Oscillation is a periodic motion where an object moves back and forth around an equilibrium point. In a block-spring system, the block oscillates between maximum displacements at x = -A and x = +A, with the equilibrium point at x = 0. The restoring force always acts to return the block to equilibrium, causing acceleration in the same direction as the force. At the turning points (x = ±A), velocity is zero, while at the equilibrium point, velocity is maximum. The total mechanical energy is conserved and equals the sum of kinetic energy (1/2 mv²) and elastic potential energy (1/2 kx²). The position as a function of time is given by x(t) = A cos(ωt), where ω = √(k/m) is the angular frequency. The period T = 2π√(m/k) and frequency f = ω/(2π). The phase constant φ in x(t) = A cos(ωt + φ) is determined by initial conditions.
Oscillation(Well Summarized)Añadido:
In this video, um, I will be summarizing oscillation, which is the first chapter of semester one.
Okay?
So, I'll be summarizing that chapter with the help of this textbook.
Uh, so this is chapter 15 of the textbook that was shared by Dr. Chilukuri.
Good. So, what is it that you have to know about oscillation?
So, I'm going to summarize this topic using, um, a block spring system.
Okay?
A block spring system.
So, when we say a block spring system, we mean uh, this system where we have a block of some mass being attached to the spring.
So, we are going to have this case where this block is at x is equal to zero. It is going to be moving just in one dimension along the x-axis.
So, in this case, the block the spring is neither compressed nor stretched. Okay?
Then, the other case, let's have where the spring has been compressed, meaning the block is now at x is equal to negative a.
Then, the third case, we are going to have where this is our third case, where now the spring is stretched, meaning the block is at x is equal to a.
Okay? So, what I'm trying to say is this motion is just along the x-axis.
So, we have negative a, zero, and positive a.
So, zero is also X is equal to zero is also known as the equilibrium point.
The equilibrium point. So, this is the equilibrium point. Okay? Now, what happens when you let go of this block from this position? What happens when you let go of this block after compressing the spring?
What happens is that the block will move in this direction.
This is going to be the direction of the motion.
Okay? So, if this is going to be the direction of the motion, it implies that that is the direction of velocity.
But, what is making this block to move in that direction? It is a force trying to restore its original position. So, the original position of the block is at X is equal to zero, which is the equilibrium point.
Okay?
Excuse me. So, now, this force is a restoring force. It will always want to restore the block to where it was.
So, if the force is in this direction, it automatically makes acceleration to be in the same direction as the force.
Okay? And when you look at these two, acceleration and velocity, when the block is moving from negative A to zero, they are in the same direction. Hence, we expect the speed of the uh block to increase. It has to speed up. So, the moment acceleration and uh velocity are pointing in the same direction, we expect the speed to increase or an object to speed up.
Okay? So, one concept is that at A and negative A. At these points, velocity of the block is equal to zero.
Why? Because negative A and A are known as turning points. So, these two are known as turning points.
Okay? That's where the object will be turning from. So, if the object is going to the right, its motion, after passing through zero, it is approaching A. Its motion is to the right. When it reaches here, it wants to change the direction of the motion by going to the left. That's why we're saying this is the turning point. And at the turning point, the velocity is equal to zero.
Okay? So, as the object is moving from negative A to zero, we expect it to increase in speed or to speed up.
When it reaches zero, this is where it attains its maximum velocity.
And then, when it passes through zero, what will happen is as it passes through zero, the force points in that direction because the force, remember, it will want to restore the original position.
If the force points in that direction, meaning the acceleration will point in that direction. But, the motion is still to the right because it is now moving from zero to A. So, these are opposing direction. So, velocity is still to the right. If they're opposing direction, we expect an object to slow down.
Okay? We expect an object to do what?
Slow down. So, this from zero to A, the object is going to slow down. So, at negative A and A, the velocity is zero. At the equilibrium point, which is x is equal to zero, the velocity is at its what? At its maximum.
That is what we have.
Okay?
Now, what must you know about this system?
How do you solve the dynamics of this system? So, the dynamics of this system, we have to solve them by using um the total energy of the system. Okay? The dynamics of this system, we solve them by using the total energy of the system.
Now, what is going to be the total energy of this system?
