Energy of spring-mass oscillators

0:00- [Instructor] If you stretch a spring and let it go, it'll start oscillating back and forth about the equilibrium position. Here A represents the amplitude; if there are no resistive forces like friction or air resistance or anything like that, then it'll just keep oscillating forever.

0:15In this video, we want to analyze the energy associated with this oscillation. So let's begin. All right, so what are the different kinds of energies we can think about? Well, we have a moving mass, so there's kinetic energy associated with it. Now, we know that real springs have mass, but let's assume that this is an ideal spring, meaning it has negligible mass compared to this object that is attached at the end.

0:38So the kinetic energy is the energy of this object attached. Okay, what are some other energies over here? For that, let's think about some forces that are acting. Well, there's one force that is the spring force that's acting on our mass, and spring force is a conservative force, which means that we can associate a potential energy with our spring-mass system.

0:59I should actually technically say a spring-mass wall system, but since this wall is big and it's not moving anywhere, we'll just call it as a spring-mass system. Are there any other forces that are doing work on this system? Well, there are no resistive forces, like we're assuming there's no friction or air resistance, but there is a force of gravity.

1:19But that does not do any work on our system because it's perpendicular to the direction of the motion. Also, there is the normal force that's balancing the force of gravity and that too is perpendicular to the direction of the motion, which means these forces also don't do any work on our system.

1:35And there are no other forces that are doing work on our system, which means the only force involved over here is the spring force, which is a conservative force and therefore the total energy, the kinetic plus the potential energy, the total mechanical energy of our spring-mass system, that remains conserved.

1:53That means as the spring oscillates back and forth, it's potential and kinetic energy change, but the total mechanical energy stays the same. Okay, so let's analyze this. Let's start analyzing from this position. At this position, the spring is compressed so we can intuitively see that there must be some potential energy in our system.

2:12But the mass is at rest, so the kinetic energy is zero, which means the total mechanical energy, which is the kinetic plus the potential energy, right now is just potential energy. And this mechanical energy stays fixed to this level, it will not change; which means right now, all of our mechanical energy is just potential energy at this location.

2:33And that is the maximum potential energy we will ever get, because after this point, the spring will uncompress, potential starts getting converted to kinetic. So the mechanical energy is also basically the maximum potential energy; this will be useful later on.

2:50So, at this particular position, our system has maximum potential energy and our system has zero kinetic energy, okay? All right, what happens as time passes by? Well, now some of its potential energy gets converted to kinetic, the mass is now moving. So the potential energy drops, kinetic energy increases, but the total sum, that stays the same, all right?

3:14Okay, as further time passes by, the mass comes to the equilibrium position. At this position, the spring is relaxed, which means all the potential energy of the system is gone, it's zero, it's been converted to kinetic energy, which means at this particular position, our system has maximum kinetic energy, but it has zero potential energy.

3:36So notice, the mechanical energy is also maximum kinetic energy. Makes sense, right? Okay, and then as you go from here to here, again, all of that kinetic energy gets transformed into potential energy and the whole thing continues forever and ever. Now this animation is great, but can we represent how kinetic and potential energies change with time without an animation on a piece of paper, for example?

4:09Yes. To do that, we need to write expressions for kinetic and potential energies as a function of time, and then we can use that expression to graph them. So let's do that. So let's pause our animation and let's see how do we write an expression for them. Let's start with total mechanical energy, because that's a constant.

4:27So let's see if we can write an expression for that. It shouldn't depend on time, right? Okay, so how can we figure out expression for mechanical energy? We could say it's kinetic plus potential, and we know kinetic is half every square, and potential, you know, spring potential energy, is half k delta x square, where delta x is just the displacement from our equilibrium position.

4:48If you're wondering where this comes from, we've derived that in our previous videos on conservative forces, so feel free to check that out if you need that. But now I can write mechanical energy as this plus this, but they are changing and I don't know how they're changing, that's what we wanna figure out so that won't be useful.

5:06But instead I can say, "Hey, my mechanical energy is basically maximum kinetic energy, right? But I don't know what the maximum kinetic energy is, so that won't be useful as well. But hey, I can write mechanical energy as maximum potential energy. And that will be useful because when will you get maximum potential energy?

5:26Because when the mass is at this location, which means when delta x is A. So when I plug delta x as A, I get maximum potential energy. So my mechanical energy, which is the same as maximum potential energy is half k A squared. This is the maximum, this is my mechanical energy.

5:49And notice, we see that the mechanical energy is proportional to the amplitude squared. This means if I were to double the amplitude, the mechanical energy does not double, it quadruples, because there's a square over here. Okay, now since I know what my total mechanical energy is, I can use this to figure out the speed of the block at any given position.

6:14All I have to do is now write mechanical is equal to kinetic plus potential, so let's do that. So mechanical energy equals kinetic energy plus the potential energy, this, you know, spring potential energy. And so, if I get rid of the half, cancel half everywhere, and if I rearrange this, I get this expression.

6:32And so, look, if I plug in the value of delta X, I can find out the value of v at that particular location. So let's test it out, let's see if this makes intuitive sense. What happens if I put delta X as A? Well, then I get A square and this becomes zero. Does that make sense?

6:51Yes, because when delta x is equal to A, we are at this particular location, we have zero kinetic energy. We already saw that, so this makes sense. Okay, what if I put delta x as zero? Well, then we would be over here at the equilibrium position, and when delta x is zero, the system's kinetic energy is maximum.

7:13So when I put delta x as zero, this goes away, there's no more potential energy, and we are at maximum kinetic energy, so that should represent maximum speed. And so, from this, we get an expression for the maximum speed. Do I have to memorize this expression?

