Potential energy and conservative forces (part 3) | AP Physics | Khan Academy

[00:00:00]We have a block attached to a spring.

[00:00:02]This represents the relaxed state of the spring. What happens if you were to stretch or compress the spring? Well, if you were to say compress the spring for example, then the spring will tend to uncompress. And as a result, it puts a force on this block. We call this the spring force. And this spring force has a couple of interesting features. First of all, the more you displace it from the equilibrium position, bigger the spring force. And the second thing is that the spring force is always in the opposite direction of the displacement.

[00:00:32]For example, if you were to stretch the spring then look again the spring force will be inwards opposite direction to the displacement. So the question I want to try and answer in this video is whether this spring force is a conservative force and if it is what is the expression for potential energy of a system when the spring is stretched or compressed by some amount. So let's figure this out. Okay. So how do we do this? Well first of all let's try and model this force. Let's call this spring force F subs and let's call this displacement from the equilibrium position as delta X. Then according to Hook's law, the spring force can be written this way. Look at what the hooks law is saying. It's saying that the magnitude of the spring force is proportional to the magnitude of the displacement from the equilibrium position. And the proportionality constant K depends upon how stiff the spring is. Stiffer the spring, more the value of K. And what does a negative sign represent? Well, the negative sign represents that the direction of the spring force is always in the opposite direction of the displacement delta x.

[00:01:42]All right, so our question now is whether this spring force is a conservative force. How do we answer that question? Well, all we have to do is find an expression for the work done by this spring force in moving the block from some initial position to some final position. If that work done by the spring force does not depend on the path taken but only depends on the initial and the final positions then it's a conservative force. But if it turns out that it does depend on the path then it's a non-conservative force. So our next step is to try and calculate the work done by the spring force. How do we do that? Well first of all let's try to write this expression in terms of position vector. In our case the block is only moving in one dimension. So we'll call that as the x-axis and we're going to choose the equilibrium position as our x=0 as our zero and then this position we can just call it as x and then look this now represents the position vector. So what's the magnitude of this displacement vector? Well it's just x. And what's the direction of this this displacement vector? Well, if we choose the right direction as our positive x direction, then we can represent the direction using ihat the unit vector in the positive x direction.

[00:03:00]So we can say delta x is x ihat. And now I have represented this in terms of the position vector. Okay. Once we do this, how do we calculate the work done? Work done by any force is the line integral of F dot dr from some initial position A to some final position B. So in our case we already have the vector expression for the spring force. Okay, the next thing we need is the expression for DR. What's DR? dr is the tiny infinite tesimo displacement vector. In our case, since we're only dealing with one dimension, dr is just going to be dx.

[00:03:40]But vectorally how do we write it? Well, we can say the magnitude of dr is just dx and infinite decimal displacement you know in the along the x direction multiplied by the unit vector along the x direction. Now would that unit vector be in the positive x direction or the negative x direction? Positive ihat or negative ihat? Well, that completely depends upon our bounds. For example, if this point was a and this point was b, then look, our displacement is in the positive x direction. On the other hand, if this was A and this was B, for example, then the displacement would be in the negative direction. So, it's the bounds that take care of the sign of our displacement vector. So, we don't have to worry about it. So, I'm just going to call this Ihat. All right, we have the expression for both the force and the displacement vector. It would be a great idea to pause the video and see if you can plug it in and compute the work done. All right, so let's plug it in. So the force over here is going to be the spring force minus kx Ihat dot dr is basically dx * ihat. So let's pull out the negative sign over here and we can separate the magnitudes which is kx dx time ihat do ihat what's i do ihat?

[00:04:51]Well, that's going to be the magnitude of Ihat, which is 1 because I hat is a unit vector time magnitude of Ihat, which is again 1, time cos of the angle between them. Well, the angle between them is 0 because they have the same vectors. Cos of 0 is just 1. So, it's 1 * 1 * 1. And so, this is just one. So, what we are left with is just this part.