Okay, so like we have said, we have um an object oscillating.
So, the object is oscillating.
It is oscillating along the x-axis. So, we have that.
This is where we have Okay? Now, the total energy of the system is going to be given by the kinetic energy plus the elastic potential energy. Okay? So, kinetic energy, why?
Because we have a block which is attached to a spring, and that block has a mass, and it will also have some speed, which is velocity.
Then, the elastic potential energy because there's a spring, and there will be some displacement. Okay? Some either extension or compression into that spring.
That's where this elastic potential energy is coming from. And uh mind you, this motion is just along the x-axis.
Okay? So, when you are dealing with the general expression for the mechanical energy, this is going to be 1 over 2 the mass of the block multiplied by its velocity squared plus 1 over 2 the spring constant multiplied by displacement squared. Okay? When do you use this formula to find the total energy? So, the total energy is also known as the mechanical energy.
When do we use this formula? So, we use this formula when we are dealing with these regions.
Okay? Negative A to 0 or 0 to positive A.
So, mathematically speaking, these are open intervals. So, open intervals, they mean the end points are not included.
Meaning, when the block is between um negative A and 0, where these two are not included, then we can use this formula to find its total energy.
When it is between 0 and A, where the end points are not included, we can use that formula. So, this is equation number one about this topic.
Okay? That is equation number one.
Then, what else are you supposed to know?
What is going to happen to the total energy of the system at the end points, at the turning points? So, at the turning points, there is something that we know. We know that the speed at the turning point is equal to 0. So, if the speed is equal to 0, which energy do we expect to disappear? So, the energy that depends on the speed of the block is the kinetic energy. Hence, we won't have kinetic energy for this case. So, we're going to have 1 over 2 K. But, at the turning point, we don't just have X. We have X is equal to -A or X is equal to +A.
But, this X is squared. So, if you substitute -A where there is X, you're supposed to square it, which is just +A squared. If you substitute positive or negative, the answer is going to still be +A. So, here, the total energy is uh just equal to the maximum. So, this is the maximum elastic potential. Why am I saying maximum elastic potential?
Because the position, the displacement X is at its maximum.
Okay? A and -A, these are known as maximum displacements.
The potential energy, this elastic potential energy, depends on the displacement. So, if the displacement is at its maximum, meaning the elastic potential is at its maximum. So, at this point, the block only experiences one type of energy, which is the elastic potential energy.
Okay?
At that point, same, only one type of energy, which is the same, the elastic potential energy. So, this is going to be our equation two.
Okay? Now, how do you find Where do you find these? So, this is the total energy at -A.
So, not just -A, so we say at X is equal to -A or X is equal to +A, at those particular points.
Then, what happens to the total energy when now we are at the equilibrium point.
So, at x is equal to zero, at x is equal to zero, what will happen?
At x is equal to zero, we have this disappearing.
The elastic potential energy depends on x. So, when x is zero, that will disappear, but this won't just be any other velocity. The velocity at x is equal to zero is maximum. So, meaning the total energy of the system will just be equal to the kinetic energy, and that is the equation.
So, under this topic, the only equations that you have to pay attention to are these equations here.
These three equations.
They are the only equations if it means mastering, they are the ones that you should master. Because any other equation that you're going to see is coming from the three equations.
Okay? Now, let me try to explain what is happening here. We are saying at point A, the total energy is just equal to the elastic potential. So, the block only has Excuse me. So, the the block only has uh the elastic potential energy at point negative A.
Okay? Then, as it begins to move, this elastic potential energy is going to be converted to the kinetic energy. Okay?
When is it going to finish that conversion? When will this completely be converted into the kinetic energy? When the block reaches x is equal to zero.
This is where the entire elastic has been converted to kinetic.
Okay? This is where the entire elastic has been converted to kinetic. As it passes through zero, the kinetic is slowly being converted to elastic potential.
Until it reaches A, where the kinetic is again entirely converted to the elastic potential.
This is what is going to happen. Now, if you compare this scenario to the scenario that we did in secondary, where we have the height and the ball placed there. So, at this height, the ball experiences the maximum uh kinetic potential energy. As it begins to fall, this potential energy is slowly being converted to kinetic. When it reaches halfway the distance, the height, we say the potential is equal to kinetic.
