3 and current is 2.5 through the solenoid.
So 2.5.
Now we have to simplify.
Now we have to simplify which is obviously uh 10 ^ of - 6 - 7 is -3. -3 + 3 is - 10. - 10 + 10 is uh - 9. So we can see a - 9 and 4 into 2.5. 4 into 2.5 is also 10.
So it will be 10 ^ of -8. So little take your time. Uh if you are not able to do it very fast, take your time understand what are all the 10 ^ values then cancel it. Okay. So -6 - 7 -3 -3 + 3 is - 10 - 10 + 1 is - 9 and 4 uh 2.5 into 4 is another 10. So it will be + 1 another + 1 so it will be 10 ^ of - 8. Now only pi and this is left. So 21 by 44 into 22x 7. Okay. 22x 44 into 22x 7. So 44 2 are uh 20 22 are 2 is 44 and 7 3's are 21. So it is basically 3x 2 into 10 ^ of -8.
So it will be 1.5 into 10 ^ of -8 uh newton m torque newton meter. So 1.5 into 10 ^ of -8 Newton m.
Next question. A conducting wire is bent in the form of a parabola Y² = 2X carries a current I = 2A as shown in the figure. This wire is placed in a uniform magnetic field B = - 4K Tesla.
The mag uh the magnetic force on the wire is so magnetic force on the wire is Q V cross B.
Okay. Uh sorry sorry sorry. This is due to a moving charge current carrying wire. Right. So it will be um I L cross B.
Magnetic force acting on a current carrying wire is given by I L cross B.
So B I sin theta that we have learned but here uh they slightly made it complicated. You need to find the uh length vector. So because it's a curved line so length vector you have to find is not very simple. But uh it is given that this is starting at point A. This is start ending at point B. So that means the length vector is basically going from this direction to this direction. Now someone during my live class was asked sir why the direction is from A to B? Because you are going from A to B. Starting point is A, ending point is B. That means the direction of vector must be from A to B. Now you have to find the length. length is very easy.
So this is a y square is equal to 2x.
Okay. And the x value here is 2. Right?
In the diagram it is given x value is 2.
So satisfy that. So y² is equal to 2 into 2 that is 4. So y = + or minus2. So this is + 2. This is minus2. So what is delta l? Delta L is basically or L vector is actually equals to -2 - -2 uh JCAP.
Okay, that is -4 JCAP. You can do it like this also or you can find the length. This is also 2. This is 2 22 4 minus 4 because it is from negative yaxis. Okay. Or you can take the find the coordinates. If you find the coordinate final point minus initial point that is also fine. Uh and that is that this is happening along yaxis. So jcap. Okay. So -4 jcap is the l vector.
So now I is given here as 2 aere length vector is -4 jcap and magnetic field vector is -4 kcap.
So - plus 4 4 are 8 8 2 uh sorry 4 4's are 16 16 2 3 into J cross K. So now uh those who are not familiar I explained it multiple times I J K always draw it J cross K should be equals to positive I so 32 ICAP so 32 ICAP is the force acting on this system okay it will be 32 ICAP next two wires of currents 2 ampere and 1 ampere are enclosed in a circular loop. Another wire with current two 3 ampere is situated outside the loop as shown. The uh magnitude B do DL around the loop is. So if you see that here this is the closed loop and here we have a current carrying wire which is outside the loop. So from the amper circuit law we know that B dot DL okay equals to mu not I enclosed. So what is the I enclosed here? I enclosed is 2 aere upward 1 a downward. So this is + 2 this is minus1. So total I enclosed is 2 -1 that is 1 upward. If you take upward as positive then you will get this as only mu not okay because I enclosed is 1 ampere only so it will be simple mu not okay now let's move on to the next question a voltmeter which can measure 2 volt is constructed by using a galvvinometer of resistance 12 ohm and that produces a maximum deflection for the current of 2 milliampere then the resistance capital R is uh so the voltmeter can uh the voltmeter can measure 2 volt that means if you apply a voltage 2 volt if you apply a voltage 2 volt okay the current in the circuit will be the current in the circuit will be 2 uh milliampere it is given right this is the maximum current that can pass through the galvanometer so maximum current should give you maximum maximum voltage as well. So 2 volt is the voltage. 2 ampere is the current. Uh in the resistance of the galvvinometer is 12 ohm. This is R. So total resistance is 12 + R. And then ohms law V is equal to I R total.
Okay. V is 2 volt. I is 2 milliampere and R total is R + 12.
Okay. R + 12. So 2 2 cancels and we get r + 12 = 1,000 or r = to 988 ohm. So it will be 988 ohm. Next question.
Current carrying loop is placed in uniform magnetic field in four different orientations as shown in the figure.
Arrange them in decreasing order of potential energies. So if you see that again potential energy is what?
Potential energy is minus MB cos theta. Same as dipole. Okay.
Magnetic dipole electric dipole only the name changes. Uh the uh the magnetic moment becomes magnetic dipole moment becomes electric dipole moment. Electric field becomes magnetic field like that.
Okay. So the the quantities only changed but otherwise formulas are same. So minus MB cos theta. So if the cos theta value is uh less than 90° okay the potential energy is negative. So potential energy is minimum for 0° and slowly it starts increasing as the angle increases and it becomes maximum at theta= to 180°. So at 180° the potential energy is maximum. At 90° potential energy is zero but zero is not minimum. Okay. So let's see here if you see here that here the angle at this point angle this is the magnetic moment vector. This is the magnetic field vector. So the angle between them is 180°. So the potential energy will be maximum at this point. Okay. Uh and if you see this case the angle is this case the angle is more than 90° angle is greater than 90°. This theta is greater than 90°. So this is also positive potential. So this will be second.
This will be second highest potential energy. Now if you see that this angle theta is smaller than 90°. So this will be negative potential energy. So this will be the lowest in the given case.
Okay. And because this is theta is 90°.
So 90° means potential energy is zero. But zero is not minimum in this case. So this will be maximum. Second this will be third and this is lowest that is fourth. Now if you see the question arrange the them in the decreasing order of potential energy. So decreasing order means this must be first, this is second, this is third, this is fourth. So 1 4 2 3 1 4 2 3. If they ask you increasing order, then the order will be changed 3 to 41. Okay? If they ask you increasing order, then you have to go in the reverse order 3 to 41. So uh so remember uh obtuse angle always potential energy is uh positive and more acute angle that is less than 90° angle it is negative and less. Now let's move on in the given figure represents a material which is a diamagnet. So if you see that this is a material and this is the magnetic field which is repelled by the magnetic magnetic field is repelled by this object. So it must be a diamagnetic substance. This must be a diamagnetic substance.
Following figures show the arrangements of the bar magnets in the different configuration. Each magnet has a magnetic dipole moment M. Which configuration has highest magnetic dipole moment.
So magnetic dipole moment net magnetic dipole moment is basically vector sum.
So net magnetic dipole moment magnitude wise is given by square root of m1 squ + m_sub_2 squ + 2 m1 m_sub_2 cos theta.
Okay. So cos theta highest value of cos theta is theta equ= to0 and as you increase the angle okay the cos theta becomes uh zero first then it becomes negative. So if you see that so that means smaller the value of theta higher will be the magnetic moment. Smaller the value of theta because smaller value theta of theta means cos theta is closer to one that means this is maximum positive. So maximum positive if you take the square root you will get maximum value right. So Mnet is bigger if the angle is smaller larger the angle uh smaller is the uh uh net magnetic moment. So here so among them you see that this angle is 90 this angle is 90 this angle is 60 this angle is 30. What about this one? So this one has a magnetic moment north to south to north.
So basically this is theta is 180°.
Okay. So this is the smallest one.
Okay.
Sorry. H I G H E S T. So this will be the highest. Smaller the angle highest is the uh magnetic moment net magnetic moment. So answer is obviously three.
Answer is three. Next question. A coil has 1,000 tons and 500 cm squared as its area. The plane of the coil is placed perpendicular to a uniform magnetic field of 2 into 10 ^ of - 5 Tesla. The coil is rotated through 180° in 2 seconds. The average emf induced in the coil in mill volt. So average emf induced is basically d5 by dt.
Okay. And because there are number of turns are involved. So it will be n d5 by dt.
Okay. So time is given 2 second. What is the change in flux? So if you see that um flux what is flux by the way? Flux is again magnetic field into area. fluxes.
So a lot of some definitions you have to remember but these are all important common formulas. So magnetic field is 2 into 10 ^ of - 5 Tesla. Area is 500 cm² that is 10 ^ of - 4 m²ared. Okay. So if you multiply 2 into 500 is 1,000. So 1,000 means 10 ^ of - 9 + 3 that is 10 ^ of - 6. This is the flux. Okay. So in when it is flat like this the flux is linked with like 10 ^ of - 6. So when you rotate it 180° that means you completely flipped it. So when you completely flipped it then what is the change in flux? So initial flux is five final flux is minus 5. Okay. Okay. So if you flip it completely the change in flux become the flux linked with the coil becomes minus 5. Okay. So the total flux is basically the change in flux d5 is equal to - 5 - - 5. So in final minus initial that is - 2 that is - 2 into 10 ^ of - 6. Okay. Now there is also a minus sign involved here. Okay. There is a minus sign involved here. So you this minus will cancel. So don't worry about it. Uh n value is given here 1,000 turns minus minus cancels. We are not writing minus. Okay. 1,000 volt. D5 is 2 into 10 ^ of -6 divided by time is 0.2 seconds. So that will be uh this will give you 10.
10 means 10 ^ of - uh 2 volt or 10 m volt. Okay, 10 m volt is the induced emmf which is option B.
Okay, clear.
The equivalent inductance of two inductors is2 4 Henry when connected in parallel and 10 Henry when connected in series. What is the value of the inductances of the individual inductors? Now remember inductors behave same as resistance when they are connected in series and parallel. So when they are connected in series their inductance is simply added.
When they are connected in parallel their inductances is reciprocally added.
So when they are connected in parallel the equivalent emf in parallel combination is given by L1 L2 by L1 + L2.
Now again when connected in series L1 + L2 this is given 2.4 and this is given as 10. So if you put this value in this equation you get L1 L2 equals to 2.4 into 10. Right? So basically L1 L2 by 10 equals to 2.4 24 or divide multiply by 10 on this side. So it is 24. Now you don't need to solve anything else. You just find the combination multiply them and check which one is giving you 24 and adding them should also give you 10 addition. So this condition and this condition both should be valid. For example 12 into 2 is 24. But if you add them 12 + 2 that is 14. So it is satisfying this. It is not satisfying this one. So definitely you cannot take 12 and 2 as as combination. So what will be the combination? If you see 8 + 2 2 is 10 but 8 into 2 is 16. 6 4 are 24. 6 + 4 is 10. Obviously this is the answer.
5 + 5 is 10 but 5 5 are 25 that is also not matching. Uh 3 7 are 21. 7 + 3 is 10. So again multiplication wise it is not matching. So multiplication and addition both should match. That is the way to solve these kind of questions.
The normal magnetic flux passing through a coil changes with time according to the equation 5 = 6 t² - 5t + 1. What is the magnitude of the induced current at t = 0.25 second and resistance 10 ohm.
So phi is given.
So we know that induced TMF is d5 by dt.
Okay. And only one loop. So number of loops are not there. Simply d5 by dt which is 12 t minus 5. Okay. Now they are asking you the time. Time is given as.25. So epsylon at t is equal to.25 25 is 12 into 0.25 - 5. So it will be uh 14 3 - 5. So 3 - 5.
So minus 2 volt but again minus value is not important because by definition there is a negative sign which we ignored. So you can add that and you can say that this will be positive. Okay.
You can keep a negative sign from the beginning. Okay. And you can say uh this will be this will be positive 2 volt. So the induced emf is 2 volt. What is the current? So current is basically voltage by uh resistance in this case induced emf.
I don't know why why the reser does not work sometimes.
So it will be emf by uh resistance emf is 2 divided by 10 that is 0.2 ampere. So 0.2 ampere is the correct answer. Let's move on to the next question. In the series LCR circuit shown the impedance is so here we have a series LCR circuit LCR all of them are shown frequency is given and they ask you what is impedance.mp Impedance is obviously formula for impedance square roo<unk> of r² + xl - x c². So if you obviously you have to calculate xc xl is 2 pi f into l that is 2 pi into f is 50 by pi l is uh 1 henry.
So pi pi cancel. So this is 100. So xl is 100 ohm. What is xc?
xc is 1 by uh 2 pi uh f into c f into c. So it will be 1 by 2 into f a is 50 by pi and c is uh 20 microfarad. Now I told you if there is a microfarad millifarad so those micro means 10 ^ of minus 6. So that 10 ^ of minus 6 will go up and become plus in the top. So do not write in the 10 to the^ of minus 6 because you are solving it for yourself not for your teachers. So don't uh avoid some extra steps if you can. Okay. If you still have doubt write that if you can you can write it directly. So uh 10 ^ of 6 divided by 100 into 20.
So it will be 10 ^ 3 by 2 which is 500.
So we have calculated XC R value is given. Now we need to calculate the zed.
So zed is 300² + 500 - 100 square. Obviously because there is a square there XL - XC or XC - XL does not matter. So you get 300 squared + 400 squared which is basically 500.
Okay. So 500 ohm is the impedance.
Correct answer option D.
Let's move on. A step down transformer is used on a 1,000 volt line to deliver 20 ampere at 120 volt at the secondary coil. If the efficiency of the transformer is 80% the current drawn from the line is so step down transformer uh used uh 1,000 volt line uh deliver 20 ampere current at 120 volt.
So this is the output these are the output. So 20 ampere and 120 volt are output uh if the efficiency of the transformer is 80% the current drawn from the line.
So input current you have to find uh now what is power input? Power input is voltage input into current input or voltage in primary into current in primary.
What is power output? Power output is uh percentage efficiency which is 80% into power input.
Right? In terms of efficiency, power output should be efficiency fraction into power input and that should also be equals to voltage in secondary into current in secondary.
Now we will put those values. So 80 by 100 80 by 100 is 8 by 10 by the way into power input is VP into IP equals to uh VS into is okay now it's basically simplification so IP equals = to 120 into 20 by 800 that is 3 aere so it will be 3 a not 30 aere not.3 a 3 aere so now let us continue with the question number 25 the RMS value of potential difference V shown in the figure is so this is a very typical type of graph and this kind of graph based questions. Uh if they ask you find the RMS value, remember RMS value is given by a very simple formula which is given by V RMS is equals to square root of V1² into T1 + V2² into TS2 divided by T1 + TS2. So what is T1? T2 what is V_sub_1 V_sub_2? So if you when the graph is given like that you first of all you have to identify the time period. So time period is t okay and in this zone whatever is the time. So up to this time the voltage is V not. So basically V is your V_sub_1.
Okay. And what is T1? T1 is your time period half time period. Okay. It can be any value. So make sure you check the graph properly. And here the second uh part it is given v is equals to0. So v_sub_2 equals to z. Now you can find if the same graph is given suppose it is given in the negative direction and the negative value is also vot then what will be the rms value? Please calculate and tell me during the live uh session right now.
So what is the v_sub_1 t1 v1 square v1 is v. So it will be square root of v² t1 plus and t_1 is tx2. So let me write that as well. tx2 plus 0 into ts2. Okay.
Second half it is zero. So t2 does not matter anyway. And uh the total time is tx2 plus tx2 that is total time is actually time period. So in the division it will always be the total time period t. Okay. So now simplify that we will get square root of t cancels and we get v² by2 that is v by <unk>2. So rms value is v by <unk>2 for this kind of graph as well. But remember it can be any value.
So if you just want to guess vx2 because that is the uh standard case for sine waves. Make sure you calculate it.
Without calculation do not answer. There might be a trap there for you. Now, next question. An alternating current in a circuit is given by I = to 20 sin 100 pi t + 0.05 pi ampere. The RMS value and the frequency of the current respectively are.
So here current is given.
We know that standard equation of uh U AC current is given by I = to I max sin omega T + 5. That is the standard equation. Now if you see if you compare that obviously we get IM is equal to 20 ampere. IM is equal to 20 ampere. So what is the IMS? I RMS is basically IM by <unk>2 that is the standard formula for sine waves. So now put that here. So that will be 20 by <unk>2.