7:30Absolutely not, we can always derive it using energy conservation. Anyways, the expression says if the amplitude is higher, the maximum speed would be higher. Does that make sense? Yes, if I stretch it more, then it'll accelerate more, and so you would get a higher maximum speed at this position, so that makes sense.

7:47If the stiffness constant or the spring constant is higher, then you have more maximum speed. That too makes sense; for a given stretch, if the spring is stiffer, the acceleration would be higher, you'll end up with more speed over here. If the mass is higher, the maximum speed is lower, that too makes sense because for a given stretch, if you attach a much heavier or much more massive block, then more inertia, and so it'll be harder to accelerate, and so you'll end up with lower speed at this position.

8:16So that makes intuitive sense. But anyways, coming back, remember our goal is to write expressions for kinetic and potential energy as a function of time so that I can write this animation as a graph. So how do we do that? Well, first, we need a coordinate system.

8:34So let's just choose x equal to zero as our equilibrium position. Then I can just call this as x, we can choose one side positive, so right side is positive and left side is negative over here. And so, this becomes half k x squared. I know how x changes as a function of time, we've derived that in our previous video, it's a sinusoidal function, which means it can be written as a sin or a cosine function.

8:59So from this, I can write the expression for potential energy as a function of time. It's just half k x squared, which means it's gonna be half k times this squared. And, look, this represents the maximum potential energy, and so, I can just write this as this particular expression.

9:18So there we have it, we have found potential energy as a function of time. All right, so can we now write kinetic energy as a function of time? We can write kinetic energy as half m v squared, but then I need to know how v changes as a function of time. We can do that, but another way to do it is, from here, we can write kinetic energy as mechanical energy minus the potential energy.

9:42And I already found out the potential energy, so I can use that to figure out kinetic energy. Great idea to pause the video and see if you can do this yourself now. All right, so let's do this. Kinetic energy's mechanical energy minus the spring potential energy.

9:54So kinetic energy is basically... The mechanical energy is half k A squared minus the spring potential energy, we just got it over here, we can plug that in over here. Half k squared comes out as common, and so, I get 1 minus cos squared, which is basically sin squared.

10:09And so, this now is the maximum kinetic energy. Okay, so let's clean this up and let's make space for drawing some graphs over here. Instead of directly trying to draw the graphs, what we can do is we can start with the graph that we've already seen, we can graph out the position function.

10:28And from this, we can intuitively think about how the potential energy graph should be. Well, the potential energy will always be a positive value, that's the first thing because there's a squared in it. So it'll swing between maximum and zero, it won't be negative, there's a square over here.

10:45At this location, we are over here and my potential energy is maximum, okay? When my mass comes to zero, that's the equilibrium position and my potential energy at this position goes to zero. And then over here, again, look, my potential energy goes to maximum.

11:02So again, at this position, it goes back to maximum, we go back to equilibrium position, so it goes back to zero, we go back to an amplitude position, so it goes back to the maximum. And so, you can kind of sort of visualize the graph looks like this. And so, that graph, the curve over here, would be cos squared curve, and that kind of sort of looks like this.

11:24Okay, now from this, can you graph the kinetic energy? It would be a great idea to pause, but use similar reasoning to try and see how this graph looks like. All right, we already know at this location, we have maximum potential energy, so we should have zero kinetic energy.

11:39At this point, we have zero potential energy so we should have maximum kinetic energy. And the same thing goes on, so it's kind of like the opposite, it's flipped over version of this graph. So it's a sin squared graph, and it looks somewhat like this. Notice that when my mass finishes one full cycle, my potential in kinetic energies, they finish two cycles.

12:03Does that make sense, why is that happening? Well, that's because when my mass swings from, you know, plus A to minus A, notice our energies don't have a negative sign, so it finishes one full cycle, all right, because it doesn't have a negative sign. And that's why our potential kinetic energies, their oscillations, they have twice the frequency, they finish two cycles when my mass finishes one, my spring mass system finishes one cycle, so it has twice the frequency as this one.

12:33And that's why cos squared or sin squared has twice the frequency of cos or sin. And we can now intuitively see why that is the case. And if you go back to the animation, we can confirm that, okay? When my mass goes from here to here, half an oscillation would be done, but for my kinetic and potential energy, one full cycle would be done.

12:53Look at this very carefully, okay? We're going back. Okay, look, one full cycle of potential energy is done, second full cycle of potential energy. Again, one full cycle of kinetic energy done, second full cycle of kinetic energy done. Can you see that? Okay, now, we can also put the kinetic and potential energy graphs together, and it looks like this.

13:19And this is, again, reinforcing the same idea that the total mechanical energy always stays, that's the sum of kinetic and potential, always stays the same, and that will be this. So this represents your total mechanical energy, that'll always be the same, it does not change with time.

13:37The final thing we can do over here is remember how we ignored friction and air resistances, and that's why, you know, this graph goes on forever. But what if there was friction, for example? Then what do you think would happen? Well, things will be more complicated, but can we imagine how the kinetic energy graph would look like?

13:55It would be a great idea to pause and just like intuitively think about this. Okay, well, because there is friction, you can intuitively think that the speed would keep reducing and it'll eventually stop, right? At some location and at some point, it'll just stop, which means that you would expect the graph to look somewhat like this.

14:15The maximum kinetic energy will also keep reducing, reducing, and eventually at some point, it'll just stop. And since the mechanical energy is being lost here, the amplitudes of the other graphs, like the potential energy graph, will also decrease. But anyways, these kinds of oscillations are called damped oscillations.

14:34But what's important is that this energy profile of damped oscillations depend on the type of resistive forces involved. So for friction, it would look something like this, but for other kinds of resistive forces, like air resistance, it could look different. But in our ideal case, in undamped oscillatory systems, things look much simpler.