[00:05:13]Okay. So, let's simplify that. We can pull out the constant k. So, we get minus k integral of x dx. What's this integral is half x^2. So we can write this as half x^2 from the lower bound a to the upper bound b. And then we first substitute the upper bound minus the lower bound which gives us minus/ k b² minus a squ. So we have calculated the work done by the spring force on this block in moving it from some initial point a to some final point b. Now the question is looking at the expression what do we see does the work done depend on the path or not and what we see is it doesn't it only depends upon the initial and the final positions for example what this means is let's say that the initial position was just the equilibrium position so let's say this was our A and let's call this as our B now the work done to get the block from here to here is completely independent of the path I mean one way to do that work is to direct directly go from here to here.

[00:06:21]Another way would be to overshoot a little bit, go back and then bring it over here. In both the cases, the work done would be the same because look, all that matters is the final and the initial positions. The path does not matter. So look, the work done by the frictional force is path independent. It only depends on the initial and the final positions. So frictional force is a conservative force. And for every conservative force, we can associate a potential energy with it. And so our next question is what is the potential energy of the system when the spring is stretched or compressed by some amount?

[00:06:55]Well, here's how I like to think about it. Remember from work energy theorem that the total work done on an object equals the change in kinetic energy. Now in this particular case, if we ignore friction and air resistance and all of those things, then the spring force is the only force acting on this block. And so this work done by the spring force is the total work done on this block. And therefore that should equal the change in kinetic energy of this block. And so I can write this expression as change in kinetic energy delta K equals this term.

[00:07:26]But remember since spring force is a conservative force I can say that the total mechanical energy which is kinetic plus potential energy. Total mechanical energy of this system should always stay the same. That means as the block moves from one position to another, if its kinetic energy increases by some amount, the potential energy will reduce by the same amount. Similarly, if the kinetic energy decreases by some amount, the potential energy of the system will increase by the same amount. In other words, the change in potential energy is always equal to negative of the change in kinetic energy. So, I can replace delta K as minus delta U. And that's how I bring the potential energy into the picture. DeltaU is a change in potential energy. And the minus sign is because when delta K is positive, delta U is negative. And vice versa. All right. So now let's simplify this. The minus sign cancels out. And we now have an expression for the change in potential energy as a block goes from some initial point A to some final point B. But this is not what I want. I want to know what's the potential energy, not change in potential energy, but the potential energy of the system when the spring is stretched or compressed by some amount.

[00:08:41]Let's call it delta x. That's what I want. So, how do I get that from this?

[00:08:46]Well, let's see what is change in potential energy. Well, it's final potential energy minus initial potential energy. So, this would be potential energy at the position B minus pot potential energy of the system when the block is at position A. And now to bring delta x into the picture, what I can do is I can call this equilibrium position as my zero. And then this position would be delta x. And so look, our point b has become delta x. Our point a has become zero. And so over here I get delta x² minus 0. So the right hand side becomes half k delta x^ 2. So look, I finally have the expression for the potential energy of this system when the block is stretched by some amount delta x. Now that value depends on the potential energy at the equilibrium position. So what we can now do is we can just assign this potential energy some value. The most convenient choice is to call it zero. That is our choice. You could have called this potential energy to be any other value you want. But the most convenient choice is to call this zero.

[00:09:55]Now if this goes to zero then look our expression simplifies to give us just half k delta x squar. So we have finally found the expression for the potential energy of the system when the block is stretched by or stretched or compressed by some amount delta x from its equilibrium position. It's half k delta x squ provided provided our convention is that we we choose the potential energy at the relaxed state in the equilibrium position to be zero. That's our choice. Remember that. Finally, this is the expression for potential energy of this system, right? But what does the system comprise of? Well, it's not just the block and the spring, but the wall is also the part of the system. Because if there was no wall then it would be very hard to stretch or compress this spring. I mean if you were to pull the block then the entire spring and this block would just accelerate forward. But it's because it's one the other end is attached to the wall. That's why it stretches. And that's also the reason why you can compress it. Which means we're dealing with not just the spring and the block. It's the spring block and the wall. That's the system we're dealing with. If you were to compare this with a system of two masses which interact gravitationally, then the block and the wall are like the two masses and the spring is like the gravitational force. It's what mediates the force between the block and the wall. So this is the expression for the potential energy of the block spring wall system.