SHM of spring-mass oscillators

0:00- [Lecturer] If you hang a mass on a spring and stretch it and let it go, you probably know that it'll start moving back and forth. It starts oscillating. The question we want to try and answer in this video is: how do we model this oscillation? How do we analyze this?

0:13Let's find out. To make our analysis a little bit simpler, let's make the oscillations horizontal. So we'll keep the entire system horizontal on a frictionless table. The reason we do that is because if you were to think about the forces acting on the mass right now when everything is at rest, well, then there's a force of gravity acting downwards, but that force of gravity is perfectly balanced by the normal force.

0:36So in the vertical, the forces are balanced, and in the horizontal, currently there are no forces. We're assuming no friction, no air drag, or anything like that. So the spring is not stretched or anything like that right now. So absolutely no horizontal forces right now.

0:49So right now, the mass over here is in equilibrium. So we can call this as the equilibrium position. Okay. Now what's gonna happen if I stretch it a little bit and then let it go? Well, because it got stretched, the spring got stretched by some amount, let's say Delta X, according to Hooke's Law, the spring pulls back with a force that's proportional to the amount of stretch.

1:18And that proportionality constant is K, which is called the stiffness constant. It tells you how stiff the spring is. If the stiffness constant is higher, if the value of K is higher, the spring is much stiffer, and for that given stretch, it'll pull harder. And of course, the negative sign over here is telling that the force will be in the opposite direction in which we are stretching.

1:38So if you stretch towards the right, the force will be towards the left. If you, you know, compress it to the left, the force will be towards the right. Which means, look, the spring force is always directed towards the equilibrium position. The spring always tries to bring it back to the equilibrium position.

1:54Such forces are called restoring forces. And in our case, since there are no other unbalanced forces acting, this itself is the net force. So the net force on this mass itself is restoring. So what's gonna happen to it? Well, before we answer that question, let's first choose our coordinate system.

2:15And since, you know, the restoring force or the net force depends on the displacement from the equilibrium position, let's just call the equilibrium position as X equals zero. That's our origin. And then we can call this displacement as just X, right? So now we can say the net force is directly proportional to the position of the mass.

2:32And while we're at it, let's also choose one direction as positive. Let's choose right side as positive, left side as negative. Okay. So what do you think is gonna happen now? Well, this mass is going to start accelerating towards the left. But as it does, notice its position reduces.

2:49And as the position reduces, the net force will reduce. But it will continue accelerating towards the left until it reaches this point, the equilibrium position. Then X equals zero, the restoring force goes to zero, right? The net force goes to zero now. But because it already has gained a lot of speed, it will overshoot.

3:09And as it overshoots, now look, the position is towards the left. And so the restoring force, the net force, will be towards the right. And so the mass will start slowing down. And eventually, at some position towards the left, it will stop, and it will start accelerating toward the right now.

3:28And the story continues forever and ever. Of course, in reality, it won't continue forever because there'll be friction and there'll be air drag that'll slow it down, and eventually the whole thing stops. But if you consider an ideal case where there are no frictions or air drag, this would continue forever.

3:43So we have a periodic motion. And this particular periodic motion is called a simple harmonic motion, SHM. What makes it a SHM? Well, notice in this particular case, the restoring force, the magnitude of the restoring force, is directly proportional to the magnitude of the displacement from the equilibrium position.

4:04That's what makes it an SHM. Now it's possible for restoring forces to be more complicated, where they're not directly proportional to the displacement, but it's still restoring in the sense the forces will still be always directed towards the equilibrium position.

4:18In that case, you'll still get a periodic motion, but it won't be SHM. What makes it an SHM is that the restoring force is directly proportional. The power over here is one, right? So it's directly proportional to the displacement from the equilibrium position.

4:33So now the main question we wanna try and answer is: how does X vary as a function of time? Now, there are a couple of features that could help us. First feature is the idea of period. Notice that it's a periodic motion. That means we can think about how long it takes for the mass to complete one full cycle.

4:52That's called the period. And in this particular case, the expression for the period happens to be this particular expression. We're not gonna derive it, but let's see if it makes sense. First of all, the S over here represents spring. This is to differentiate it from periods of other oscillating systems.

5:07Anyways, this expression says that if the mass increases, the period increases. Does that make sense? Well, yeah, because if the mass increases, it'll have more inertia. It becomes harder to accelerate it and turn it around and stuff. And so we would expect it to, you know, take a longer time to complete one full cycle.

5:27So that makes sense. You also notice that if the spring constant, the stiffness constant, were to increase, the period reduces. Does that make sense? Yeah. I mean, if the spring got stiffer, then the forces would be much higher. The restoring forces would be much higher.

5:41And so now it'll accelerate much quicker. It'll turn around and stuff much quicker. So we'd expect the whole thing to be much quicker. So we would expect the period to reduce. So that kind of makes sense. The second feature about this periodic motion is you can see that this is the maximum displacement, you know, over here.

6:00And we call that the amplitude. And we represent that with A. So if we choose the right side to be positive, then we can call that as the positive amplitude, and this is the negative amplitude. So amplitude is the maximum displacement from the equilibrium position.

6:13You can also think of it as half the distance between the extreme position. So it's half of this total value, right? Okay. So how do we now... How does X vary as a function of time? A cool way to do this is by attaching a spray paint can over here, and we have a paper behind.

6:35And, you know, as it oscillates, we'll just steadily move this paper, say, upwards. And then it'll literally paint a picture of how X varies with time. Right? So the vertical over here will represent time. So let's do that. Okay? So let's stretch it, let it go, and here we go.

6:59So look, it's painting a picture of how X is varying with respect to time. Isn't that beautiful? Okay, so let's look at this picture a little bit more carefully. So it's turned around, and here we go. So we now have a plot. So this represents our X, that's our position, and this represents our time.