Right? The potential is equal to kinetic. And just before it strikes the ground, the entire potential has been converted to kinetic. So, for this system, for this system to be specific, this system we are saying at the distance halfway, the potential equals the kinetic. And we proved these uh type of scenarios many times. Okay? Now, this be the case for the block spring system.
What do I mean?
Is it that or at this point where we have negative A over two, or at this point where we have A over two, the potential is going to be equal to the kinetic? That is the question. I- is that going to be the case? Because here, when the height is half, we expect potential to be equal to kinetic. Now, would that be the case for this scenario?
Let us find out.
Let us find the potential. If you find the potential at negative a over 2 or uh positive a over 2, it will be the same. So, let us find the potential and see if it will be equal to um it will be equal to the kinetic energy.
Okay? So, what do we know?
The potential energy is given by 1 over 2 kx squared.
But, at which position in the x-axis have we been taught to find that potential? So, we're trying to find the potential at either this point or that point. So, just pick one. If you pick that one, all these points are known as half the maximum displacement.
Okay? Half maximum displacement.
These points.
Okay? So, where there is x, you're putting negative um a over 2 or positive a over 2, which will be the same because x is squared.
So, here you're going to have 1 over 2 k then negative a over 2 squared, which will give you that potential energy at that point.
Okay? Potential energy at that point is equal to So, negative squared positive a squared then 2 squared is 4. So, you're going to have 1 over 4 multiplied by 1 over 2 k a squared. I've just splitted the two, but this is multiplication, so it will be the same.
You can agree with me.
Now, excuse me.
Now, when you look at this equation, when you look at this expression for the um potential energy at that particular point, we can see this as k a squared. So, 1/2 k a squared is the formula for the total energy.
Here, this formula here. So, this is representing the total energy. Okay? So, all these formulas, if someone uses equation one to find the total energy, and another person uses equation two, the answer should be the same because all these equations are used for the same purpose, which is finding the total energy. Okay? So, there we can say that our potential energy is just 1/4, a quarter of the total energy.
So, if the potential energy at x equal to negative a over two or positive a over two is just equal to a quarter of the total energy, can we say it is equal to the kinetic energy? No. Why? We know that the total energy of the system the total energy of the system, let me use a different ink.
The total energy of the system is given by kinetic plus potential.
So, if we have we are looking for kinetic, where the potential we now know it's e over four.
Meaning, our kinetic will be equal to e minus e over four when this one goes the other side, which is 3/4 of the total energy. So, which one are these two the same? Because this is the kinetic energy at that point. This is the potential energy at that point. So, if let's say the value of the kind the total energy of the system is 100 J, okay? What is going to be the value of the potential energy at that point? You say 100 / 4, which is going to give you what? So, if you divide this by 4, you have a 25 J.
Meaning the uh 75 is going to be for the kinetic. So, this is not the case.
At half the displacement, the potential energy is not equal to the kinetic energy for the block-spring system.
But, what values do these energies have?
At that particular point, the potential energy is just a quarter of the total energy.
Whereas, the kinetic energy is just 3/4 of the total energy.
Okay?
This is what we have at that point, and this is fixed as long as it is at the point which is a -a/2 or positive a/2 at the point halfway the maximum displacement.
Hope that is clear. So, how do you then know the position where these two are the same?
How can you know the position where these two are the same? Well, you just say kinetic and potential, they should be the same.
And this is 1/2 uh kx² and that is 1/2 mv².
You just have to look for the x, so meaning this is going to be um the root of m/k then v.
That's what we have. So, in an event where you've been given the mass, the spring constant, and the velocity, you can know the exact position where the object uh should be in order for the kinetic energy to be equal to the potential energy.
That is the implication.
Okay. Now, with that being said, excuse me.
With that being said, how how can we how can we find the function describing the the motion of the block?
How can we find the function describing the motion of the block?
Okay? So, the motion of the block this motion of the block Okay, of the block.
So, the motion of the block can be described by this function.
Remember, we said this block is just moving uh between -A and positive A, where we have a zero somewhere there.