So it will be 10 <unk>2. Now if you see that 10 <unk>2 is there in two options.
So you have to calculate the frequency as well. But if you find that after calculating the first half if you get only one match then you can be sure that that would be the answer. So no need to calculate. But in this particular case you have to calculate. So what is frequency formula? Frequency formula is omega by 2 pi. Now if you compare these equation okay so omega actually is equals to omega actually equals to 100 pi.
Okay. Uh so this is not minus by the way just a mistake. So omega is equals to 100 pi by comparing. So we can write this will be 100 pi by 2 pi that is 50 Hz. So it is 50 Hz right. So 10 <unk>2 ampere and 50 Hz. So option C is correct answer. Next question. The electric field associated with an electromagnetic wave in vacuum is given by E = 40 cos KZ minus 6 into 10 ^ of 8 T icap where E Z and T are volt per meter meter and second respectively. The value of wave vector k is so standard equation of electro electric field is given by uh eax or e not sin k x kz whatever is the direction minus omega t that is the standard equation of electric field in case of electromagnetic waves. So if you compare that obviously omega is this now you have to find the value of k. So omega = to 6 into 10 ^ of 8 radian/s. Radian is dimensionless. It is basically uh the unit will be per second. Okay. Now from here what is the dimen what is the value of k? So k is actually omega divided by speed of light. Okay. Because electromagnetic wave speed of light is constant which is obviously 3 into 10 ^ of 8. So we have to divide that. So 10 ^ of 8 10 ^ of 8 cancels and you get two.
So basically K equals to 2. So 2 m per meter is the answer. Okay.
Next question.
An electromagnetic wave propagating along north has its electro uh electric field vector upwards. Its magnetic field vector point towards. Now electromagnetic wave remember this electromagnetic wave is basically E vector cross B vector. Okay the direction of electromagnetic wave can be calculated using this E vector cross B vector. So basically you have to find a proper way to find to find the direction of B. Now I will explain how to find the direction of EM uh vector. So first the electric field here we have uh northeast west as well as up and down direction.
So I will draw it like this. Uh so if you just give me a uh time to draw it.
So this is suppose your north south direction.
Okay. Uh then we have our east west direction. So let me move it.
And then we have our uh top and bottom direction.
Okay.
So this is my uh our direction three-dimensional figure obviously. So you can see this is our east, this is our west, this is our north, this is our south, this is our up and this is our down.
Okay. Now they ask you that electromagnetic wave is propagating in the north direction. So this is the direction of electromagnetic wave and the electric field is act in the upward direction. So this is the direction of electric field. Now what could be the direction of uh B. Now as I mentioned it will be E cross B. There is no vector division. So you cannot directly find it. So you have to do it indirectly. So you have first of all how to find the cross product? Direction of cross productduct always use your hand to find the suppose we are trying to find the direction for two vectors say a vector and b vector. How to find a cross b direction for a cross b. So first you point your fingers point your fingers in the direction of the first vector that is a vector. Okay.
A cross b. So curl your fingers towards the second vector. Okay. A cross B. And then your thumb will point the direction of the resultant vector. Okay. The cross vector. Okay. So if it is B cross A, if it is B cross A, so order obviously matter. If it is B cross A, so B cross A is basically first you point your finger in the direction of the first vector.
Curl your fingers towards the second vector in the shortest way possible.
Okay. then you will get the direction of the uh uh product right so in that case it will be away from so direction does matter the order of cross product also matters in case of when you are finding the cross productduct right so now here E cross B now first obviously so we understand that our thumb points the direction of the resultant vector or the crossroduct vector so obviously E vector is the resultant vector so it will be uh so first we point our finger in the direction of electromagnetic wave. Then we point our fingers in the direction of E vector. Now which way should we curl so that we get the direction? So only way to curl is if we curl it towards the east direction. If we curl our fingers towards the east direction, we get the uh electromagnetic vector. So very important. Uh so remember that always the thumb points the direction of the cross productduct vector. Okay. And uh first vector should be started with your fingers and then you have to curl towards the second vector so that the thumb points the direction. So here the direction is uh northward and electric field is upward. So only possible way is that the magnetic field vector is along east. So magnetic field field vector has to be along the east direction. So correct answer is east. Okay. Now let's move on to the next question. A ray of light is incident normally on the prism.
N is equal to 3x2 immersed in a liquid as shown in the figure. The largest value of the angle alpha so that the ray is totally reflected at the surface AC is 30°. The refractive index of the given liquid. Now for that we will firstly use uh two uh diagrams. First diagram is the uh diagram which I will explain what is happening. So if the ray is actually totally internally getting totally internally reflected then okay so this is the incident ray. So when it is falling on this prism what is happening? It is totally internally reflected. Totally internally reflected means it will reflect like this. Okay it will reflect like this.
Right now, so this is the typical case what is happening here. So alpha value is given here. So from there we can calculate that if this angle is alpha, this angle is 90°. So that means this angle will be obviously uh 90° minus alpha. Now if you plot the normal at this point, if you plot the normal at this point, okay, so that angle of incidence will also be alpha. Okay, because this angle is also 90°. So total um angle will be so this angle will be also alpha.
Okay, now why this is important? We will see that in a while. Uh now for this incidence if the angle is slightly less than alpha, what will happen? The ray of light will total the ray of light will pass across uh just a second. the ray of light will pass across the surface like this. Okay.
If you if the angle is slightly more than this value, slightly less than this value, the ray will pass between the surface. Clear? Now this is a prism obviously. Now we will be so the uh and in that case the angle of uh refraction will be 90°. Okay. So incident angle on the surface is alpha and the refracted angle is 90°.
So this refractive index is given by 3x2 and this refractive index say n. Now we will be applying the snail's law. Now remember we have cheated little bit.
What cheating we have done? We have assumed that the angle of incidence is slightly less than alpha. Slightly less than alpha. So this is the maximum possible value. So maximum possible value means uh anything less than that the ray will pass back to the second medium. So for this angle alpha total internal reflection is actually happening. Now total internal reflection is happening. Now we obviously cannot use the total internal reflection information. We have to go a slightly smaller angle so that it is actually passing between the medium so that it is called the critical case. Right? So alpha is almost critical angle but slightly small. So that cheating we have done so that we can apply the law of refle refraction.
Okay. What is law of refraction? N1 sin I equals to N_sub_2 sin R. Okay. Okay. So what is n1 here? n1 is 3x2 3x2 sin alpha.
Okay. So if you are still confused how this angle become alpha just let me explain once more because I know some of you still might be confusing sir why this angle is alpha. So let me explain once more. So this angle is alpha. So this is a right angle triangle. So right angle all triangles have all angles equals to 180°. So if this angle is 90° obviously if this is totally internally reflected this angle will be 90° that means the other angle will be 90 minus alpha this will be 90 minus alpha right so that sum of all the angles become 180° now again this is another 90° this is another 90° right so this is 90 minus alpha this is total angle is 90 so this angle will be 90 - 90 - alpha that is alpha Okay. So remember that this is a very typical geometry. So this angle is alpha. If is a right angle triangle. If this angle is also right angle then the angle actually becomes equal. And from law of reflection angle of incidence should be angle of reflection. So from that we get the angle of incidence to be alpha. Clear? So those who have had doubt sir how you get this angle to be alpha. So just for them. Now what about n2? N2 we do not know N_sub_2 value we do not know but sin R which is when it is coming out of the surface the angle is 90° so it will be sin 90° okay now from here we can calculate the value of N_sub_2 okay sin 90 value is 1 obviously and sin alpha is 30° so sin 90 is 1 is 1. So let's put that. So 3x2 sin 30 = n_sub_2 or n_sub_2 = 3x2 into half that is 3x4.
So the refractive index of the medium is uh the liquid is 3x4.
Next question. A concave mirror of focal length f produces a real image n times the size of the object. the distance of the object from the mirror is. So basically what is the formula for magnification in a mirror? Magnification of a mirror is given by basically uh m is equal to minus v by u. Now this in this case you have to use the sign convention of u v etc. And if the image is real then we can write this must be equals to minus n. So n * n * is the magnification and it is mentioned that it's a real image. So if you read carefully real image for real image the magnification itself is negative. Okay.
So this negative this negative cancels.
Now let us put the sign of v and u. V's sign we do not know. U is always negative by uh geometry by new cartian sign convention U is always negative. So from here we can say V must be equals to minus N U. Okay. Now we will go back to our mirror equation. Mirror equation is 1x V + 1x U equals to 1x F. Put the values. So 1 by - n u uh + 1 by - u = 1 by - f. Concave mirror focal length is negative. For concave surfaces focal length is always negative. For convex surfaces focal length is always positive. Okay. Now all the negative signs actually cancels. Now if you take 1 upon u as common you get 1 upon n + 1 = to 1 upon f okay or 1 upon u = n by 1 + n = to 1 upon f okay so now you will find the value of u so u cross multiply just cross multiplying okay u = to f into n + 1 by n. Okay, cross multiply 1 + n was there. I written it n + 1. No harm done, right? So, it will be n + 1 f by n. So, correct answer is option c. Very careful about sign convention. Without sign convention, you might get us last option as well. So, make sure you are applying the sign convention in each and every step. A double convex lens made of material of refractive index mu1 is placed inside two liquids of refractive indices mu2 and mu3 as shown. Now muub2 is greater than mu1 and mu1 is greater than mu3. Okay. A wide parallel beam of light is incident on the lens from the left. the lens will give rise to. So here if you see that this is a convex lens normally it's a convex lens having refractive index mu2 and mu3 as well.
Now if the outside uh medium has a less refractive index then it will behave like a convex lens. For example standard convex lens placed in air. Okay. So air means the refractive index is less than that of glass. So, so that means it will behave like a convex lens, right? But if you change the surrounding uh refractive index such a way that the resultant uh um outside medium's refractive index is more than that of the refractive index of this medium then it might be it will behave as a concave lens. Okay. The nature of the lens changes because it the focal length becomes negative. Okay.
Okay. So this is very very important to remember. So the bottom portion where the refractive index of the medium is more than that of uh refractive index of the medium is more than that of the surrounding medium, it will behave as a convex lens. Okay. So this will behave as a convex and this will behave as a concave.
Okay. So top portion will behave as a concave mirror and a concave lens and bottom portion will behave as a convex lens. Okay. So if parallel beams falls this ray which are part falling on the concave mirror they will diverge. They will diverge and the ray which are falling on the convex lens they will converge. Okay. So we will get a convex uh converging as well as a diverging uh beam. So we will get a convergent and a divergent beam. So correct answer option D. Very uh thoughtful question. This is one of the difficult question. um I avoided it in the past. Now let's solve it anyway. Uh a ray of light is incident normally on the first refracting surface of the prism of refracting angle A. The ray of light comes out at a glazing emergence. If one half of the prism shaded portion is knocked off, uh the ray will emerge not emerge emerge at an angle. So basically either you have to figure out whether the ray will come out or not or the ray if if it is coming out what will be the angle okay emergent angle that is our question. Now here obviously first first let us understand what is happening. Okay so here this ray is falling. So so what is the angle of incidence? This is a this is a normal incidence. So in case of normal incidence we know that angle of incidence is actually zero. So this angle on this surface is zero. Now what is happening when it is coming out of the surface when it is coming out of the surface it is mentioned that it is having a just critical angle right if I'm not mistaken.
So it is mentioned that a ray of light falls on the first refracting surface.
The prism of the refracting angle is a the ray of light comes out as a grazing emergence. Grazing emergence means obviously the angle is critical angle.
So this is the grazing angle. Right?
Now uh this is the first case. If you knock off it, we will see at see that in a while. But first let us see what is happening here. So in this case obviously the angle of incidence is zero. Angle of incidence is not 90. Make sure angle of incidence is not 90. If we always measure the angle with respect to the normal and it is falling along the normal. So angle of incidence is zero.
If I is zero, R is also zero. Okay. So R is also zero. Now what about the other angle? So suppose this is the other angle. So this angle say R1. So this angle is R1 and the emergent angle in this case is 90°. Suppose the refractive index of the medium of the prism is N.
Okay. The refractive index of the medium of the prism is N. Now what? So now what we can do is apply the prism principle that angle of the prism which is given by A. Okay.
Okay. So angle of the prism basically given by R1 + R2. Right? This is a standard formula. Now here R1 is zero and R2 is R1. Okay. So we can say that A must be equals to R1. Now we will apply Snail's law again at this point. Okay. Snail's law at this point. So what is the formula for Snail's law? So snail's law says that n sine of angle of incidence in this surface angle of incidence is r1. Okay. So sin r1 equals to outside medium is air. So refractive index is 1. So 1 sin 90° right sin 90 is 1. So from there we can say that n must be equals to 1 by sin r1 or 1 by sin a right because r1 equals to a. So this is the first part this information we will keep it with you with us and we will move on with the uh second part where the uh lens uh sorry the prism is knocked off. Okay. So when the prism is knocked off, let's see what how does the prism looks like. So prism looks like this, right? Uh let me change the color.
Okay, so this is how the prism looks like now. So half of the prism is knocked off. So what will be the angle of the prism? So angle of the prism now will be a by2.
Okay. Now the ray which was falling normally the ray which was falling normally in the p on the prism so it will still fall like this. So angle of incidence is still zero.
Okay but what will be the angle of emergence that's uh that is our question. So angle of emergence obviously it will not be a grazing angle it's it can be any some other angle that angle we have to find. Okay. And um this angle is also not the same angle.
Okay, this angle on this surface is also not the same. So this angle is say rdash. This angle is still zero. So here the angle of incidence is zero.
So angle of emergency uh incident on the inner part is still zero. Okay. So these are all zero.
Okay.
>> Now what we can say that this angle is Rdash. Now uh this angle of emergence say this is E dash.
Angle of prism is changed but refractive index of the medium of the prism should not change. Right? Refractive index of the med medium of the prism should remain same. So now we will apply the snail's law again. Now for this condition okay and before applying that prism's rule prism rule says that angle of the prism must be equals to sum of the uh internal angles that is 0 + rdash. So rdash is actually equals to a by2 okay so the angle here also it changes to a by ax 2. Now we apply the snails law. So what is snails law? N sin R dash should be 1 sin E E dash. Okay, we are trying to find this angle E dash. Okay, now N value we have already calculated in the previous part which is 1 by sin A. Okay, so let's put that uh 1 by sin A.
And what is Rdash? Rdash is A by 2. So sin A by 2 and that should be equals to sin E dash. Okay. Now we have to simplify this and find the value of E dash. So sin A by 2. Now let we can write it as sin A by 2 divided by 2 sin A by 2 cos A by 2. Right? We can write it sin a is equal to sin 2 into sin a /2 right sin 2 theta formula sin a by 2 sin a by2 cancels we get half sec a by 2 and that is equals to sin e dash. So e dash equals to sin inverse half se a by 2. Okay. So sin inverse half sec a by 2 sin inverse half sec a by2 correct answer option a long question uh lengthy question uh but good question in my it gives you a satisfaction if you are able to solve it two sources of light of wavelength 2500 anstrom and 3500 anstrom are used in young's double slit experiment simultaneously which orders of fringes of two wavelength patterns coincide. So here remember that if nth order of first wavelength coincides with mth order of the second wavelength then always we can write n lambda 1 is equal to m lambda 2 that is uh now lambda 1 by lambda 2 equals to m by n.
Okay. So, uh what is uh uh lambda 1?
Lambda 1 is 2500.
Lambda 2 is 3500.
And that is equals to m by n.
0 0 cancels. 5 5 are 25. 5 7 are 35. So, m by n equals to 5 by 7. So, if m becomes 7, n should be 7. So fifth order of the first and uh sorry uh first one is nth right. So n seventh order of the first seventh of first with fifth of the second.
Right? Remember one more important part is that higher the wavelength order should be smaller. Higher the wavelength order should be smaller. So in between one and two second one has highest wavelength. So the order for second should be smaller than that of the first wavelength. Okay. So seventh order of the first and fifth order of the second.
