Potential energy and conservative forces (part 3)

0:00- [Instructor] We have a block attached to a spring. 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.

0:17We 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.

0:32For 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 wanna try and answer in this video is whether the 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?

0:58So 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 sub s, and let's call this disposition from the equilibrium position as delta x. Then according to Hooke's law, the spring force can be written this way.

1:16Look at what the Hooke's 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. Stiff for the spring, more the value of k.

1:32And what is 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. All right, so our question now is whether this spring force is a conservative force.

1:48How 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.

2:11But 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.

2:25In our case, the block is only moving in one dimension. So we'll call that as the x-axis. and we're gonna choose the equilibrium position as our x = 0 as our 0. And then this position, we can just call that as x, and then, look, this now represents the position vector.

2:43So what's the magnitude of this displacement vector? Well, it's just x. And what's the direction of this displacement vector? Well, if we choose the right direction as our positive x direction, then we can represent the direction using i hat, the unit vector in the positive x direction.

3:00So we can say delta x is xi hat. 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.dr, from some initial position, a, to some final position, b.

3:21So 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 that tiny infinitesimal displacement vector. In our case, since we are only dealing with one dimension, dr is just going to be dx.

3:40But factorially, how do we write it? Well, we can say the magnitude of dr is just dx, an infinitesimal displacement, you know, 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 i hat or negative i hat?

3:58Well, 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.

4:12So 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 gonna call this i hat. 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.

4:28All right, so let's plug it in. So the force over here is going to be the spring force minus kxi hat, .dr is basically dx times i hat. So let's pull out the negative sign over here, and we can separate the magnitudes, which is kxdx times i hat.i hat. What's i hat.i hat?

4:51Well, that's going to be the magnitude of i hat, which is 1, because i hat is a unit vector, times magnitude of i hat, which is again, 1, times cos of the angle between them. Well, the angle between them is 0 because they're the same vectors. Cos of 0 is just 1.

5:06So it's 1 times 1 times 1, and so this is just 1. So what we are left with is just this part. Okay, so let's simplify that. We can pull out the constant, k. So we get -k integral of xdx. What's this integral? It's 1/2 x squared. So we can write this as 1/2 x squared from the lower bound, a, to the upper bound, b.

5:31And then we first substitute the upper bound minus the lower bound, which gives us -1/2 k b squared minus a squared. So we have calculated the work done by the spring force on this block, and moving it from some initial point, a, to some final point, b. Now, the question is, looking at the expression, what do we see?

5:53Does the work then 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 as just the equilibrium position. So let's say this was our a, and let's call this as our b.

6:12Now, 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 directly go from here to here. 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.

6:35So 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 this system when the spring is stretched or compressed by some amount?"

6:55Well, here's how I like to think about it. Remember from work energy theorem that the total work done on an object equals a 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.

7:11And 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. 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.

7:38That means as the block moves from one position to another, if it's kinetic energy increases by same amount, the potential energy will reduce by the same amount. Similarly, if the kinetic energy decreases by same amount, the potential energy of the system will increase by the same amount.

7:53In 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 -delta U, and that's how I bring the potential energy into the picture. Delta U is a change in potential energy. And the minus sign is because when delta K is positive, delta U is negative, and vice versa.

8:16All 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 changing potential energy, but the potential energy of the system when the spring is stretched or compressed by some amount, let's call it delta x, that's what I want.

8:44So how do I get that from this? Well, let's see. What is changing potential energy? Well, it's final potential energy minus initial potential energy. So this would be potential energy at the position b minus potential energy of the system when the block is at position a.

8:59And now to bring delta x into the picture, what I can do is I can call this equilibrium position as my 0, and then this position would be delta x. And so look, our point b has become delta x or point a has become 0. And so over here, I get delta x squared minus 0.

9:19So the right-hand side becomes 1/2 k delta x squared. 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.

9:37So what we can now do is we can just assign this potential energy some value. The most convenient choice is to call it 0. 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 0.

9:55Now, if this goes to 0, then look, our expression simplifies to give us just 1/2 k delta x squared. So we have finally found the expression for the potential energy of the system when the block is stretched or compressed by some amount, delta x, from its equilibrium position.

10:12It's 1/2 k delta x squared provided, provided our convention is that, as we choose the potential energy at the relaxed state in the equilibrium position to be 0. That's our choice, remember that. Finally, this is the expression of potential energy of this system, right?

10:30But what does the system comprise of? Well, it's not just the block in 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.

10:48But it's because 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 are dealing with not just the spring and the block, it's the spring block and the wall, that's the system we are dealing with.

11:03If 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.