Okay? So, the motion of the block is going to be described by it is moving in the x-axis. So, as this block is moving, you can find the position at a particular time. So, if we say find where this block is when t is equal to 2 seconds, which function are you going to use for finding the position of the block at a particular time? So, you're going to have x of t, which is equal to A cos omega t.
Okay? So, this is the function describing the um the position of the block as it oscillates between these two regions.
Okay? So, the generic formula the generic formula has a phase constant in it. Okay? Has that in it. But, for simplicity, I will start with this, then I talk about the phase constant later.
Okay? So, this is the equation, the position the position as a function of time describing the position of the block as it oscillates. What quantities do we have here? So, x is the position, which is depending on time. A is the displacement, then omega is the angular frequency. So, omega is the angular frequency, which is defined as the root of k over m.
That is our um angular frequency. Okay? Now, >> [clears throat] >> what's going to happen to this equation? Or what does it mean?
So, we are saying this is describing the position of the block at any given time. So, if you want to know the velocity, you just have to differentiate this with respect to time.
This gives you the velocity as a function of time. So, what is going to be the velocity? The velocity, you differentiate this, okay? Which will give you omega. So, you're going to have omega A, then derivative of uh cos is negative sign. So, negative omega A sin omega t. This is the velocity uh function, velocity as a function of time describing the block. Then, if you want to know the acceleration, it is just the second derivative of the second derivative of what? The position with respect to time, which is just the first derivative of velocity with respect to time. You're just differentiating this and this gives you acceleration as a function of time, which will be another omega is coming there. So, you're going to have omega squared A. Then, derivative of sine is cos.
That is going to be the acceleration.
Okay? So, if that is the acceleration if that is the acceleration how are these equations going to be when you say time is equal to zero? At T is equal to zero. How are those equations going to be? So, at T is equal to zero the position we said position as a function of time is equal to A cos omega T. So, at T is equal to zero this changes to X of zero, which is just going to give you A because cos zero is A.
Cos zero is a one, then one times A is A.
Then, velocity as a function of time we've said it's negative omega A sine omega T. So, V of zero, velocity at zero will just be equal to zero because omega times zero is zero, sine zero is zero. Okay? And then, acceleration as a function of time we said it's negative omega squared A cos omega T. So, at T is equal to zero, this is T is equal to zero T equal to zero and T equal to zero, we are going to have A of zero, which is going to give us negative omega squared A.
Capital A.
So, negative omega squared A.
Now according to what we said, this acceleration at t is equal to 0.
The This is just the acceleration. But, if I say, "What is the magnitude of the acceleration at t is equal to 0?"
Meaning, this has to be put into the modulus. So, the magnitude of the acceleration at t is equal to 0, it means that which is just going to be equal to omega squared A.
Okay? So, where is this value going to be maximum?
Where is this value going to be maximum?
So, clearly, you can see that when you put positive A or negative A here, this is still going to give you this.
Meaning, the magnitude of the acceleration magnitude of the acceleration Okay?
is going to be at its maximum. So, it's going to be is A max at the turning point.
What is the other name for the turning points? So, at the maximum displacement.
Maximum displacement.
That's what we have. So, here, this is the now the final key.
So, the final key that we have is that um we have this the x-axis, negative A positive A and then 0.
Okay?
That's what we have.
Good. Now, here, we are saying v is equal to 0.
v is equal to 0.
And then here, we have v max.
But, here we are saying the magnitude of acceleration is equal to max as long as it is t is equal to zero.
Then even there, the magnitude of acceleration is equal to maximum as long as it is at t is equal to zero. Then here, that's where you're going to have minimum value or that's where you're going to have a zero value. So, acceleration here magnitude is zero.
Okay? According to this because x is a at those points, but here x is zero. So, that becomes zero.
All right. Now, how is the plot of this going to be?
Before we even go to the plot, let's go to uh the simplest question ever.
How can you find the following?
If I say a block spring system is described by this equation, then find omega, frequency, period, and a.
Find these. How can you find that? So, you just have to compare this to the standard one. The standard one is that cos omega t. Okay? So, that cos omega t, what does it imply? It implies that it implies that our number here is our amplitude. So, amplitude is just 10 m. Then the coefficient of t is omega, which is 20 radians.
So, rads per what? Per second.
Rads per second.