So correct answer is option B.
Okay. Now next question. If I not is the intensity of principal maximum in the single slit defraction pattern then what will be its intensity when the slit width is doubled. So intensity of light intensity of light is proportional to square of the slit width. So if you double the slit width uh the intensity will be four times intensity will be four times. Okay. So correct answer is four I not. Next question. Two identical waves propagating in the same direction have a phase difference delta. After they superpose the intensity of the resulting wave will be proportional to.
So the resulting wave for identical wave the resulting wave intensity is given by 4 I cos² 5 by2. So here it will be cos² delta by 2. Okay. So because the phase difference is delta phase difference is phi the resultant wave becomes 4 I cos square 5 by2 here it will be delta x2 the correct answer is is proportional to cos² delta by 2.
Next a transparent thin plate of polar polaroid is placed on another similar plate such that the angle between their axis is 30°. the intensity of the emergent and uh the unpolarized incident light will be in the ratio of. So here we have two polariz polarizer one is say uh polarizer one another one say polarizer two.
So unpolarized light say intensity I incident on the polar polaroid. Okay. So as we have discussed in the past that the out uh emergent light the intensity of the emergent light after just after polarization will be half of the incident beam. Okay. If the incident light is unpolarized the after the first level uh the intensity will be I by2.
Okay. And for the second one we have to use Malu's law and it says that the this intensity of light into cos² theta that should be the final emergent light.
The final emergent light say I -ash is given by I by 2 cos² 30°. Now cos 30 is <unk>3x2 cos 30 is <unk>3x2. So it will be 3x4 <unk>3x2 whole square. So that is 3x4.
So it will be 3 I by 8. Now they ask you what is the ratio of unpolarized light emergent and unpolarized light. So emergent and unpolarized. So it will be Id -ash by I which is 3x8. If it is opposite of that then it will be I by Idash. makes sure that the order in bas in in case of ratio based questions order is correct.
So correct answer here will be 3 is to 8. Okay. Now next question figure shows the variation of photocurren with an anode potential for a photosensitive surface for three different radiations.
Let I A, I IC be the intensities and new A, new B and new C be the frequencies for the curves A, B and C respectively.
So what happens is that in this case the ray which are falling here it is C.
Okay, it is B, it is A. Now if you trace them back you will see that they all meet at this point. So B and C they all B and A sorry B and A they all meet at same point. So if they're meeting at the same point that means they must have same frequency same frequency only have same cutff voltage. Okay. So that means uh frequency of A must be equals to frequency of B. Okay. And for C the cutff voltage is much higher. Okay. In the negative side the cutff voltage is much higher. That means new C should be greater than both of them. New C should be greater than both of them. Now if you consider the current part, current part means the Y-axis part. So if you see the Yaxis part, uh the C and B they are matching. C and B they are matching at this point. That means they must have same intensity. So I C must be equals to I and because their saturation current is higher than the that of A that means their intensity should be higher than intensity of A. Okay. So IC equals to I and that should be greater than that of I A. Now if you read the options uh IC= to I is not given anywhere but new A is equal to new B is given. So if you see new A is new B and I A is not equals to IB. Obviously I A is this one. So which is smaller than that of IB. So new A equals to new B and I A is not equals to IB. Okay. So exact relationship is not being asked just they are asking you that which one is are equal which one are not equal. So this is the correct match. Now next question. Figure represents graph of kinetic energy of most energetic photoeleron kmax in electron volt and frequency new for a metal used as cathode in photo electric experiment. The threshold frequency of light for the photoelectric emission from the metal is. So here if you see that photoelectric emission the cutff frequency is actually this one but in the graph this point is not shown. This point is not shown in the graph. So what you can do you have to calculate only one point of information given to you that when the kax is 3 electron volt the frequency is 10 ^ of 15 Hz. No other information is given. So obviously we will take help of Einstein's photoelectric emission K max equals to K max equals to H new minus H new KN it is generally the work function but work function is equals to H new so we will use that so from here we can take H as common and we write new minus new not we are trying to find this new not so we have to separate it somehow Okay. So, K max is given in this graph as 3 electron volt. So, new - equals to K max divided by H. Right? So, what is K max 3 electron volt? Now 3 electron volt you have to convert it into jewles which is 1.6 into 10 ^ of -9. There is no shortcut here. Divided by H. H is 6.63 into 10 ^ of -34 which will be so now if you calculate this using a calculator it will turn out to be uh 0.8 into 10 ^ of 15 hertz.
Okay. Now from uh comparing this so new not should be new - 0.8 8 into 10 ^ of 15 and new is given by new is given by 10 ^ of 15. So we can write it as 1 into 10 ^ of 15 - 0.8 into 10 ^ of 15 or that will be 0.2 into 10 ^ of 15 or 2 into 10 ^ of 14. So the answer is 2 into 10 ^ of 14 Hz. So if you see the answer it will be 2.1 slightly more than that. That's fine. Now next question.
Clear? So this is a very simple straightforward question. Before going just let me uh discuss this question once more. So this is a very straightforward simple question. Okay. I ran out of space so that's why I had to squeeze it here. Uh let me write it a little bigger.
So that will be equals to 0.2 into 10 ^ of 15 = to 2 into 10 ^ of 14. Okay. Now uh so just in some cases only small informations will be given. You have to figure out which equation equation to use directly and using that equation you have to solve it. Okay. Now let's move on with the next question. According to the debrogley hypothesis, the wavelength associated with the moving electron of mass m is lambda e. Using the mass energy relation plank's quantum theory, the wavelength associated with the photon is lambda p. If the energy of the electron and photon is same, then the relationship between lambda e and lambda p r. So now uh we know that for a photon energy is given by h new that is h c by lambda. Now because it is given as lambda p we are writing it as lambda p. So energy of photon is h c by lambda p. For electron debrogley wavelength is given by h upon p. P is momentum right? for electron. So from here if we calculate momentum we will get h by lambda e. Now relationship between energy and momentum if you remember energy is given by p² by 2m kinetic energy basically when the electron is moving obviously its energy is kinetic energy. So kinetic energy k is given by p square by 2m p is given by square root of 2 mk. So from there it is this. So obviously energies are equal.
Energy of photon is equals to energy of electron. Right? So this is energy of photon. This is energy of electron. So to put the values p² by 2m equals to h c by lambda p. Now what is p²? p² is h² by lambda e². So we can write uh h² by 2 m lambda e² equals to h c by lambda p right. So basically if you compare these are all constants. So only this and this are important. So lambda p is proportional to lambda e squared.
Right? Both were in in division. So obviously when we take the in proportionality relation we have to take uh lambda p proportional to lambda a².
So from that we can say correct answer is this one.
Next a proton accelerated through a potential v has dbrrogly wavelength lambda. Okay. Then the debrogly wavelength of alpha particle when accelerated through the same potential v. So when you accelerate a charge okay uh through a potential difference what is the energy gained by it? So u the energy gained by it is actually given by so here we are talking about kinetic energy. So let me write k is e into v.
Okay. So the kinetic energy gained by the uh charged particle when it is moving through a electric field uh it is given by q into v. Okay, not E sorry uh it will be Q into V. Q into V is actually the work done. Okay, and from work energy principle the gain the gain in kinetic energy is basically Q into V, right? Gain in kinetic energy is the work done. So gain in kinetic energy is equals to Q into V. Okay. So when you accelerate an electron, when you accelerate an electron, what will be the gain of the kinetic energy by the electron? It will be E into V. E is charge of the electron V is potential.
So what will be the Dragly wavelength?
Dubra wavelength will be h by p which is h by uh square root of 2 mass of electron into uh okay so notice that this is not electron uh my mistake this is proton this is proton okay okay it is proton But proton also have chargy. So wavelength of proton is h by p proton which is h by square root of 2 mass of proton into uh kinetic energy that is e into v. Okay.
Proton pot momentum is 2 m k. Okay. So from here we calculated the wavelength of proton. Now let us go back and calculate the wavelength of alpha particle. So alpha particle has mass of alpha particle is actually four times mass of proton. Okay. And charge of alpha particle is twice of proton. Okay.
It has two protons. So alpha particle basically mass of alpha particle is 4 MP and charge of alpha particle is 2 e. So now we have to use these information and calculate the uh wavelength of alpha particle. Now everything will be same. Okay. H by square root of so instead of MP we have to use mass of alpha particle. Right? Do not do it. Do not overdo it or do not do it separately. Just you need to replace the mass of proton with mass of alpha particle. Rest will remain same. So mass of uh alpha particle into this charge term should be replaced by charge of alpha particle which is 2 e into v rest everything should remain the structure of the the equation should remain same. So this is one important thing. So once you have done uh something for one item, the second item you just replace the corresponding terms and you will get the second item as well. Okay, this is applicable for many other questions as well. So uh lambda alpha is now you take this four and two out of the square root you get square root of 2 MP EV.
Right? If you take this four and two out of the square root. So what is 4 into 2?
4 into 2 is 8. So it will be root 8, right? We have taken that out. Why?
Because this quantity is nothing but wavelength of proton. So from here we can write alpha lambda alpha is equal to 1x roo<unk>8 uh lambda proton and 1x roo<unk>8 is what? 1x roo<unk>8 is 2<unk>2.
Okay. So, correct answer for this question is this.
Now, next question.
In a hydrogen atom, the radius of nth bore orbit is Rn. The graph between log RN by R1 and log N will be. So, we have to find Y = MX + C or Y= Y²= 4 AX. like that we have to find the corresponding equation to understand what is the relationship between the x and y coordinates. So here we know that rn is given by n² r1 for board's orbit rn is equal to n² r1. Now if you uh try to find rn by r1 you will get equals to n². Now if you take log of both side so log of rn by r1 equ= to log of n right. So x² log is 2 log x. So this is the log equation when you take the log from both sides. So basically this is our yaxis. This is our yaxis and this is our x-axis.
Right? So from that we can say it must be y = 2x. What is this equation? This is an equation of straight line and not only any straight line. It's a straight line passing through origin. So the straight line must be passing through origin and from here only one graph which is passing through origin as a straight line which is first one. So correct answer is first one. Okay.
Next. Total energy of electron in an excited state of hydrogen atom is minus3.4 electron volt. The kinetic and potential energy of electron in this state. Now remember that if total energy is given kinetic energy is actually modulus of total energy. Kinetic energy is exactly modulus of total energy.
Potential energy is twice of total energy including sign. Okay. So basically this is always true for electrons orbiting in b orbit or a satellite orbiting around earth. The same formula goes. So the total energy is 3.4 minus 3.4 e. So the kinetic energy should be plus 3.4 e. Only one option is with plus 3.4 e which is this one. And potential energy automatically matches which is minus 6.8 electron volt. So correct answer is option B.
Right. Next question. First bore radius of an atom with Z is equal to 82 is R.
Radius of its third orbit is now um R N normally is N² by Z R1.
Okay, for hydrogen zed becomes 1. So it is simply N² R1. Now if zed is constant, if zed is constant, okay, the zed plays no role over here.
So because R1 also have a quantity which is which is basically having uh au involvement of zed. So zed is in independent as long as uh the zed value is constant. So it will be n² r only.
you don't need to consider um zed value because zed is constant. So n is basically r for first orbit it is given. So one is given and you are finding third orbit. So third orbit means 3² r. So this is 9 r. So correct answer 9 r. Remember this is only valid when zed is constant. If zed is not constant and then ask they ask you which one is bigger or smaller you have to use n² by z.
Now next a nucleus with mass number 240 breaks into two fragments each of mass number 120. The binding energy per nucleon of the unfragmented nuclei is 7.6 6 me while that of fragments is 8.5 me. The total gain in the binding energy in the process is so you have to find the total binding energy. How to find that? First we have to find the binding energy total binding energy for the uh parent nucleus which is nothing but 240 into uh 7.6 okay uh 240 into 7.6 and six. This is the parent. So this is the unfragmented.
Unfragmented means the parent nuclei.
Okay, that is the uh nuclei with mass number 240 uranium basically. Now what is the total uh binding energy for daughter?
Total binding energy for daughter is basically given by uh how many uh nuclei each have 120 each have 8.5.
So each have uh 120 each have 8.5. So basically what you are doing 120 into 8.5 plus 120 into 8.5. This is for the first one. This is for the second one.
What is when you add two same quantities you can instead of adding you can write it as multiplied by 2.
So now 2 into 120 is 240 into 8.5.
Okay. So now what is the uh what is the gain in energy? The gain in energy is given by the uh EB for parent.
My uh sorry EB for daughter. daughter is now heavier minus EB for parent. Okay. So EB for daughter is uh 240 into 8.5 and this is 240 into 7.6.
So it will be 240 common 8.5 - 7.6 6 which is 240 into 0.9 which is approximately 216 very close to.9 means very close 90%. Very close to 240. Okay. So in obviously you have an access to a oncreen calculator but otherwise even after solving up to this much you can guess from the given option as well. Next question. the approximate ratio of nuclear mass uh densities of 19779 AU and 10747 AG. So gold and silver. So nuclear mass densities of almost all elements are almost constant. So their ratio will be 1 is to1. Okay. For all stable nucleus it is almost same because nuclear mass density is a constant quantity. That's that means the gold and silver both have same nuclear mass density.
Next the energy equivalent of 1 g of substance. So what is the energy equivalent? Energy equivalent is delta m c². So delta m is 1 g. 1 g means 1 into 10 ^ of -3 kilog. And what is c? C is 3 into 10 ^ of 8².
Obviously C square so uh 9 C² is 9 into 10 ^ of 16 - 3 13 okay so 9 into 10 ^ of 13 Jew easy question let's move on a full wave rectifier circuit along with the output is shown uh in the figure the com the contributions from the diode one is are so if you see that uh here the diode one is the bottom one. The diode one is the bottom one and uh the input signal here is first is positive, second is negative. So this is positive half cycle, this is negative half cycle.
This is very very important to observe the input as well. Now we know that this diode conducts when this side is positive, this side is negative. this diode will conduct when this side is negative, this side is positive. Okay.
So this becomes the uh conduct uh conduction path. So that means diode one conducts during the negative half cycle and this diode say uh two this diode conducts during the uh positive half cycle. So they have just changed the order. Generally when you write the same question in your exam we start with the first diode then we go back to the second diode. they have reversed the order to check. Now remember this question. So obviously the answer for this question is negative half cycle which is C and B and D. So B and D are actually conducted by diode number one in this question. Again as I am mentioning that you have to be careful the uh input wave is important. If they reverse the input wave then you have to change the order. Remember for a negative half cycle the bottom diet conducts. For positive half cycle the top diode conducts. So if the order is inverted. So suppose the input signal is now given. They have given you an input signal like this say like this. So in this case this will be for the diode number one. This will be for diode number two for this case.
Okay. So be careful about the input signal. If the input signal is given based on input signal you need to change your answer. Okay. So it will be B and D which is correct answer. So option C is correct.
Now next question of the diode shown in the following figures which is reserve reversed bias. So reversed bias means N side should have a higher resistance than that of the P side. So if you see that here P this is P this is N P is -5 N is -12 so -12 is smaller than minus 5 so this is a forward bias so this is a forward biased what about this so here if you see that P side is zero N side is minus 10 again P side is higher than the N side so this is also forward biased now let's check this now here the this side is zero O this side is + 5. So 0 is less than + 5. So this must be the reverse biased. Okay. And this is our answer. You can check the other on last one also. Here again P side is higher voltage 10 volt. N side is 5 volt. So that that is again a forward bias. So this is also forward biased.
Okay. So only reverse biased is this one.
Next. Assume that each diode shown in the figure has forward bias resistance of 50 ohm and infinite reverse bias resistance. Uh the current through the resistance of 150 ohm is so we have to find the current through this 150 ohm resistance. Now it is mentioned in the question that forward bias resistance is 50 reverse bias resistance is infinity.