7:27And again, the period over here is the time it takes to complete one full cycle. So after this, it repeats itself and repeats itself. So that's our capital T. Okay. So what is X as a function of time? Well, we can see it right in front of our eyes. It's a sine graph, right?

7:47But of course, X won't be just sine of T; that's why I've left some space over here. Because, for one, sine swings between plus one and minus one. We want our graph, you know, our X to swing between plus A and minus A. So how do we do that? Well, we have to scale this by A.

8:06So now as sine swings between plus one and minus one, look, X swings between plus A and minus A. So that takes care of the amplitude. But the second thing to worry about is what goes inside over here. You can't just write as T because, for one, sine takes in values in radians, right?

8:23But our T is time. So it would be, for example, in terms of seconds for us. So the units won't match. But even if you forget about units, sine will repeat itself after exactly two pi radians, right? So if we just have little T in the argument, then this will be two pi, meaning the period would be two pi, which is not what we want.

8:45We want the period to be capital T, right? So how do we ensure that? Well, we multiply by two pi divided by capital T. Let's see how this makes sense. When T equals zero, this whole thing is zero, and we are over here. Okay? But now what happens when you substitute T equals capital T?

9:05Well, that capital T and this capital T cancels and you get sine two pi, which is zero, which means we come back over here. And so indeed now when time equals capital T, the sine resets, right? It would have completed one full cycle. If you put T equals two capital T, then you get four pi.

9:24That means you would be over here. It would have finished two cycles. Exactly what we want. And even the units work out now. The units of time cancel out, and we are left with radians. But one question we should have is that sine and cosine graphs actually look the same.

9:40They're just shifted a little bit. So couldn't we have called this a cosine graph as well? Yeah. It all depends upon what we choose our T equal to zero to be. If at T equal to zero, the mass is at the equilibrium position, then we get a sine graph. But at T equal to zero, if the mass was already at its positive amplitude, that's what we choose our T equal to zero to be, then we would get a cosine graph.

10:05And again, you can see, if you put T equal to zero, cos zero is one. So we are over here. So X would be plus A. And then if T equals capital T, again, that'll cancel out. You get cos two pi, which is again one. So you get again, X equals A, so we will be over here.

10:22So whether it's a sine or a cos purely depends on what the position of the mass is at T equals zero. So let's just go with this one. Okay? Now, because one over T is frequency, the reciprocal of period is frequency, we can also write this as A cos of two pi F times T.

10:42And we can also now substitute the expression for period over here. And if we do that and simplify, we'll get this. They're all the same things. All right, now that we know how exactly the position varies with time, here's a question. Can we figure out how the velocity would vary with time and how the acceleration would vary with time?

11:03First of all, since the position varies periodically, we would expect the velocity and the acceleration also to vary periodically. So we would expect them to also have maximum values, both in the positive and the negative direction. But how exactly would the graph be?

11:17Well, one way to do that is to remember that velocity is basically the slope of the position-time graph. So let's figure out the slope at few key points. Let's start with over here. At this point, what's the slope of the graph? Well, we can see the slope is zero, which means the velocity at that point must be zero.

11:38Okay, as we go ahead at this particular point, look, the slope is very high and negative. In fact, this is the maximum slope you'll ever get. So you have maximum velocity, but in the negative direction. Then a little later, the slope goes back to zero, then the slope becomes positive maximum.

11:57The slope goes back to zero, and then you get negative maximum, and the whole thing repeats. And so we can now kind of sort of see the graph you would expect would be something like this. Now of course, just from these points, we cannot predict the exact shape, but it happens to be sinusoidal, which means the graph will look like this.

12:16Okay, what about the acceleration time graph? Acceleration is the slope of the velocity time. So it'll be a great idea to pause the video and see if you can analyze the slopes over here and predict what the graph of acceleration time looks like. All right? So at this point, look, the slope is already negative maximum.

12:33So you start with negative maximum, then the slope goes to zero. So the acceleration goes to zero. You get a positive maximum slope. And then the slope goes back to zero, negative maximum, and goes back to zero. So you can kind of see the graph looks like that.

12:51All right, now let's bring it all together by going back to the horizontal spring. This is still the horizontal spring. I've just placed it vertically over here so that we can relate it better with the graphs. And this time we're gonna choose left side as positive.

13:05I mean, we can choose any direction as positive we want, okay? So we'll choose left side as positive because then notice this will be positive amplitude and this will be negative amplitude, just like what we have in our graphs. All right, so right now we are over here at the positive amplitude.

13:23That's the position. And the velocity is zero. This spring is at rest, so its velocity is zero. And because it's compressed to maximum, it'll have a maximum force in the negative direction. So maximum acceleration in the negative direction. Now let's release it, and here we go.

13:46Right now we are at X equal to zero, the equilibrium position. But what about its speed? It's the maximum because until this point, it was accelerating in the negative direction, in the same direction as it was moving. So it was speeding up. But after this point, it'll accelerate in the positive direction, in the opposite direction as it will be moving.

14:10And so it'll slow down, which means this is the maximum speed in the negative direction. And what about its acceleration? Well, it is at the equilibrium position, so there is no net force and therefore the acceleration is zero. Okay, if we let it play, the spring overshoots and comes to the negative amplitude.

14:31It comes back to rest again, and this time it has the maximum stretching. So it'll try to unstretch by putting a maximum force in the positive direction. And so you get a maximum acceleration in the positive direction. And then the whole thing will repeat in the positive direction.

14:51We are over here, back to the equilibrium, maximum speed, but in the positive direction. And again, zero acceleration. And eventually it'll come back to the original position that we started with. And under ideal situation, this will continue to go on forever.