Then, how do you find frequency? We know that omega omega is equal to 2 pi f.
Meaning f is equal to 2 omega. f is equal to omega. You're just dividing by 2 pi. omega over 2 pi which is measured in Hz.
Then how do you find the period? Period is the reciprocal of frequency, meaning it will be 2 pi over omega.
Okay? So, there is an important a very important point to note.
Which point is that? When you're making calculations on this topic, specifically on the block spring system, what you need to observe here is this.
This is the angular frequency, which is measured in radians per second. And then, this all thing is acting as an angle, but here we have time measured in seconds.
Meaning system.
If you multiply the two, multiply these two, you're going to remain with a number that is in radians.
Okay?
Hence, when making these calculations, ensure that your calculator is in radians.
Okay? Your calculator should be in what?
Radians. So, how do you convert the the calculator into radians? You just go on to mode and see which which is saying rad. If it will be on position number three, you click on three and on the screen, you should see this. Not degree.
Okay? So, degree uh means the calculator is in degrees. Then that it means that you've successfully converted it into radians.
How do you prove that, okay, it is correctly converted. You have to punch cos pi This should give you -1. Then sin pi over 2 it should give you positive 1.
Okay? And then if you punch sin 90 If this calculator is in degrees, this should be 1. But if you've converted it into radians, this should not be equal to 1. I think it should be less or something. Yeah, it should be less.
Okay? So make sure that you convert your calculators into radians. Do not forget that.
Okay? Make sure you do not forget that.
Take note. Okay? Now With this being said You know, these formulas here can be expanded. These two formulas can be expanded. What do they mean? We mean frequency Frequency is equal to 1 over 2 pi.
Then omega is on top, but we know omega is equal to k over m.
Then period is just going to be equal to 2 pi m over k.
So they can also ask you questions like what happens the period when both the mass and uh when maybe the the mass is quadrupled meaning the mass increases by the factor of 4. So you just put you say 2 pi square root of 4 m over k. But this is a perfect square, so this becomes what?
2 multiplied by 2 pi the root of m over k. This 2 is coming from outside the root. And this is the odd period, meaning the period is going to be doubled. Okay? You You know how to answer such type of questions.
All right. Now, let us talk about the Um let's let's talk about the phase constant.
The phase constant, how do we determine the phase constant?
How do we determine the what? The phase constant.
Perfect. So, the phase constant is going to be determined from this.
Let us sketch this uh graph.
The function that is describing the block oscillating at those points.
Okay?
So, we said it's x of t is equal to a cos omega t.
This is the original function.
Okay? That is the original function. You should know how does it look. So, the original function is in this manner. Okay, I can do better.
So, the original function is in that manner. So, we have those points. So, this is negative a, that is positive a. This is the x-axis.
That is representing time.
Okay? So, what we are going to have is at t is equal to zero, we have a. So, at t is equal to zero, we have a. So, we are going to have something like this.
Okay, something like that.
And then, it moves like that.
So, it is a cosine function. The same way you know how to sketch cosine functions.
So, that will be the original function.
The This is where the complete wave will be because we know from here from here to there. So, from this point to that point, we have 1 over 4. From this point to that point, okay, we have 1 over 4.
Then 1 over 4, and then here we're going to have 1 over 4. So, that is where a complete wave is.
Now, what must you know about the phase shift?
What must you know about this phase shift?
Okay? So, if you're given this, let's say we never knew anything about this, and then I give you uh this graph.
I say, "Okay, the block that is oscillating is give is described by that graph."
The block oscillate Okay, so here, don't mind the drawing. The drawing is not perfect, but I can do better.
Let me see.
The oscillating block has something like that.
Yeah, anyway, still, but yeah, we understand. So, we have got two waves of the same magnitude.
Okay? We've got two waves of the same magnitude.
Now, how do we describe these?
How do we describe those waves? So, here is what we have. We've been given that as the graph representing the block spring system, of which we know that it should be a cosine, but this is a sine.
So, what is it that has been done? Cuz remember, I can as well extend this.
I can extend it and say, "This is still going like that.
This is still going like that.
And then it goes up and so on, okay? On the left side. So, what happened? What is it that they did to the original function?
So, what they did to the original function is that it was shifted to the right, okay? So, this this point here came somewhere here.