So which diode is forward biased? Which diode is reverse biased? So the the battery is trying to send a current in this direction. Okay. Through the circuit and it will only pass through this diode because this diode is showing the arrow this arrow in the direction of current. Basically this uh triangle small triangle actually shows the direction of current. So if the current is flowing in the direction of arrow the current will pass. If the current is flowing towards this barrier, current will stop. So this diode is reverse biased. Reverse biased means the infinity resistance. Infinity resistance means no current flowing through it. No current flowing through this as well. So this is basically disconnected. This portion is disconnected. Only the diode which is connected is this one. And only resistance we need to consider this resistance, this resistance and diode resistance. Do not forget that because diode also have a resistance 50 ohm in the forward bias. Okay. So that resistance also you need to consider otherwise even after solving the entire question correctly you will get zero marks because you forgot to calculate the diode resistance. Diode is not ideal but and ideal diode have zero forward resistance. Here it is mentioned forward resistance is 50 ohm. So total resistance is 50 + 50 + 150.
So it will be 250 ohm.
So total resistance is 250 ohm. The voltage that you have applied is 10 volt. So what is the current? current is uh 10 volt by 250 which will be uh so let me uh consider it will be uh 1x 250 means 1x 25 4 by 100.04 okay so it will be 0.04 104 ampere.
Now next when a P and junction diode is connected in forward bias its barrier potential decreases and less current flows in the circuit decreases. So when the P and junction diode is connected in forward bias its barrier potential decreases and current flow is more. So barrier potential decreases and more current flows in the circuit. Basically forward biasing means we are allowing the current to flow by decreasing the barrier potential. So barrier potential is basically the like a dam which is pro preventing the flood. Okay. Uh so if the uh if you open the floodgate the uh water will start flowing. So basically the forward biasing is like the opening the floodgate. So basically the barrier potential decreases and more current flows in the circuit. Next question. A charge is uniformly distributed over a large iron plate. The electric field at a point close to the center of the plate is 40 volt per meter. If the iron plate is replaced by a copper plate of the same geometrical dimension and carrying the same charge then the electric field at the same point will be. So here we have an iron plate.
Okay, we have an iron plate and we have a copper plate of same dimension. Okay.
Uh let me use the tool so that uh my drawing is uh pathetic. So but I can use some tool to show the uh this is an iron plate suppose. Okay. And we have a same copper plate.
Okay. So we have an iron plate.
Iron plate.
and we have a copper plate. Okay. So if and the dimensions are same we know that uh electric field near a conductor is given by sigma by 2 epsylon KN. It does not depend on what is the material. It only depends on the charge density and charge density is basically total charge divided by area. Now charge is same area is same. So the electric field should also be same. So if the electric field for iron is 40 volt per meter for copper it will also be 40 volt per meter. Okay next question. How many 6 microfarad 200 volt condensers? Condensers are nothing but capacitors are needed to make obviously this capacitors I have added it was not there in the question itself. How many 60 6 microfarad 200 volt condensers are needed to make a condenser of 80 18 microfarad 600 volt.
So if you see that the voltage rating of the given capacitor is 200 volt but the uh one we are trying to find is 600 volt which is more than this rating. If it is less than this rating we could not do we cannot decrease the voltage rating. We can however increase the voltage rating.
How can we increase the voltage rating?
To increase the voltage rating, we have to connect in series. So when the devices are connected in series, their respective voltages gets added. So if you add three such six microfarad, If you add three such six microfarad, what will be the voltage rating?
Voltage. This one will take 200. This one will take 200. This one will also take 200. So total voltage rating will be 600 volt. So first stage is to check how many capacitors we require in series. These kind of questions are common. So if you see this kind of questions, first step is to find how many capacitors we need in series. To do that first voltage analysis 600 volt divided by 200 volt which is three.
Suppose it is say 1,000 volt. So 1,000 volt means 1,000 by 200 will be five capacitors in series. That is the first step. Once you have done the first step, what is the equivalent capacitors of all these capacitors in series? In equivalent when the they are connected in series, equivalent capacitors becomes any one capacitors divided by uh uh the total capacitors uh sorry number of capacitor total number of capacitor. So here we have 6 microfarad divided by 3.
We will calculate that uh in a while.
Now let uh let us assume obviously this is not giving us our required 18 microfarad to increase the capacitance we always have to add capacitors in parallel. But first step is to adjust the voltage. One voltage once voltage is adjusted then we will connect them in parallel. So parallel means we have to take these combination of three capacitors and keep on adding them in parallel combination. Okay. So that the voltage rating remains same and we are keep on adding more and more capacitors so that the uh total capacitors becomes uh 18 microfarad. So now we have say such in such row in such row each row having three capacitors. So in such row each having three capacitors. So total how many rows are there? n. So what is the total capacitor? Total capacitance should be n into c by3 and that should be equals to our required capacitors that is 18. Okay. So because first step is voltage. Voltage can be added only.
So voltage we have added capacitors. To increase the capacitors we have to connect them in parallel and we are using the shortcut method of parallel and series combination. Okay. So now n we are trying to first our step first step is to find n.
Okay. So n into c is 6 by3 = 18 or n = 9. So we require nine such rows. Okay.
Nine rows each row have three capacitors. So total number of capacitors should be 3 into n. Okay.
Nine such rows each row have three capacitors each. So 3 into 9 that is 27 capacitors. So we require not one not two but 27 capacitors to uh achieve the combination.
Next a wire when connected to 220 volt main supply has power dissipation P1.
Now the wire is cut into two equal pieces which are connected parallel to the same supply. Power dissipated in this case is P2. What is the ratio of P P2 and P1? So in this case we have a wire of length L.
Suppose the resistance of this wire is R. So what is the power dissipated?
Power dissipated is V² by R. Right?
Simple. Now you cut this wire into half.
Okay. So when you cut the wire into half, the resistance also becomes half because resistance is proportional to length. Okay. Now you have connected them in parallel. Now you have connected them in parallel. So what is the equivalent resistance? Equivalent resistance becomes R2 RX2 divided by two. Two equal resistances connected in parallel. We don't need to use R1 R2 by R1 + R2. we can directly calculate. No need to waste our energy in calculating simple things.
So parall combination divided by two series combination multiply by two.
Right? We did that for capacitors just few moments ago. Only thing you have to understand what is happening in series combination resistance increases parallel combination resistance decreases. So equivalent resistance in this case is Rx4. So what is the power dissipated now? Power dissipated now is same source. So v ² by r equivalent which is v ² by rx by 4 that is 4 v ² by r or 4 p1. Now they say what is the ratio of p2 is to p1. So p2 by p1 equals to 4. So the ratio is 4.
Okay. 4 is to 1 actually. Now next question.
The graph showing the variations of the magnetic field strength uh B with distance R from a long current carrying conductor is. So we know that magnetic field strength due to long current carrying wire is given by mu by 2 pi I upon R. So obviously magnetic field is inversely proportional to the distance and inverse proportionality graph is this one. Simple question. Okay, simple question. Do not try to overthink. Just uh in some cases you might need to write the formulas. In some cases you might need to find the formula and find the relation. Here it is direct formula based. You know the formula find the graph. Obviously that is the answer. If a uh if a diamagnetic substance is brought near the north or south pole of the bar magnet, it is repelled by the north pole and attracted by the south pole. Attracted by the north pole and repelled by the south pole, attracted by both poles, repelled by both poles. A diagonal substance always moves out of the magnetic field. So that means it will be repelled by both poles. It has to be repelled by both poles. stagary substance. It's not unlike that it's a monopole or like that. It will be repelled by both poles. Very careful. It does not being repelled by one and attracted by one. It is always repelled by both poles. Okay. Next. If a wire of length 2.5 kilometers and resistance 35 ohm has fallen from a height of 10 m in earth's horizontal field of 2 into 10 ^ of - 5 tesla the current through the wire is uh so length of the wire is 2.5 km so length is 2.5 kilm which is not SI unit so we to convert it into SI unit which is 2500 m.
Okay. Uh resistance is 35 ohm. Okay.
Resistance given here is 35 ohm and it is falling in earth's horizontal field. So first step is to calculate the induced emf. What is induced emf?
Induced emf is B v. But sir V is not given in this question. So V is not given but what is given is height through which it has fallen and it is falling under gravity right. So when the wire is falling under the gravity okay the gravitational acceleration is g. So after it falls by a distance of 10 m what is its velocity that we have to calculate. So v uh squared equals to u² + 2 g s obviously u is zero. So v² = 2 into 10 into s is 10.
So v = 10 <unk>2.
So v = 10 <unk>2. Okay. Now we have to put that back in this formula. epsylon sorry uh the induced emf is magnetic field 2 into 10 ^ of -5 into now you can put the value of <unk>2 also roo<unk>2 is almost 1.4 four. So it will be almost equals to 14. Almost slightly more than that but for all practical purposes we will take this as 14. Okay. So V is 14 and length is 2500.
Okay. So this is the first step. Now we have to find the current. The question is asking question asking about the current. So what is current? Current is obviously emf by resistance. So emf is 2 into 10 ^ of 5. You can uh reduce it but uh slightly later 2 into 10 ^ of -5 into 14 into 2500 divided by 35.
5 7 are 35 5 are 25. Now multiply 2 into 500 is 1,000. So 2 into 10 ^ of - 5 + 3.
Okay. 2 into 500 is,000. So it will be 2 into 10 ^ of -2 that is 0.02 ampere. So the current through the circuit is 0.02 ampere. Option C is correct. Slightly longer lengthy question but not so difficult.
A dialectric constant of air is 1.006.
The speed of electromagnetic wave traveling in air is a into 10 ^ of 8 m/s where a is about. So for air and vacuum uh the speed of light is almost equals to 3 into 10 ^ of 8. So correct answer is 3 into 10 ^ of 8. The dialectric constant is almost close to one. Okay.
Okay, if it is not close to one, then you have to consider the dialectric uh uh uh constant for air. But for air, the speed of light is almost very close to uh three. Okay, actually it is 2.9998.
But who who cares? It's three, right?
Now here B is uh the value of A cannot be more than three. Remember that this is a cosmological limit, a maximum speed limit. Okay, cosmic speed limit sometime it is called uh so that value is restricted to three. It cannot be more than three. It can be less than three but that for for it to be 2.5 the dialectric constant should be very very high.
Okay. A generator produces a voltage that is given by V = 240 sin 2 120T where T is in seconds. The frequency and RMS voltages are. So again voltage is given. So what is the frequency?
Frequency is given by uh omega by 2 pi.
We have solved another question just few moments back. So omega is 120. So 120 means 120 by 2 pi.
So this will be 60 by pi.
Pi is 3.14.
Okay. So if you consider this to be exactly three, it should be very close to 20. It should be very close to 20 but slightly less than 20. What about the uh uh RMS value? RMS value of the voltage is RMS value of the voltage is V by <unk>2. So vot here is 240 by <unk>2.
So it will be uh approximately 120 <unk>2 which is 120 into 1.4.
So it will be uh 12 into 14 12 into 14 means 140 + 28 168 approximately 168.
Okay. So if you see the option obviously the correct match among all the option is 19 Hz 160 volt. This is the closest match. Right? Now let's move on.
A boy of height 1 m stands in front of a convex mirror. His distance from the mirror is equal to its focal length. The height of uh his images. So boy height of the boy is 1 m. Okay. in front of a convex mirror. Okay, distance from the mirror is equal to focal length. So we know that so u is actually equals to focal length. Convex mirror both are all are negative here. So v if you calculate 1x v + 1x f equals to 1x f you can use the sign convention s minus - will cancel anyway uh and you will get or you can add both of them as minus. Don't worry uh I am not cheating it will be fine. So 1x f minus 1x f so it will be zero.
So v will be 1 by 0 and that will be infinity. The object uh should be h sorry sorry this is a con uh sorry convex mirror. My mistake this is a convex mirror. This is not concave mirror. So concave mirror focal length is positive. Sorry my mistake. So 1x v + 1x - f = 1x f or convex mirror focal length is positive. Sorry I was solving so many questions so I misunderstood.
Sorry. So 1x v + 1 1x - f is equal to 1x f. So 1x v = 1x f + 1x f which is equals to uh 2x f.
Okay. So from here we get v = fx2.
So what is the uh magnification?
Magnification is uh minus v by u. Okay, magnification is minus v by u. Now uh for all practical purposes we will take uh this as minus fx 2 by - f u is given as u is given as focal length. So minus minus cancels and it will be half. Okay. So mag uh the magnification for this particular problem is half. So the height of the image should be half into height of the object that is half into 1 m that is.5 m.
Okay. So the height of the object is image is.5 m. If it is a concave mirror then uh infinity is correct answer.
Okay. Sorry I just misunderstood the problem. Now next question. Two coherent monochromatic light beams of intensities I and 4 I are superimposed. The maximum and minimum possible resulting intensities. So maximum possible intensity is basically sum of the given intensities and minimum possible intensity is maximum value minus minimum value that is 3 I. So maximum is sum of them, minimum is difference between them.
Okay. So correct answer is 5 I and 3 I.
Okay. Always.
Next. The momentum of electrons having wavelength 2 anstrom. Okay. Given the value of H um mass of electron is given.
So you have to find the momentum of electron when the wavelength is given.
We know that wavelength is given by H by momentum. Okay, mass is not required.
Only thing you require is momentum equals to H by lambda. If they ask you kindinetic energy then mass would have been required. So H is 6.626 into 10 ^ of - 34. We wavelength of electron is 2 anstrom. 2 anstrom means 2 into 10 ^ of - 10. Okay. Okay, angstrom means 2 into 10^ - 10. So now you cancel it. You will get 3.313.
So 3 uh 313 into 10 ^ of -4 SI unit. So 3.313 into 10 ^ of -4 kg m/s.
Okay. First bore radius of an atom with zed is equal to 20. I think I solved this question. Sorry for that. So bore radius is actually proportional to n² if zed is constant. So for nth orbit it will be n² first orbit which will be 9r.
Okay. Sorry I have solved this question but again no harm in doing one more question. We have 101 question. So don't worry. The binding energy of dyerin is 1.15 mega electron volt per nucleon and of an alpha particle has binding energy of 7.1 me per nucleon then the reaction of two doeran merging to become one helium the q energies so basically you have to find again the total energy of the product and total energy of the uh reactants okay so reactants are two helium atoms Okay. So, and it is binding energy binding energy of dyer per nucleon is given. Okay. So, per nucleon means total energy of one deran total energy of one deran is given by how many nucleons are there? Two nucleon each of them have 1.15 right that is 2.3 me.
Now when the in the reaction you see that how many dioons are there 1 + 1 that is 2. So total energy total reactance energy reactance will be 2 into 2.3 that is 4.6. Now don't ask me why I am multiplying by two. Obviously you can add 2.3 + 2.3 and adding two same quantities multiplying by 2 is same right so that's why we are doing it so reactant is 4.6 6 me I know some of you are will may be confused and ask me sir why uh you have done this multiplied by two because here we have two deterons so 2 into 2.3 now now let us go to the product so what is the product how many product are there only one product which is helium so and uh binding energy per nucleon is 7.1 so total binding energy total binding energy of the product. So this is the binding energy of the reactants.
This is not parent or daughter because parent daughter is when one atom breaks into two. Here two atoms are fusing into one. So for product it is basically 4 into 7.1. So four nucleons are there and each of them have 7.1 me. So it will be 28 4 me.
So 28.4 me is the energy of the product.
So to so energy difference or Q value energy difference is Q value which is basically 28.4 - 4.6 that is 24 23.8 8 23.8 okay so correct answer is 23.8 8 MAV.
Next question.
Two identical PN junctions may be connected in series with the battery in three ways as shown in the figure. The potential drops across the two P and junctions are equal in. So if you see that there are three circuit okay uh and they are asking in which case the potential drop across each P and junction will be equal. So let us understand the diagrams first. So if you see that the this is positive this is negative this is P this is N this is N this is P. So this should be forward biased this should be reverse biased. So in this case both uh there should not be any current flowing or even if there is some current is flowing the voltage drop across this should be uh very small and voltage drop across this should be very high.
