Energy graphs for simple harmonic motion

- [Instructor] What I have drawn here, is a mass sitting on a frictionless surface that is attached to a spring that is attached to the wall, and what we're going to do is we're going to compress the spring. We're going to get the mass to position A. Right now it's at position zero. We're going to get it to position A. And then at time equals zero, we are going to let go. And so you can imagine what's going to happen, especially with this mass on a frictionless surface. It's going to oscillate between position A and position negative A, and we have that depicted right here on this position versus time graph. It'll start at position A, and then it will oscillate to the left to position negative A and then oscillate to the right to position A again, on and on and on forever, if we're dealing with a world that is frictionless. Like I said, frictionless surface, and let's also assume no air resistance. So that is all very interesting, but what we want to think about in this video is how that might relate to energy. So given the information that I've just given you, let's start thinking about elastic potential energy. Remember, at time equals zero, the box is at position A, so our spring is compressed. And we're dealing with a box-spring system, so the combined system of the box and the spring. And we're going to assume that there is no added energy that's added to or taken away from this system. So right at time zero, when we have the spring compressed, that box-spring system is gonna have some elastic potential energy. And so let's put that right over there. And then what's going to happen when we let go? Well, the box is going to be pushed by the spring towards the left. Actually, is going to be accelerated to the left. And right when the box crosses the x position of zero, which we see happens at time equals one second, all of our potential energy is going to be converted to kinetic energy. And so our potential energy is going to be right over here. And then what happens is the box starts getting decelerated by the spring, and it gets to position negative A. Well, at position negative A, which we see happens at time equals two seconds, well, then we are back to having our maximum potential energy again. So we're back to having our maximum potential energy at time equals two seconds, with is associated with being at position negative A. And so you can see where this is going. At three seconds, all of that potential energy is back, converted to kinetic energy. At four seconds, we are back at position A. It's back into potential energy again. And so the graph of our elastic potential energy is going to look something like this. It's going to look something like, this is a hand-drawn version of it, but you, I think, get the general idea of what's going on here. Notice it is not getting negative, and so it would look something like that. Now what about kinetic energy? Well, I've already made some reference to it, but let's think about how that would trend over time. At time equal zero, when the box is at position A, right at that moment, we aren't going to have any kinetic energy just yet. But then the box is going to be accelerated as that potential energy is turned into kinetic energy, and we are our maximum kinetic energy when the box crosses position zero. Well, the first time it crosses position zero is at time equals one second. So we have our maximum kinetic energy right over there. And then when we get to time equals two, our box is at position negative A. We no longer, for a moment, we won't have any velocity, and our kinetic energy is gone. And so you can see how this is going. We keep switching between potential and kinetic energy as the box keeps oscillating between position A and position negative A. Once again, this is my hand-drawn depiction of it. So that is our kinetic energy. Now, when we first introduced ourselves to energy and the law of conservation of energy, we saw that, hey, look if we are in a closed system and there are no dissipative forces and we're not adding energy or taking away energy from that closed system and if we're just dealing with mechanical energy and this, and non-dissipative forces, well, then mechanical energy should be conserved. If we say that the total mechanical energy of the system E is equal to our potential energy, which, in this case, is all elastic potential energy, plus our kinetic energy, this should be constant. And it is indeed the case. If, at any point in time, you were to add these two curves up, you would get something that looks like this. It would just be a constant line, and that would be the graph of our mechanical potential energy. Now, an interesting question is, what if we did have dissipative forces? What would things look like then? Well, if we had dissipative forces, say friction or air resistance, well, then the box might start at position A, but then it wouldn't get all the way to negative A. It might look something like this. It might start here, but it might not get all the way to negative A. And then it would get even not as far this time, and then it would get, and I'm trying to draw it as best as I can, and then it would get even not as far that time. And if we think about it in terms of energy, the total mechanical energy would decrease. Where is it going? Well, it is being transformed into thermal energy by the dissipative forces of friction and air resistance. So the total energy would decrease, and then this would define the envelope for the oscillations for the potential energy and the kinetic energy. So, for example, the kinetic energy in that situation would look like this. It would look like, it would look like this, where the peaks are going to be bounded by this total, by this total mechanical energy. So I will leave you there. Hopefully this gives you a sense of how potential energy, kinetic energy, especially when you're dealing with a spring-block system, how they relate to each other, especially in relation to the law of conservation of energy.