Then this point here is the one that came at the origin for us to have this.
Because this 1/4 the 1/4 here, the one that we are seeing here is this one. So, it has come to this and the 1/4 that is to the left side is the one that we have here now.
Okay? So, this 1/4, if you have a circle which is 360 a quarter of this is 90.
Okay? Which is pi over two. So, the shift here, this graph was shifted by 90 to the right.
How do you know that it is the right?
Well, this is uh A and that is negative A. From zero, the object is going up, meaning according to what we had here negative A, zero, positive A. If it is going to positive A, meaning it is going to the right. So, here it is going to the right.
And then it goes to the left and so on.
So, this graph is showing that we have X of T Okay? We have X of T being equal to A cos omega T with the phase constant of what?
Of pi over two.
That is the phase constant. Because the general equation that incorporates the phase constant is given by x of t is equal to a then cos omega t plus the phase constant that that is the general equation.
So, from here from the graph, you can tell. But, what if they don't give you the graph? They give you the initial conditions.
Okay? What if they give you the initial conditions?
How can they give you the initial conditions? Well, if they say at t is equal to zero Okay? At t is equal to zero x is equal to zero Okay? When the object is moving to the right.
Those are the conditions. Okay? Now, how do you find the phase constant? So, the phase constant here how do you find it?
You say um According to the equation, this is the general equation. A cos omega t plus the phase constant.
So, at t is equal to zero, x is zero. We have a then cos here we when we put zero there, we have omega times t which is zero.
Uh so, we're just going to have cos phi.
So, we have cos phi uh which is the phase constant is equal to dividing by a, you're still going you're still having a zero. So, what immediate angle are you going to have in order for you to get a zero for cos? So, you've got two possibilities. You've got negative pi over two and you've got positive pi over two. Okay? So, which one should be the correct answer? The correct answer here should be positive pi over two. Why positive?
Because they said it is moving to the right. So, meaning our phase constant is nothing but pi over two.
All right. So, if you want to gain more understanding of phase constant, you can use this same book.
The book that was shared by um Dr. Chiluksha.
Okay? And then, you go to this. This is page page 417.
Initial conditions, the phase constant.
So, you go through this.
Uh you see that now we have the phase constant.
And you can, you know, at least try to gain an understanding. But, basically, that should be it about uh oscillation.
And unless otherwise, but uh this is more than enough. So, remember, you can play around with the three equations of energy to obtain any given equation.
What do I mean? If you want to obtain the equation for maximum velocity, so you've got three equations. E is equal to 1 over 2 mv squared plus 1 over 2 kx squared.
Then, E is equal to 1 over 2 ka squared. And then, E is equal to 1 over 2 mv max squared.
Okay? So, if if you equate the two equations, if you equate the two equations, what is it that you're going to have?
Equating these two because they are used to find the same thing. So, equation one equation two and equation three. So, if you equate equation one, equation two and three to find V max, you're going to have 1/2 M V max squared is equal to 1/2 K A squared.
The half the half cancels. Then, V max, you divide by M you're going to have K over M A squared. And then, you square root.
So, V max is plus or minus of square root of K over M and then A. Of which, square root of K over M is nothing but the angular frequency. So, this is plus or minus omega A.
That is your V max. Okay? So, you can equate any given equation and you should be able to get the answers correctly.
Okay? So, um make sure you do that and go through these systematically. Gain an understanding.
Okay?
And um well let us try to, you know, encourage each other as we study for nothing is impossible with God.
Anything that you do as long as you even just uh root your education into the wisdom of God you should be able to perform wonders and your impact is going to be felt.
Okay? So, first Daniel um Daniel chapter 1 verse 7 says that to these four young men God gave knowledge and understanding of all kinds of literature and learning.
So, God gave Daniel and the people that went outside, okay? Into the foreign country, he gave them the knowledge to gain that understanding, to be able to grasp information that was totally new and strange to them. And if you just believe and root your education into the wisdom of God, you should be able to do wonders and your education is going to mark its territory. You'll be able to leave a mark, okay? There will be an impact. So, just make sure that in whatever that you're doing, you involve God and try by all means to live a righteous life.
Okay, thank you.
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