Now here uh this case the uh if you see this one again the same uh positive and negative this is forward biased this is also forward biased so voltage drop from these two will should be small that is around 7 for a silicon diode so this is also should be small so these voltage drops both of them should be small value so circuit number two must be uh having the same voltage drop Now if you go back to the other one. So this diode is reversed bias. This diode is reverse bias. So whatever voltage drop is happening across this uh that voltage is not translating to the second diode. You might say sir both diodes are reverse bias. Both of them should have very high voltage drop across them. Actually no.
Suppose you have applied say 10 volt across here. This diode itself will absorb all the 10 volt because it will not allow any current to pass. So that means this diode is completely disconnected. And because this diode is completely disconnected, no current will flow. So that means the voltage drops will not be same when you connect two diodes in reverse bias. But when you connect two diodes in forward bias because current is flowing uh the voltage drop across them should be uh same small and same. If one is forward one is reverse. Obviously that is not the case. Okay. One in one case even if the some current is passing through that the current voltage drops should be very very high for reverse biased and very small for forward bias. In case of both diode reverse bias, first diode itself will stop the almost entire current.
Second diode will not get a chance to get any current through it. So that is also not the case. So very interesting question. It I think slightly older question but very interesting question.
So in this case only circuit two is the answer.
I think circuit two is not given. So that is uh one thing that you need to careful. Circuit two should be the answer. Okay. And one more part is that if they say that okay this case voltage current is not flowing ideal. Okay. So they are identical P and junctions but this is not mentioned whether they are ideal diodes or not. If they are ideal diodes then voltage drop across each case for these two should be zero. Okay.
Because no current will flow. So circuit one and circuit three if they are ideal diodes no current flows through the circuit at all because this is forward bias. But that second diode is completely disconnected. So in that case circuit one and circuit three but that is mentioned here. But in that case it has to be mentioned that the diodes are uh identical. Ideal P and diode. Ideal P and diode means it conducts in forward bias voltage drop is zero. Okay. And in reverse bias voltage drop is uh infinity. Okay. voltage drop is uh across them is infinity so that no current passes through the circuit. So that can be the answer if they are saying they are ideal doubts. If they're not saying ideal layout, this should be the answer. Tricky question, slightly older question and I believe there are slightly uh not 100% correct uh question. Anyway, so now let's move on to the next question. So now let's move on to question number 65. The value of resistance is 10.845 ohm and the current is 3.23 23 anstrom on multiplying we get the potential difference is 35.02935 volt. So they have already multiplied it so you don't need to multiply it. The value of potential difference in terms of significant figures would be. So what we are doing? Current is given, resistance is given and voltage equals to uh so what we have we are doing we are basically doing V is equal to IR. So I V and I and R are given. So multiplying that you get this result. Now how to find the significant figures in case of multiplication? So I told you during addition the decimal number matters.
Okay. After decimal how many numbers are there for addition and subtraction. For multiplication and division you have to find the minimum significant figure. For the given data. So in the given data the minimum significant figure is three.
Okay. 1 2 3. For this number it is 1 2 3 4 5. Okay. between five and three.
Obviously three is the less significant number. So in the answer also there should be only three significant figures only. It does not matter how many decimals are there. Decimals are important for addition and subtraction.
Whereas for multiplication and division least number of significant figure is important. So here least number of significant figure is three. So our answer should also be up to three digits only. So 1 2 3. Okay. Obviously this zero is significant. After decimal any zero is there that is significant. So option will be correct answer will be 35.0. You don't need to do anything else. No multiplication is required. In even if they have not given you this value they ask you V is that I is that R is that V is that what what will be the value? So remember that in case of division and multiplication least number of significant figure is important. The values will be almost everywhere it will be same. So make sure that you understand that for addition uh decimal number is important. For multiplication least number of significant figure is important. Two are different things.
Okay. A body is projected vertically upwards. The times corresponding to the height h while ascending and while descending are t1 and ts2 respectively.
Then the velocity of projection is so here we have it is not a normal projectile motion. It is basically you are throwing an object upward. Okay with some velocity say v. Okay. And what else is acting on it? The gravitational acceleration acting downward. Suppose there is a height h.
Okay. The object will go up through height h it will go to the maximum position and then it will come back again. So the corresponding times when going up it is t1 while coming back it is t2. Okay. And they are asking you what is the initial velocity. Now at first glance it might look a difficult problem but only one thing that you need to remember is the equation of motion and second thing is quadratic equation solution. Okay. So first it is a motion under the gravity. So the uh distance traveled will be given by half uh sorry not half u t minus/ gt² obviously we are trying to find not v uh let me write it u so we are trying to find the initial veloc initial speed u okay so h is equal to ut +/ gt² multiply by 2 every term we get 2 h = 2 u t minus gt² Now uh reorganize gt² - 2 u t uh + 2 h = 0. So obviously this is an quadratic equation. Quadratic equation means a x² + bx + c =0. So this is the standard quadratic equation. So if you see this is a quadratic equation of time t. Now you can write so if you solve this equation you will get t1 and t2.
that solution is not required. We will use a much more shortcut method which is the either we will find the sum of the roots first. We will try sum of the roots otherwise we will try to find the multiplication of the roots. Okay. What is sum of the roots? Sum of the root is minus b by a. Okay. minus b by a is sum of the root and multiplication of root is c by a right so now let's put that value over here uh so what is b here b you can see is 2 u b is 2 u so let's put that so t1 + t2 is equal to minus 2 u by a is what a is g and sorry there B is not 2 U. B is minus 2 U because here we have a positive sign. Quadratic equation generally comes with a positive sign. So it will be minus 2 U. So it will be minus 2 U by G.
So it will be minus 2 U by G. So minus minus cancels. We get 2 U by G. So from this equation we can easily calculate U. U will be G into T1 + TS2 by 2. So correct answer is option B. Now in case we do not find using T1 plus T2 you have to use the multiplication as well. But here it is not required obviously. So if you see the options either multiplication or additions are there. Okay. In fact three of the terms involving multiplication only one two terms involved uh addition. So in some cases you might be required to use multiplication and addition for example option D but here it is not required.
You can directly get the value of U by only adding the roots. Okay. So so it will be just by adding the roots you get the you get the answer right. So correct answer option B and make sure you understand the problem. Okay. Basically when some object is accelerating it's a quadratic equation. Okay when an object is accelerating with some initial speed especially under the motion under the gravity it's a quadratic equation and if you solve the quadratic equation you will get two values of t okay except for the top position you will get two values of t and that actually gives you a an idea that how to solve a problem. So this is a very typical problem uh you have to be little careful to solve. Next question. A particle starts moving from point 2 0a 1. Uh displacement for the particle is 8 icap minus 2 jcap plus kcap. The final coordinates of the particle is so if a particle moves from say x1 y1 z1 2 x2 y2 z2 then what is the displacement vector r? So displacement vector R is actually given by X2 - X1 icap plus Y2 - Y1 JCAP plus Z2 - Z1 KCAP. So this vector is actually given here. This vector is actually given here. So what we will be doing? We will just compare this x iCap term with this term, jcap term with this term and kcap term with this term. Now sign has to be included here and x1 uh y1 and z1 is given. Okay. So this is the starting point x1 y1 z1. So just put those values. So x2 minus 2 should be equals to 8. Okay. So this term is basically this and x1 is given. So this is x1 x1 y1 z1 this is given right. So just put the value. Okay this is 2 not z. So this will be x2 will be equals to 8 + 2 that is 10. Similarly, y2 - uh y2 - 10 = -2. Here it is -2. Okay. So, y2 will be 10 - 2 that is 8. And z 2 - 1 = 1. So, zed2 equals to 2. So, what is the final point? Final point is 081.
So final point is sorry not 0 10 81 10 81. So correct uh 1082. Sorry sorry sorry 1082 1082 is the answer. So 1082 So obviously option A is correct answer.
Okay. So just follow the step and you will find the answer. This is a easy question. Next two vectors are given. A vector equals to icap + 2 jcap + 2 kcap and b vector equals to 3 icap + 3 jcap + 2 kcap.
Okay. Another vector c has same magnitude as b but has the same direction as a then which of the following vector represent c. So a and b are given and it is mentioned that c vector has same magnitude as b but same direction as a. So this is the question.
Basically you need to find a vector which looks like this. Now how to do that? Obviously magnitude we can calculate but how to find the unit vector. To find unit vector obviously what you need is so this is unit vector you need to write the original vector divided by magnitude of this vector right this is unit vector so basically what we are trying to find we are trying to find b vector by a vector modulus both of them modulus into a vector so a vector is there now if you are smart if you see that so the vector form should be Icap 2 JCAP plus 2K cap multiplied by some constant. So if you see that that appears in two options. So either this option or this option. Now if you don't want to solve, if you want to guess the answer and move on, you can guess one of these two answers and move on because these two answers would be wrong. Okay?
Uh and uh if you want to guess, my um opinion is that because 7x3 is common, you can guess 7 by3. But that could be wrong also. So don't trust those things in comet K. It works in case but in comet K does not work. But anyway, so first uh anyway we can find the magnitude of B and check which one is matching. So magnitude of B is basically square roo<unk> of 3 squar + 6 square + 2 square right square of individual terms and square root. So it should be uh square root of 9 + 36 + 4. So it will be 49.
Okay. 49 square root. So this should should be seven. So by solving only one we see that yes uh this is the correct answer. So the common term is actually is correct. Common term of magnitude and common term of vector. vector we have figured out that obviously this should be the correct vector. So that is how you even if you have not are not solving it you can guess it like that. You can check the other magnitude as well.
Magnitude of a is given by square root of 1 square + 2² + 2² that is uh 9 square roo<unk> of 9 which is 3. So 7x3 is the correct answer. So first step is to identify what vector we are trying to find. Second step is to find the magnitudes and calculate the problem. Very easy standard problem. Uh not a time-taking problem at all. Two projectiles, one fired from earth with 5 m/s and another fired from a planet with 3 m/s trace identical trajectories. If the uh acceleration due to gravity on earth is 9.8 m/s then the acceleration due to gravity on the planet is so basically we have two projectiles each are tracing identical paths that means the angle of projection is constant obviously same angle of projection. Now to solve this question you can either use the range formula or you can use height formula does not matter because the trajectory is identical. So that means height should also be identical.
So any one of them I will use the range formula. So range for projectile one is obviously u1² sin 2 theta by g and for another planet u2² sin 2 theta by gdash. G dash is the obviously the gravitational acceleration of the planet. So sin 2 theta sin 2 2 theta cancels. So we get gdash equ= to u2² by u1² into g that is u2 is 3 squared that is 9 and u1 is 5 that is 25. Now for all practical purposes even though it is mentioned 9.8 8 let us use 10 only that actually helps us to calculate faster okay so it will be two and we will have five so 18 x 5 18 x 5 means 3.6 okay 3.6 m/s squared so correct answer is 3.5 m/s squared okay so we have calculated the closest value obviously because we have instead of taking 9.8 date we have taken uh 10 and you have a small on-screen calculator if you want to use it to find the exact value you can do it or you can find the approximate value without using calculator without wasting time because clicking on the calculator putting the values takes some time so if you can solve a problem without using that calculator you should definitely try that okay because the using the calculator for every and each and every step because it is not a physical calculator it takes time okay three forces on a body are shown in the figure. To have the resultant force only along the y direction, the magnitude of the minimum additional force needed along o x is. So this is not a lamis theorem problem. You might be thinking sir three fources are acting why we are not applying lamis theorem. First of all this is not a lamis theorem problem.
This is a resultant calculation problem.
Okay. So here we have 1 2 3 forces given. They are saying that what should be the minimum force along this direction so that the resultant is only along y direction. So basically what we have to do we need to find x component of the given vectors and find the resultant x component. Once we find the resultant x component just opposite of that we require on the uh opposite side either on o x actually they're saying o x so it should be along this direction okay so w should be in this direction so we have to find the resultant force in the negative x-axis and then we will say that okay we require this much force along the w not not an issue so what is the x component of this vector x component of This vector is 1 Newton cos 60°. Okay. 1 Newton cos 60°. Now if you see that this angle is not given. This angle is not given. So how to calculate that angle? So for that uh if you see that this is a straight line. This is a straight line. Right? So the sum of all the angle should be 180°. Sum of all the angles should be 180°. So let's do that as well.
So here this is 30 this is 60. So 90° and this angle is also 90. So this is 60 means this should be also 30°. Right?
Any doubt here you can ask me during the class. So here it is 60. Here it is 30.
So 60 + 30 is 90. 90 + uh uh 30 is 120.
So this angle will also be 60°.
Okay, just apply basic geometry you will be able to find the angle. So this angle is 60°. So this component will also be cos component. So this will be 2 cos 60. Okay. Obviously adjacent angle matters. This is the uh adjacent side.
So it will be 2 cos 60. So this is the all the force along x. Okay. Ox. So this will be 1 into half + 2 into half that is uh 3x2. So if you add this this will be 3x2 along O X.
This force is along O X. Now what is the force along the negative X-axis? O Xdash. Okay, that is this component.
This is the O X dash. This component again this angle is 30. So this angle will be 60°. So that means this angle uh the o x dash component will be 4 cos 60.
So all angles are actually 60°. So 4 into half that is 2 Newton. So obviously the force along the negative x-axis is more compared to positive xaxis. So how much force extra force you need to add?
The extra force that you need to add is uh 2 Newton minus 3x2 Newton that is uh 0.5 Newton. So 0.5 Newton you have to add along ox so that uh the uh resultant force remains only along yaxis. You don't need to calculate anything along y-axis. So the these kind of uh confusing language questions are common in comet K which where they are trying to confuse you. We will see through one more problem. Uh but here they are saying the resultant force should should be along y-axis only means the x component should be zero. That is the uh essence of this question. Okay.
You don't need to do anything about y-axis.
Right. Now let's move on to the next question. So this is another question where the language can be confusing if you don't understand it properly. An object with mass 5 kg is acted upon by a force F = - 3 icap + 4 jcap newton. If its initial velocity at t =0 is v vector equals to 6 uh 6 icap minus 12 jcap the time at which it will just have a velocity along x-axis. So velocity along x-axis means the y component of velocity is zero. So that is the problem. Nothing to do with x component at all in this problem. Only thing that you need to be careful about what the question is saying that only along x-axis means y component is zero. So we only need to calculate everything related to y component. Okay. X component has nothing to do with the sol solution of this question. Okay. So what is the y component of force? y component of force is four. So what is the acceleration of y component? Acceleration of y component is force y by mass which is 4 by 5. This is the acceleration uh along y component.
Right? What is the initial velocity of y component? Initial velocity of y component is -12. Okay. And what is the final velocity along y component? Final velocity along y component is zero. We have to find the time. So this is the problem basically. Okay. So uh we know v equals to uh u minus 80.
Okay. Obviously uh um okay u + 80. Okay not minus do not consider it to be deceleration plus 80. So let's put so v y is zero uy is -12.
Okay. A y is 4x 5 and into t. So 4x 5 t = 12 or t = 12 into 5 by 4. So 3 uh 12 are 3 4 are 12. So it will be 15. Okay.
So 15 seconds. The time after which the uh x component will only x component will remain is uh 15 seconds. Now, now the same question can be asked if after how many time second only y component will left then you have to solve everything related to x component. Okay, that is a homework for you. Solve that and we can discuss if you are not able to solve it. So everything you need to consider for x component x acceleration x force first x force then x acceleration mass is same for mass does not depend on uh x-axis or y axis. Okay.
So x uh component of acceleration then x component of velocity and then you calculate the time required for which after which the uh x component veloc y component will remain that is x component will be zero. So here v component will be zero y component will be zero in the system shown in the figure the acceleration of 1 kg block is now here the system is not uh coming I don't know why I added the image so this the image is very simple okay uh we have actually solved this question so the image is very uh this uh um let me change the color first.
So this is a this is a pulley. Okay. Uh there is one more pulley here, right? So you remember this diagram. So this is basically a 4 kg mass. This is a 1 kg mass. Okay. And they are saying that what is the uh what is the uh I think this is uh this is fixed. This side is fixed also.
This side is fixed also.
Okay. So this is the system through which a two masses are hanging and they're saying that in the system shown in the figure the acceleration of 1 kg block is. So here if you see this diagram this 4 kg mass will definitely accelerate downward because of its heavier mass and this 1 kg block will accelerate upward because of its smaller mass. Now the accelerations are not same. Okay, these two accelerations are not same. Now if you uh uh uh uh consider this mass first. So when this mass is going down suppose the both uh uh so if this mass is going down what will happen? It require a length from this side it will also require a length from this side. Okay. Now length of the wire is fixed right. So who will provide? So how much length is required?
So suppose it goes down by a distance of delta x. So the the the the system goes down by a distance of delta x. So this system going down by a distance delta x refers to we have to add a length of wire on this side as delta x as well as on this side delta x. So entire system has to go down by delta x. So delta x on this side delta x on that side as well.
So total displacement of the wire or this string which is connecting this system. The total displacement of the string should be 2x. Okay. Now there is only one source one single string is there. So if 2x is required on the left hand side that means that 2x will be coming from the uh right hand side. So del 2 delta x should go up right. So this 2 delta x will go up and that 2 delta x will be redistributed among the uh right and left side of this pulley. So this is very very critical thinking uh level question. Okay. So when this pulley decreased by a distance of delta x on right and left hand side this object lifts by 2x. Okay. So because uh this pulley has two arms.
This pulley has only one arm. Okay.
Actually effective one arm. When this mass going down both arms are getting affected. But here if this mass is going up up only one arm is directly getting affected. So this arm is actually supplying arm. These two arm are actually consumer arm. So they are actually consuming the length provided by this. So that means if delta x is changing here two delta x is changing on the uh right hand side. So if you divide that by time square you actually get the acceleration. So if the acceleration of this is a then the acceleration of this will be 2 a okay because acceleration is d² x by dt² okay so here basically even though it is a double derivative but you should remember x is actually 2x here so that two will come out here so acceleration of the uh if the acceleration of this object is a the acceleration of this object will be 2a because the rate with which it is falling it is moving up with a double rate. Okay. So that is why the acceleration of mass 1 kg is 2a and mass 4 kg is. Now we will use our normal concept. So this is tension.
This is also tension. Okay. So the total tension of this block here is 2t.
Okay. And for this is the tension is t up upward. And obviously mg downward and this one is also having an mg downward total tension here is 2t. Right? So now we have found the free body diagrams and respective accelerations. Now we have to use the free body diagram equation. Now I told you that first when you are solving the question you have to remember what is the direction of acceleration. Direction of acceleration is very very important. So if you see that direction of acceleration of this object here is a downward. Direction of acceleration for this object is a downward. So which force is richer? The rich force is this one. Remember when you see a free body diagram? Okay. Some forces are acting say f_sub_1 f_sub_2.
Okay. First thing you need to look for what is the direction of acceleration.
Whoever has the direction any force which is acting in the direction of the acceleration that force is called rich force right and the force which is acting in the opposite direction they are called poor force okay now rich minus poor rich minus poor equals to ma okay rich minus poor equals to ma so here again the same concept so tension is upward mg is downward for this object. A is downward. So mg must be bigger. So mg must be bigger. mg minus 2t should be equals to m a.
Simple, right? Mg - 2t should be equals to m a. So from here we can calculate 2t = m g minus a or t = to m by 2 g minus a. Okay. Now if you put the value of m g and a keep it as it is for now. So it will we'll get t = to 2 g minus a. So this is the first object. Okay. So this is the equilibrium for first object. Now go to the right hand side. the uh uh uh the smaller mass. Now the acceleration is upward. Acceleration is upward. The weight is acting downward. The tension is acting downward. So who is rich in this case? Obviously T is rich and mg is poor for this object. Right? So rich minus poor. So T minus MG should be equals to ma here A is 2 A. So it will be M into 2 A. Okay. So uh now put the value of tension here. Tension is I think there uh tension is 2 into g minus a minus what is m is 1 kg. So it will be simply g and that should be equals to 2 a. M is equal to small m= to 1. Right? Now simplify that.
Simplify that uh 2 g - 2 a - g = 2 a. Okay. Or uh 2 g - g = 2 a + 2 a. So g = 4 a or a = g by 4. So the acceleration of uh this 1 kg block is g by4 and it is accelerating upward direction. So gx4 in the upward direction is the answer. Now again analysis of this question if you can analyze the free body diagram the question will not take long time to solve but analysis itself is tricky. So that that you should be little bit careful about.
Next question. A ball of mass 1 kg collides with the wall with speed 8 m/s and rebounds on the same line with the same speed. If mass of the wall is taken as infinite, the work done by the ball uh ball on the wall is so here we have a we have a wall. Okay, the ball is actually hitting the wall and coming back back. Okay, so it is applying some force. No doubt about it. Okay, you can calculate the force. How to calculate the force? So, initially the uh momentum is MV. Final momentum is minus MV. So, change in momentum is uh MV - MV. So, it will be MV 2 MV. So this is the change in momentum and force is delta P by delta T. So this kind of question also comes in some cases here it is not required though. Now once you realize that but what is the displacement of this wall? If you throw a ball on the wall what is the displacement of the ball? Displacement of the ball is there but the wall is not getting displaced.
That means work done by the ball on the wall is zero. Okay. a big big big zero.
Okay. So, uh because the wall is not moving, the work done by the ball on the wall is zero. So, don't get confused uh with the language. Uh the part I have shown here is basically for from the chapter Newton's laws of motion.
Sometimes based on that you might get a question. A spring of spring constant 5 into 10 ^ of 3 newton per meter is stretched initially by 5 cm from unstretched position. Then the work required to stretch it further by another 5 cm. So initially the string is stretched from its equilibrium position by 5 cm.
Now you stretched it further even further by another 5 cm and they said that what is the work required. So what is the energy stored inside a spring? So energy potential energy stored inside the spring is half k x² right and initially it was 5 cm finally it is uh what is the this is initial potential energy final potential energy is half k x2² so this is x1 this is x2 squared okay so what is the change in potential energy change in potential energy is obviously u2 minus u1 that is k x2² - x1² very simple everything is provided and that change in potential energy is actually work done that is the definition of work done so it will be half k is 5 into 10 ^ 3 into x2 is 5 cm now 5 cm cm 5 cm means 5 into 10 ^ of -2 m okay so and second one this is x1 what is x2 x2 should be 5 + 5 that is 10 right and both of them will be added with a square that means 100 - 25 into 10 ^ of -4. Okay. So if you square this you will get 25 10 ^ of -4. If you square this you will get 100 into 10 ^ of -4. So that 10 ^ of -4 is common and we are just finding the difference. So we get half into 5 into 75 into 10 ^ of -1 10 ^ 3 10 ^ -4 10 the^ -1. So it will be 5 into 75 is 25 125 125 into 3 375. So 375 or 37.5 by 2. Okay. If you consider the 10 ^ of minus 1, it will be 37.5 by 2. It will be uh 16 point some value.
Okay. So it will be uh 16 no um 18 some value. Okay. 18 to 36. So it will be 18 some value which will be 18.75. You can check that out. It will be 18.75 Newton.
meter or jewles.
Okay, 18.75 Jew. So, make sure the problem is basically simple. The main idea is obviously what is the formula that you are using and the unit conversion. Other than that, the problem is very very simple. Now, next question.
The relationship between the force and position X of a body is as shown in the figure. The work done in displacing the body from x= to 1 m to x= to 5 m is. So here you have to find the work done to displace the body from x= to 1 m x = to 1 m to x= to 5 m. So remember what you are doing here.
You are trying to find the work done from here to here. This triangle is not part of the calculation. This we are not trying to find x equals to0 okay so we are only finding from here to here very very important okay to understand this kind of questions sometimes they will give you some extra on this side sometime they might give you some extra on that side so as soon as you see the question don't start calculating the area and adding it no just take your little bit sweet time and then solve it now if you see that here the work done is positive and the this is one this is Five. Here the work done is negative.
This is also one and this is also five.
So that means this work done and this work done they will cancel each other.
They will cancel each other. One is positive, one is negative. So they will not have any uh effect. So only you need to calculate this area and this area and add them. So that problem even though it looks like a lengthy question actually it is a very easy question if you have the basic understanding of work done. So this is positive work done and the area of this 1 into 5 the area of this is 1 into minus 5. Okay. So both of them have same areas and then that means they will cancel each other. So we will not go for these two rectangles at all. We will only calculate the area of this rectangle and this rectangle. Work done by a variable force is area under the graph. Right? So that everyone I guess everyone knows. So we and the problem alsoh tells you begs you calculate the area but which are all the areas that you need to calculate is you have to be careful about. So from 1 to two the work done from 1 to 2.
So work done from 1 to 2 is given by this uh time distance that is 2 - 1 that is 1 and the force is 10 newton. So 1 into 10 is 10. Okay. And the work done from 4 to 5 is half into base into height again five. So 15. So total work done total work done is uh 10 + 5 that is 15 that is 15.
Okay. So question is easy but there are traps. So avoid those traps. You should be fine. Now next three point particles of masses 1 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4 cm, 3 cm and 5 cm as shown in the figure. The center of the mass of the system is at point. So here we have three points. No coordinate system is given and they are asking you where is the center of mass. Masses are all different. So in that case what you have to do? You have to make your own coordinate system. Okay. Uh so and they're asking you center of mass of the system is at a point. Uh so and if you see that all the options are giving 1 kg 1 kg 1 kg 1 kg. So this 1 kg mass is important. Okay. So we will take this as our origin. We will take this as our origin.
And that means this will be our x-axis.
This will be our y-axis.
Okay. So we have to find xcm and ycm.
Very simple.
So and because of this structure xm and ycm are very easy as well. So, XCM is mass mass into position plus mass into position plus mass into position.
Y-axis means x coordinate is zero, right? Y ais means x coordinate is zero.
So, and divided by total mass. Total mass there is nothing uh no shortcut.
you have to calculate the total mass as usual. So you get 4.5 divided by uh 345.
So it will be approximately 0.9.
So 0.9 towards right. Uh okay. So 0.9 cm right. So and if you see that 0.9 cm right is only one option is there. 0.9 cm right only one option is there. So option C should be the correct answer.
If there are multiple correct option and you may need to calculate YCM as well.
In case of YCM obviously one will still be zero. 1.5 will have zero ycoordinate and y-coordinate for 2.5 is 4 and divided by mass of all that is five. Okay. It will be 10 x 5 that is 2 cm. So this is 2 cm. So 9 cm right 2 cm up. 9.9 cm right 2 cm up somewhere here is the center of mass.
Okay. So uh don't uh be afraid to solve such questions. Okay. It's a easy question.
Time taken by the projectile to reach from A to B is T. Then the distance A equals to. So this is a projectile motion. The angles are given this angle is 30°. This angle is 60°. And it is mentioned that time taken by the object to reach from here to here is small T.
So this might look confusing sir. I don't know how to calculate this distance this distance square root. No.
See the problem has enough data to solve for you to solve the question without much difficulty. Okay, first thing that you need to consider is uh you have to calculate this height. You don't need to calculate this. Okay, you don't need to calculate this.
This height will you can adjust it using Pythagoras theorem or triangle theorem.
Okay. So if you see that this has a 30° angle. So if you can find this distance, if you can find this distance, basically it's a uh cos theta cos theta angle. Okay, cos is base by hypotenuse. So base if you can calculate base you can find the hypotenuse which is this. Okay, so now remember when a projectile is moving the why we are talking about this because when the projectile is moving the uh horizontal component the horizontal component remain constant. So horizontal component of the projectile remain constant. So we only need to calculate the horizontal component of the projectile motion. So what is the horizontal component of projectile motion? It is basically velocity into cos angle of projection. So velocity is uh u okay velocity is u and angle of projection is 60°. So u cos 60° which is u uh into half that is u by2. So this is the horizontal speed. Now when the particle is moving by a time t second. So what is the distance traveled? So the distance traveled here, the distance traveled along the horizon along the horizontal direction is given by the speed into time which is UT by2.
So this distance we have calculated. So this distance we have calculated. We want to calculate the hypotenuse. So what is the formula? um base by hypotenuse equals to cos theta. Here angle is also provided to you 30°. So from here hypotenuse equals to SH by cos 30. So it will be UT by 2 into <unk>3 by 2. So it will be UT by<unk> 3. So the answer is UT by <unk>3 option D is correct answer.
Okay.
The pulley and strings shown in the figure are smooth and of negligible mass. For the system to remain in equilibrium the angle theta should be.
So here this system is in equilibrium.
So equilibrium problems are easy. Okay.
Equilibrium problems are easy. So if you see that obviously this object is having an weight downward this object is also having a same weight downward. So the tension here should be same as tension here. Okay. So this is a symmetrical part. Okay. So if you consider the symmetry this mg should be equals to this mg right. So tension should also be same. So the tension actually translates linearly. So tension on this wire will be T. Tension on this wire will also be T.
Right. Now how can this roo<unk>2m will be balanced? So the roo<unk>2 m also having a <unk>2 mg acting downward.
Right? Now how can this be balanced?
Only if the total tension uh total component of tension along this direction is actually same. So what is this component? So this is t this is t.
The angles both angles are theta. So this should be t cos theta plus t cos theta. that is t 2 t cos theta. So you have to take both components t cos theta from this side t cos theta from this side. So t cos theta plus t cos theta.
So total upward force is 2 t cos theta.
So this 2 t cos theta should balance this weight. Now we will obviously apply the formula for one of this object and this central object. That's that's it. Okay.
So for this object obviously system is in balance that means t must be equals to mg no acceleration. So t must be equals to mg. Now for this object 2t cos theta must be equals to <unk>2 mg right 2t cos theta should be equals to <unk>2 mg. So now put the value of t which is mg cos theta equals to <unk>2 mg mg mg cancel. So cos cos theta = to <unk>2x2 that is 1 by <unk>2. So theta must be equals to 45° or<unk> by 4. Okay. So 45° or<unk> by4 should be the angle of uh angle.
Next question.
Block B lying on a table weighs W. The coefficient of static friction between the block and the table is mu. Assume that the chord between B and the knot is horizontal. The maximum weight of the block A for which the system will be stationary. So again this problem also you are not supposed to apply lamis theorem because lamis theorem talks about the direct angle okay here direct angle is not there you are finding a part of the angle so if you apply the lamis theorem again the problem will be lengthier okay so those who don't know lamis theorem I might explain it at the end of the lecture please or you can ask me during the doubt session if you want to know about it okay so this mass a is basically accelerating downward trying to accelerate downward. So this will be mg. So there this object has a weight here mg or w.
So this here in the question it is given that both blocks have weights w. So w's are given mg is not m is not given. So if this block is having a weight w the normal reaction force should also be W.
That means the force of friction should be mu W. Okay. So force of friction the maximum force of friction should be mu into W. Right? Now there will be a tension on this knot. There will be a tension on this knot. This tension will give you two components. One is along this component. uh one is along this axis. Okay. One is along this axis. One is along this axis.
Okay. So this is the angle theta. So that means this is t cos theta and this will be t sin theta. Adjacent components are always cos components. Perpendicular components are always sine component.
Right? Now this t cos theta should balance this force of friction and this t sin theta should balance the weight.
Okay. So t sin theta should be equals to weight and t cos theta should be equals to uh f is mu into uh w. Now in the question they have said that maximum weight of the block A for which the system will be stationary. So basically what we are trying we are trying to find the uh weight of the um okay so block A's weight is not given. So just uh one important thing the weight of the block A is not given.
So let me write it as W dash. Okay. So the block of the weight A is not given.
B has a weight w. Okay. So T sin theta is actually W dash. T sin theta is actually W dash. Now you what do we have to do? You have to just divide them.
Obviously to eliminate T the easiest way to divide them. So sin theta by cos theta right tt cancels you get w ddash by mu w or w ddash equals to mu w tan theta okay mu w tan theta easy question now why we have not applied lis theorem because this angle is given okay and we are not trying to make the problem complicated Next.
Oh, I think this problem we have already solved. Next. The work done by a force acting on a body is as shown in the graph. The total work done in covering the initial distance of 20 m. Again, you have to be careful. Uh the problem is talking about only 20 m. That means you uh need to be calculating the problem only up to 20 m.
Okay, right hand side is excluded. Okay, so work done again work done is area under the graph. That is not an issue.
Okay, you need to calculate the area of this triangle. You need to calculate the area of this rectangle and you need to calculate You need to calculate the area of this rectangle uh this triangle and this trapezium. Okay. So this is how you need to solve it. So what is the area of this rectangle?
Area of a rectangle uh sorry area of a triangle is half base into height. Area of a rectangle is width into base. And for um trapezium we will see that formula for trapezium as well. Formula is there you don't need to consider triangle rectangle direct formula you should use.
So work total work done is first the triangle half into base is 0 - 5 5 - 0 that is five right I guess you are able to read the graph. Uh so this is zero this is five. So it will be five only.
height is 10. Height is 10 plus the area of the rectangle area of the rectangle is 15 - 5 that is 10 into 10. So it is actually a square plus area of the trapezium. Area of the trapezium is half half. Sum of the parallel sides. Sum of the parallel sides that is 10 + 20. 10 + 20 into the distance between the parallel sides. So distance between the parallel side is five. Okay, this is the formula for area of a trapezium. So five 10 are five. So it will be 25 + 100 + half into 30 into 5. So it will be 15 that is 75. So 25 + 100 + 75 which is 200 Jew.
So the work done in this problem is 200 Jew.
Okay. Now, next.
A body initially at rest and sliding along a frictionless track from a height h as shown in the figure just completes a vertical circle of diameter ab equals to d. The height h is equals to. So here object is falling and it is making a vertical circle. So we know that to make a vertical circle at this point at the minimum point the minimum velocity required is given by square root of 5 GR.
Okay square root of 5 G r R is the radius of the vertical circle. Now here instead of radius diameter is given. So we will write this as 5g D by 2. So this is the minimum velocity required by the uh object to have a to make it a complete vertical circle. So if the vertical circle is completing the minimum velocity at the bottom required is uh 5g r. Now now we will go back and we will apply the conservation of energy principle at the beginning of this his journey. What is its potential energy? Potential energy is mgh.
Okay. And when it comes down here entire energy is kinetic energy. So kinetic energy is/ mv².
Okay. So mm cancels. So v = to square root of 2 gh. Now if the object completes the vertical motion. So this v should be equals to v m. So minimum value of v should be equals to v mm. So square roo<unk> of 2 g h is equals to square roo<unk> of 5 g d by 2. So if you take square on both side 2 gh equals to 5 g dx2. So gg cancel. So h becomes 5 dx4.
Okay. So it will be 5 dx4. So option d is correct answer.
Next question.
Two masses m1 equals to 1 kg and m2 is equal to 2 kg are connected by a light inextensible string and suspended by means of a weightless pulley as shown in the figure. Assuming that both masses start from the rest, the distance traveled by the center of mass into second is. So first of all, so first step is to find the acceleration of this system and acceleration of the center of mass and then we will be able to solve the displacement by doing the displacement is equal to/ 80². So these are the steps. So first step is to understand what is happening here. Only one single pulley is there. So no uh need to find think any anything out of the box. So this is a heavier box. So this will go down. This is a lighter box. This should go up. So the acceleration here say a.
So here we have mg here downward and mg here downward as well. Tension is same.
Only thing that is same here is tension.
Okay. And the acceleration can in fact directly be calculated using a shortcut formula. So only double pulley system when only one single pulley double mass is there. Single pulley double mass the acceleration is given by so both objects will accelerate with the same value. So this one will go up this one will go down. The acceleration is given by difference in masses.
Okay. Difference in masses divided by sum of the masses.
Okay. So difference is 2 - 1 sum is 2 + 1 into gravitational acceleration.
Always remember this. Okay. So this will be 1/3 into uh gte take 10 it mentioned. So uh so acceleration of the uh system is 10 by3 leave it do not try to simplify do not make it 3.33 like that keep it like this. Now next we have to find the center of mass acceleration of the center of mass. So acceleration of the center of mass is what?
So anything which is moving up we let let us consider that positive. Anything going down uh so no sorry. So this object is heavier. So anything going down let us say positive. Anything going up let us say negative. It will depend.
It will it depends from system to system because this is heavier mass. It is expected that center of mass will stay closer to the heavier mass. So that's why we are doing it. Now how to do it?
So very simple a1 m1 + a2 m_sub_2 by m1 + m2. Now here a being an acceleration it can be positive or negative based on the direction. So here as I mentioned uh downward we are taking positive. So this one is 2 kg acceleration is 10x3 plus this one is 1 kg acceleration is minus 10x3 okay so if this is going down it is positive going up it is negative now that positive negative again I'm saying that there is no fixed rule of this positive negative depends on if you are seeing the heavier object object always take the acceleration of that heavier object to be uh positive. Generally heavy object goes down. So generally we take downward as positive generally divided by mass sum of mass 2 + 1. So it will be uh 2 - 1 that is 1. So it will be 10 by 3 by uh 3. So it will be 10 by 9.
So 10 by 9 is the acceleration of the center of mass which is downward. Now we have to find the uh distance traveled by the center of mass in 2 seconds. So distance traveled by the center of mass in 2C is given by um half a t².
So that is half into 10 by 9 into t² t² is 2 seconds. So 2² so it will be uh 4 by uh by 2. So that is 20 by 9. So 20 by 9 m. So 20 by 9 m is the displacement of the center of mass in 2 seconds. Okay. And in fact you can use the same shortcut that we have used for this side as well. So ACM you can write ACM is same difference in mass plus addition of mass into A. This is also okay. Okay, this is also okay. This way also you can calculate center of mass.
Now next question.
A disc is rotating with angular velocity omega about its axis. A force F acts along a point whose position vector with respect to the axis of rotation is RCAP.
R vector the power associated with the torque due to the force is given by so remember that torque torque is given by R cross F normally this is the torque vector. Now if you go to power, power normally is given by force dot velocity.
Power is given by force into dotproduct of velocity. Now here in the rotational axis power can be written as the force will be replaced by torque and velocity will be replaced by angular velocity.
Okay. force will be replaced by torque and uh velocity will be replaced by angular velocity. Okay. So that means it will be R cross F dot omega R cross F dot omega. So correct answer is option A.
Okay. Now next a body weighs 45 Newton on the surface of Earth. what is the gravitational force on it due to the earth at a height equal to half the radius of the earth. So we know that weight is given directly so we know that gravitational acceleration changes.
So if the weight of the object on the uh earth is 45 then weight of the object at a height uh of some distance is given by uh w divided by 1 + h by r² right this is the formula same formula applies for gravitational acceleration as well if you divide both side with mass you get the acceleration but here because the Question is asking about weight. So direct weight is given. So we will use the direct weight formula. No need to worry. Same formula you can calculate. So 45 instead of g is actually not involved in this question.
That's it. So 45 / 1 + rx2 by r². So if you see that here it is mentioned height equal to half of the radius. Okay. So rx2. So it will be 45 by 1 + uh half square that is 45 by uh 9 by 4. So 45 into 4 by 9 5 4 are 45 that will be 20. So 20 Newton is the 20 Newton is the uh weight of the object. Next question. GE and GP denote the acceleration due to gravity on the surface of the earth and on another planet whose mass and radius are twice to that of the earth. Then what is the relation between G and GP? So G on the earth is given by GM by R².
Right? Now what will be GP?
It is said that the mass of the planet is twice. Okay, mass of the planet is uh twice. Okay, and radius is also twice.
So, it will be uh gm by 2 r². Obviously, one of the two will be cancelled. Just take a little time if you uh don't you are not able to do it at one go. So, it will be half G. So GP is actually half G. Okay, just take your time. If you are not able to do it one time, you might see sir how it is like that how it is done. So you need to do it faster. Okay, because you are not having enough time in the exam. Okay, so that one step you have to be little careful about. Okay, so that step if you do it little faster you can save lot of time. Next figure shows stress and strain curves for three different materials taken to fracture.
Which curve best shows the behavior of a copper wire and which one best shows the behavior of a glass fiber? So if you have seen a glass fiber, glass fibers are um stiff. Stiff means they usually do not uh bend but they suddenly breaks right? If you have a glass object okay they generally do not bend like metals.
If you hit a metal before breaking it will first bend okay if you continuously beat it then it will break.
For a glass it will not bend. Bending is not happening in glass. It suddenly breaks. So that means it it can withstand a high amount of stress but it suddenly breaks. Okay. So stress is very high but it breaks suddenly.
Strain is not varying here. So that means this object represents glass.
Okay. Which can withstand a high amount of stress. Okay. But it breaks suddenly without any curve or without any uh bending. So whereas if you see this graph here it is not only having high amount of stress it is also bending this c this b this bending in the stress train curve actually shows the material is before it it breaks it bends okay so it bends it becomes little uh thinner then it breaks. So this happens in case of metals only. This happens in case of metals only. This one on the other hand is like a very low stress tolerant.
Okay. But it can have a very high strain for low stress. This is like likely to be a rubber band or uh uh rubber uh rubber type materials. Okay. Elastic materials. Okay. So that means we can say that number one is glass and number two is copper. Okay. First one is copper, second one is glass. So 2 and one is our answer. 2 and one is our answer. So option D. So whenever order based ratio based question comes always make sure read your question twice. So first one is copper, second one is glass. But in the diagram glass is given first, second one is given as copper.
And the option first option is also given one two. So you might be carried away and answer it. Okay. to make sure you read the question and answer. A thin liquid film formed between a U-shaped wire and light slider, okay, supports a weight of 1.5 into 10 ^ of -2 Newton as shown in the figure. The length of the slider is 30 cm and its weight is negligible. the surface tension of the liquid film is. So here remember uh as I told you if you are considering say a surface it has uh when you are considering a film film surface film means uh so if you I gave you the example of bubbles right so when you have a soap bubble confined in a wire okay that is called a film soap bubble confined in a wire you have not blown it yet okay so that is called a film so this is like this. So there is a liquid film entrapped in this wire frame. Okay.
So how many surfaces are there? So it's one surface you can see from this side.
There is one surface from behind. So two surfaces are there. It is not one single surface. One surface you can see one surface is behind it. So that surface you cannot see. Basically like your hand. So one if you show your hand basically this is one surface but there is one more surface behind as well. So two surfaces are there. Okay. So if the surface tension which is acting on this um wire is T then T into 2 L 2L comes from the fact that surface is actually two. One is this surface one is surface which is behind. Both surface will have their individual surface tensions. Both surface will have their individual surface tension. So total force is actually 2 into TL. Total force is 2 into TL and that should be equals to W.
That should be equals to W. So surface tension is actually equals to W by 2 L.
So w is given here 1.5 1.5 into 10 ^ of -2 divided by 2 into the length is given by 30 cm 30 cm means 30 into 10 ^ of -2 m 10 ^ of -2 10 ^ of -2 cancels we get 1.5 by 2 into 30 okay so just we have to simplify it little bit more. So this will be uh 15 uh 300 so 20. So it will be 1 by 40. 1 by 40 is the answer. So if you multiply five on top and bottom you'll get 5 by 100 uh 5 by uh 200 sorry. So that is 0.025 0.025 025 Newton meter. Right? So 0.025 Newton meter is the correct answer.
Next question.
Water rises in the capillary tube to a height of 2.0 cm. In another capillary tube whose radius is one/ird of it, how much the water will rise. So capillary rise is actually proportional to one by radius not square not cube only simply 1 by r okay so smaller the tube diameter or smaller the tube radius larger it will rise okay so it is rising 2 cm for a say for r for a capillary tube having radius one/ird of it the height will be 3 * the original uh so this is h1 1. Okay, it will be 3 * h1. Okay, inversely proportional. So if radius decreases, the height will increase. So that will be 6 cm. So 6 cm is the correct answer.
Next, a pan filled with hot food cools from 94° C to 86°C in 2 minutes when the room temperature is 20°.
The time taken for the food to cool from 86°C to 74°C is. So first case is given always this is a Newton's law of cooling based problem. So first steps will be given we will find the constant and we will use the constant in the second problem. So 94 to 86 in 2 minutes. So 94 to 86 in 2 minutes. Okay, this is the so first step is d theta* dt which is 94 by 86 by 2 which is 8 by2 that is four 4° C per minute okay so that is how it is losing its heat now what is this Newton's law of cooling says that d theta* dt equals to k someone told me some of you only told me the formula you sir use this formula it should be uh easier okay so let me use this formula I was using uh uh uh individually calculating this and this and then calculating it you said no sir directly use the formula so okay let me use the formula directly so k value we don't know t1 is 94 + 86 by 2 - 20.
Okay. So it will be K into 94 + 86 is 100 180 by 180 by 2 that is 90 - 20. So it will be 70K.
So K value is 4 by 70. So K value is now calculated. Now we will use the same method for the second part as well. Only thing that in the second part DT is not given. We are trying to find dt. D theta. If you see actually d theta is same 86 to 74 again another 8 uh 8° right? So d theta is uh 86 minus 74. Oh okay. Okay. Uh have I made a mistake?
Uh no.
Uh 80 94.
Yes. This no. So here this theta is not same. Sorry. d theta is 12° here d theta is 12 not for generally they give the same range here the d theta range is different it will be 12° okay and we are trying to find dt so 12 by dt equals to so same formula again okay 12 by dt equals to k into here the temperature values are different temperature values are 86 and 74 divided by 2 - 20 room temperature is same. So 12 by dt equals to k is uh 4 by 70 into so 86 + 74 is 90 and 160 by 2 that is 80 - 20. So it will be 12 by dt = to 4 by 70 into 60. Right? Uh it will be uh 60. So 0 0 cancels. Now you have to just simplify this dt equals to dt equals to uh so the cross multiply this will go up uh this will go up 12 into 7 divided by 4 into 6 will come in division 4 into 6 will come into division. Okay. Uh so it will be 6 2 are 12 uh 2 2's are 4. So it will be 7 by2 minutes 7 by2 minutes. But in the question everything is given in terms of seconds. So you have to convert this into seconds. So 7 by 2 into 60 seconds.
So it will be 30. So 210 seconds 210 seconds. So dt is 210 seconds. Okay. So this is a very interesting problem.
But but the methodical problem if you approach Newton's law Newton's law of cooling based problems methodically you should be able to solve them no problem.
A compound slab is made of two parallel plates of copper and brass of the same thickness and having thermal conductivities in the ratio 4 is to1.
Okay. The free phase of the copper is at 0°C.
The temperature of the interface is at 20°C.
What is the temperature of the free face of the brass? So here we have a copper wire, a copper plate and a brass plate.
Brass is an alloy. So no shortcut there.
Okay. And um same thickness.
The thickness is same. So the conductivity of copper is four.
Conductivity of brass is one.
Okay. Now we know that heat that is transferred. So this is said 0°. This is said 20° and this is say t°. We do not know that temperature. Now heat which is being transferred is proportional to conductivity into change in temperature.
Okay. So that is K into DT if everything else remains constant. The heat that is being transferred is proportional to conductivity in into DT. Now obviously heat is an energy. So if Q heat is flowing through the copper and it is going towards the brass the same heat has to conduct through the brass as well. So we can say that the conductivity of copper into the temperature difference of copper okay should be equals to should be equals to conductivity of brass into temperature difference of brass.
Okay. Conductivity of copper is given four. 4 is to1 ratio is given obviously because it is an equation we can uh we don't need to consider the uh constant.
So delta t is 20 - 0 20 - 0 and brass is 1 and suppose it is t so it will be t minus 20. Now from this problem it is clear the heat is traveling uh uh sorry uh just a second I made a mistake. Heat is traveling from uh this side to this side. Okay heat is traveling from uh this side to this side.
Obviously heat will travel from higher temperature to lower temperature. So heat is traveling from higher temperature to lower temperature. So T minus 20. Okay.
So this is heat being transferred.
Okay. So edge of the copper is zero and interface is 20°. So heat is transferring from brass to copper. Okay.
Now we have to simplify 4 into 20 = to uh t minus 20. So it will be t = to 80 + 20 that is 100°.
So the uh interface it has a temperature of 100 and the other end of the blast is at a temperature of 100°C.
Okay. Next question. An ideal gas is taken through a cycle A to B, B to C, C to A as shown in the figure. The net the net heat supplied to the gas in the cycle is 5 ghou. The work done by the gas in the process C2A is. So here it's a cyclic process. Cyclic process means very important thing. As soon as a cyclic process is given to you, you should remember that in cyclic process change in internal energy is zero. As soon as a cyclic process is given to you, change in internal energy is zero.
That means whatever heat that you supply whatever heat that you supply is actually sum of the entire work done in the process. So if you find the work done for individual processes A B B B B B B B B B B B B B B B B B B B B B C C and CA that should give you the heat that is supplied. So in this in this problem the heat supplied is already given five jewles. Okay. So if you can if we can calculate the work done in all the processes and add them that is total work done that should be equals to 5 G right. So this is the basic problem. It is not about the work done by the system by the way work done by the system uh is area under the graph etc etc but that is for if the graph is proper being asked to you here you have the values are obviously not properly given what is this value what is this? So you cannot actually calculate the area under the graph. Okay. Instead the work done itself is given 5 Jew. Total work done is 5 JW is given. Okay. Now uh we will be using delta Q is equal to sum of all the work done. So here if you see this process first process is AB. So, wab if you see wab the pressure is constant at 10 pascals. Okay. Uh the pressure is constant at this 10 pascals or newton per meter squared and the volume is changing from 1 to 2. So volume is changing from 1 to 2. So work done is v_sub_2 minus v_sub_1. P delta V work done in ideal gas is P delta V.
So P is constant at 10 obviously and uh volume is changing from 1 to 2.
So final is two initial is 1. So work done is 10 jew.
Now what about WBC?
Work done from B to C. If you see here the uh pressure is changing even though that is not given but the volume remains same. The volume remains. So delta v is zero. So delta v is 0 means work done is also zero. So now we will come back here and we will say that the total heat that we supply delta Q that is 5 ghou that should be equals to W AB plus W BC plus WCA okay W AB is 10 JH WBC is zero WCA is what we are trying to calculate so WCA is actually 5 - 10 that is - 5 Jew. So work done from the by the gas to reach from uh C to A is -5 Jew. So option A is correct.
Next question.
1 mm cube of a gas is compressed at one atmospheric pressure and temperature 27° to to 3 6 27°C.
What is the final pressure under adiabetic condition? Okay, we know that P V to the power gamma equals to constant in adabetic condition.
But here temperature is given. So you have to use the formula P 1 minus gamma T to the power of gamma equals to constant. Okay. So this is the another formula that you need to uh know. Okay.
Okay. Now it is mentioned that uh gamma is given 1.5 uh initial pressure is one atmospheric pressure. Final pressure we try to find we will find the pressure in atmosphere atmospheric pressure first then we will go for the second part. Okay. So now we can uh write this as p t to the power of gamma divided by 1 minus gamma equals to constant.
Okay. Uh so gamma is 1.5 that is 3x2 and 1 - gamma equals to uh -/ 1 - gamma is 3x2 -/ 1 - 3x2. So that means it is minus/. So gamma by 1 - gamma is actually uh 3x 2 by -/ -3 to the power of -3. Okay. So in this case pressure t the power of -3 is constant. So p1 t to the power t1 t the^ of minus 3 is equal to p2 t2 to the^ of minus3. So P2 is actually equals to P1 T1 by TS2 to the power of minus3 or P1 T2 by T1 to the power of 3. This is for this particular problem. This is not a general solution. We just simplified it so that it have some easiness to solve.
Now P1 is 1 atmospheric pressure. Okay.
T2 is 627°C.
627°C means 900 uh kelvin.
Okay. Adds 273. Now what about uh uh T1?
T1 is 300 kel.
Temperatures has to be converted in Kelvin. If you just add the normal degrees CC temperature, it will be wrong. So it will be 3 whole cube. 3 whole cube is 27. Now this is in atmospheric pressure. Okay. And one atmospheric pressure is approximately 10 ^ of 5 Newton per meter squared or newton or pascals. Okay. So the pressure is 27 into 10 ^ of 5 Newton per meter squared.
Next question.
Pressure versus temperature graph for an ideal gas is as shown in the figure.
Density of the gas at point A is row not. Density at a point B will be. So density at point A is row not. And density at point B we are trying to find. Okay. Now here pressure versus temperature graph is given. Okay.
Pressure versus temperature graph is given. Now what is the relationship with pressure temperature and density? So density is actually mass by volume.
Density is actually mass by volume. Now mass obviously is remain constant. So if you can find the volumes you can actually find the densities. Okay. Uh so we know that P1 V_sub_1 by T1 equals to P2 V_sub_2 by TS2.
Okay. So if you see that so basically volume is actually M by row. Volume is M by row. So substitute that here you get P1 M by row 1 T1 equals to P2 M by row 2T2. Okay. So mm cancels. So we are trying to find row 2. So, row 2 equals to p2 by ts2 into uh p2 by t2 into t_1 by p1 into row 1, right? Just cross multiply. So, this row two will come up.
Everything here will go up. Okay, that's it. So, p2 by t2. What what is p2? P2 is 3 p. TS2 is 2 T. So it will be 3 P by 2T KN into T1. T1 is T, P1 is P. So T by P and the density is row. Okay. So everything cancels you get 3x2 row KN. So it will be 3x2 row. Just apply very basic equations of uh kind of of ideal gas and have a basic idea that density is mass by volume. You should be able to solve it. Now you can find the relation between volume. You can substitute back here or directly substitute volume into this equation and then you solve it. That is much simpler and easier to solve without any complication.
Internal energy of N1 moles of hydrogen at temperature T is equal to internal energy of N_sub_2 moles of helium at temperature 2T. Then the ratio of N_sub_1 and N_sub_2 is so internal energy of a gas is given by CV DT.
Okay, internal energy of a gas is given by CV DT. Now total internal energy this is the change in internal energy. DeltaU is change in internal energy. Right? Now if you integrate both sides you get the total internal energy which is CV into T. Okay. CV what is CV? CV for a gas is given by its degrees of freedom divided by 2 into ideal gas constant. Now here first case it is internal energy of N1 moles of hydrogen. For hydrogen hydrogen is a diatomic gas. For diatomic gas uh the degrees of freedom is five. So it will be 5x2 R. And for helium so this is CV 1. For helium which is a monatomic gas the uh CV value is 3x2R because degrees of freedom for monatomic gas is 3. Degrees of freedom for diatomic gas is 5. So for hydrogen it is 5x2. For helium it is 3x2. Don't make mistakes.
Okay. So it is said that the internal energy uh of n uh and one more thing n1 n has to be included. Okay. Because for one mole gas it is cbt. So for n mole gas it will be n cbt or what you can say that okay here you keep it cvt only.
Here you add uh n okay here you add n that is also okay.
So you have to add the ends.
So this is basically molar specificate.
That's it. Molar specificate at constant volume.
So it will be 5x2 5 N1 by 2 R. It will be 3 N_sub_2 by 2 R. Okay. Um so basically the internal energies are same. Uh so we have to compare C V1T equals to C V2 into 2T. Okay. It is mentioned 2T. For helium the temperature is 2T. For hydrogen the temperature is T. Okay. So we get n uh 5 n_sub_1 r by 2 into t equals to 3 n_sub_2 into r by 2 into 2t. So obviously tt cancels rr cancels these twos are cancelled you get 5 n1 = to 6 n_sub_2 3 2's are 6 so nsub_1 by n_sub_2 = to 6 by 5 so it will be 6 by 5 okay now if you are still confused so n uh this is molar specific heat if you are using molar specific heat you can multiply n over here if you are using normal specific heat you have to multiply n over there. Okay. Basically you use this here that is much better.
Okay. Uh internal energy is CV DT always total internal energy CV into T. Now if the internal energies are same so CV1 T1 should be CV2 into TS2. Now T1 is in this question is given T2 in this question is given 2T. So that values we have put in this equation and basically simplified it. Okay. Now let's move on to the next question. 96th question. A gaseous mixture consists of 16 g of helium and 16 g of oxygen. The ratio of CP by CV of the mixture is. So there is a very uh straightforward formula that the CP by CV directly is can be written gamma equals to N_sub_1 F_sub_1 + N_sub_2 F_sub_2.
Okay. Plus 2 N_sub_1 + N_sub_2 divided by N_sub_1 F_sub_1 + N_sub_2 F_sub_2. Now what are all those?
Obviously N1 is the mole number and and F_sub_1 are the respective degrees of freedom. Okay, respective degrees of freedom. So for helium mole number is 16 by4. So the uh molar mass is 4 g for helium. So mole number is four. Okay.
Degrees of freedom of helium. It is a monatomic gas. So it must be three. For oxygen the mole number is 32. Mole number for oxygen is not 16. It is 32. Okay. Oxygen is a diatomic molecule. So 16 is for one atom. 16 into 2 32. So it will be 0.5.
And the degrees of freedom because it's a diatomic molecule. The degrees of freedom is five. Now we have to put this data into this equation and simplify.
Nothing else is there. So n1 is 4 into 3 + uh 0.5 into 5 + uh 4 + 0.5 into 2. So there was a two here. I forgot to write it in the beginning. So I have am adjusting in the last. So 3 4 are 12. Now in the division if you see that this quantity and this quantity are same. So 3 4 are 12 plus.5 into 5 is 2.5.
Okay. So that will be 14.5. So 14.5 + uh 4.5 into 2 that is 9 divided by 14.5.
Okay. So it will be uh 23.5 23.5 by 14.5.
Now for this to calculate exactly you need to use the on screen calculator and you will get the answer as 1.62 62 because do not try to calculate or guess the answer. You might get a wrong answer. All the options are very very close. So in such cases where the all the options are very very close and you cannot get it directly by calculating do not forget to use the onscreen calculator. There will be an onscreen calculator use it. Okay. Next is the x versus t graph of a particle undergoing simple harmonic motion is as shown in the figure. The acceleration of the particle at t = 4 by3 second is. So what is how to calculate acceleration?
So we know that acceleration of a particle is given by minus omega² x. Okay. Acceleration of a particle is - omega² x. So first we have to find the omega then we have to find the x and then we have to find the acceleration time tx 4 second uh 3x4 uh 4x3 second sorry 4x3 second.
So if you see that here the equation of the graph it is showing here it is a uh the time period is obviously 8 second.
If you see the time period of the uh motion is 8 second. Okay. So what is the and amplitude is one. If you see amplitude is 1. So amplitude is 1 m 1 cm. Okay. So what what is omega? Omega is equal to 2 pi by t. Omega value is 2 pi by t. So that will be 2 pi by 8 that is<unk> by 4. So omega value is<unk> by 4. Now how to calculate x? X standard equation in simple harmonic motion X is given by A sin omega T right so A value is 1 sin omega is<unk> by 4 and time here is given 4x3 so 4 cancels you get sin<unk> by 3<unk> by3 means 60° sin 60 is <unk>3x2 2 right. So what will be the value of acceleration? Acceleration will be minus omega² is<unk> by 4² and x is <unk>3x 2. So it will be<unk> minus sin<unk>² by 16 into <unk>3x 2 or - <unk>3<unk>² by 32.
So minus uh <unk>3<unk>² by 32. So option D is correct.
A particle executes a linear simple harmonic motion with an amplitude of 4 cm. At the mean position, the velocity of the particle is 10 cm/s.
What is the displacement of the particle when its speed becomes 5 cm/ second. So let's see. So amplitude is 4 cm and at mean position the velocity of the particle is 10 cm. Now normally the velocities formula is given by omega square roo<unk> of a² - x². This is the standard formula for velocity. Now at the mean position at the mean position x becomes zero. Okay. So v at the mean position becomes a omega.
In this particular question mean position velocity is given as 10. A is given as four and omega is not given. So omega is constant for the motion which is given by 10x 4 that is 5 by2. So omega for this motion is basically 5x2 radian/s. Now for the second part it's mentioned that what is the displacement of the particle when its speed becomes 5 cm/s. So when the speed becomes 5 cm/s what is the displacement? So 5 equ= to omega into a² - x². So basically we need to find the x this x we need to find a is constant obviously. So now let's put the value.
So 5 = to omega is 5 by uh 5 by 2 into a² - x² or a² - x² equ= to uh uh cross multiply this. So it will be 5 into 2x 5. So 55 cancels it becomes 2.
Right? Cross multiply this becomes here.
This comes there right? So now uh square both side you get uh 4² - x² = 4. Okay because amplitude is 4. Now simplify 16 - x² = 4 or x² = 16 - 4 that is 12.
So x = 3 4 are 12. So 2<unk>3.
So x value is 3<unk> uh 2<unk>3 2<unk>3 cm is the displacement. Clear?
Now let's move on.
First overtone frequency of a closed pipe of length L1 is equals to second harmonic frequency of an open pipe of length L2. Then ratio of L1 and L2.
First overtone means first means n harmonic becomes number of overtone + 1.
Always overtone number plus one that is the harmonic number. We always write the formula in terms of harmonic. Okay, our formula always are based on harmonics.
Now we know that for one side open pipe, one side closed pipe or one side open pipe whatever you say the fundamental frequency is v by 4 L. This is fundamental frequency for nth order it becomes NV by L. So obviously for this is for open pipe. So N1 L1 and second harmonic for the closed pipe.
Okay. uh sorry uh for open pipe it is 2 n -1 v byx 4 l1 because only odd number of harmonics are present okay odd harmonics are present so n equals to 2 in this case so you have to put n is equal to 2 this is for open one side closed pipe for open pipe all harmonics are present so it will be nv by 2 l2 okay so these two frequencies are same okay and n value is two for both of So this is coming from this first overtone and second harmonic n = 2. Okay. So 2 into 2 -1 into obviously v sub velocity of sound in both medium are same uh into 1x 4 l1 equals to second harmonic into 2 into l2.
Okay. So it will be 2 2's are 4 4 - 1 3x 4 L1 = to 22 cancels 1 by L2. So now cross multiply. So 3x4 equals to L1 by L2. Okay they ask you L1 by L2 which will be 3 by 4. So 99th question. So some of you are if you are still watching till the end. Uh one more questions go to go after this figure shows the shape of a part of a long string in which transverse waves are produced by attaching one end of the string to a tuning fork of frequency 250 Hz. What is the velocity of waves? So velocity of wave is given by v is equal to uh frequency into wavelength lambda.
Okay. Okay. So if you can calculate the wavelength lambda from this given graph, you can calculate the velocity as well.
So if you see the graph here, so wavelength lambda is basically if you draw a line from here, a line from here and if you find this distance, this will give you the wavelength lambda.
Right? So this is actually the wavelength lambda. So if you see this is 0.1, this is 0.5. So wavelength lambda is 0.4 4 cm.5 minus.1 right now. So wavelength is now calculated frequency is given in the question. So velocity is 250 into 0.4 cm. So 5 4 uh uh 100. So it will be uh 10 10 into 10 that is 100 100 m/s 100 cm/s sorry 100 cm/s.
Why cm/s? Because this value of uh lambda is in cm. So that means 1 m/s.
So if you see the answer will be 1 m/s.
So conversion of unit is very very important. So only trick was converting the unit. Now this is the last question.
All the best. Uh if you have made it to this point and if you have any doubts, any questions obviously uh as I mentioned in the beginning of the lecture, you should note it down and ask me during the live doubt class. Three equations of a progressive wave can be given by y = 15 sin 660 pi tus 0.02 02 pix cm. The frequency of the wave is only frequency you need to calculate.
Now the standard equation wave equation is uh amplitude sin omega t minus kz. Okay. So here omega value you can see is this. So because this is associated with time t that means the remaining constant should be omega. So omega is 660 pi. What is frequency? frequency is omega by 2 pi which is 660 pi by 2 pi which is pi pi cancels so it will be 330 herz so it will be 330 herz which is the frequency of the wave okay so this is the last question and congratulation you have completed the entire 101 questions and uh and I will meet you tomorrow during the live doubt class Whatever live doubt class live doubt you have please note it down and uh we will be opening this PDF and we will be discussing the doubts that you might have conceptual or uh some problem specific. Okay. So, see you tomorrow.
Till then, bye.
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