Pendulums

- [Instructor] So, as far as simple harmonic oscillators go, masses on springs are the most common example, but the next most common example is the pendulum. So, that's what I wanna talk to you about in this video. And a pendulum is just a mass, m, connected to a string of some length, L, that you can then pull back a certain amount and then you let it swing back and forth. So, this is gonna swing forward and then backward, and then forward and backward. It oscillates just like a simple harmonic oscillator and so that's why we study it when we study simple harmonic oscillators. And technically speaking, I should say that this is actually a simple pendulum because this is simply a mass connected to a string. It's not complicated. You could have more complicated examples. Let's say you connect another string, with another mass down here. This gets really complicated. In fact, it gets, what physicists call chaotic, which is kind of cool. If you've never seen it, look up double pendulum, it's pretty sweet. But really complicated to describe mathematically. So, we're not gonna bother with that. We've got enough things to study by just studying simple pendulums. We can learn a lot about the motion just by looking at this case. So, what do we mean that the pendulum is a simple harmonic oscillator? Well, we mean that there's a restoring force proportional to the displacement and we mean that its motion can be described by the simple harmonic oscillator equation. So, if you remember that was described by an equation that looked like this, X, some variable X is a function of time was equal to some amplitude times cosine or sine, I'm just gonna write cosine, of two pi divided by the period, times the time and you can if you want add a phase constant. I'm not gonna write it 'cause usually you can get away with not using that one. So, this is the simple harmonic oscillator equation. So, how would I apply this equation to this case of a pendulum? Well, I wouldn't use X. The far more useful and common example of using a variable to describe a pendulum is the angle that the pendulum is at. So, consider the fact that this mass is gonna be at different angles at different moments in time. So, it starts over here, maybe it's at like, 30 degrees and it swings it's only at like 20 and then 10 and then zero 'cause we're measuring angles from the center line. And then it swings through, maybe it's at negative 10, negative 20, negative 30 and then this whole process repeats. So, instead of using X, we're gonna use theta. So, this is gonna be an angle as a function of time. So, I'll write theta as a function of time is gonna equal some amplitude, but again, since I'm measuring theta, my amplitude is not going to be a distance in X, or a displacement in X, this is gonna be not the maximum regular displacement, it's gonna be the maximum angular displacement from equilibrium right here. This line here would be equilibrium 'cause if you put the mass there and let it sit it would just continue to sit there, there'd be no net force on it. Only when you displace the mass from this equilibrium position does it have a restoring force. So, this would be the maximum, I'll just call it theta maximum, 'cause this is the maximum angular displacement when you pull this back, the maximum angle you pull it to, whatever that it. Maybe it's 30 degrees, maybe it's 20, that would be the angle that I plug in here. And then we'll multiply by cosine and it will have the same argument in here. Two pi over whatever the period is, and the period is the time it takes for this pendulum to reset or to complete a whole cycle and we always have to multiply by T, that's our variable, that's what makes this a function, it's a function of time. Alright, so I gotta come clean about something now. Technically speaking, the simple pendulum is not a perfect simple harmonic oscillator, it's only extremely close to being a simple harmonic oscillator. In fact, for small angles, this will only be off by very small amounts, like less than a per cent. So, because of that, we often treat a simple pendulum as a simple harmonic oscillator, but technically speaking it only works really well if you're less than say a certain amount, say 20 degrees. As you get to larger maximum amplitudes, this is gonna deviate more and more. It'll still be reasonably close, maybe within like 20 per cent, but only for small angles is it extremely close. But if you are at small angles. So, if you're considering a pendulum that has small angles. Like, maybe this is only 20 degrees or less, that pendulum would be described really well by this equation because it would be extremely close to being a simple harmonic oscillator. Alright, so let's assume we're in that small angle approximation where this amplitude is small. What can we say? Well, one question we can ask is what's the period of this pendulum gonna depend on? Right, this period here, what could we change that would change this period here? So, what might this depend on? My first guess might be, well, maybe it's the mass. So, let's think about this. If we increased the mass on this pendulum, do you think that would increase the period or decrease the period or leave it the same? Some people might say, well, I think an increase in mass would increase the inertia of this system. Right, it's gonna be harder to move. When the mass of something goes up, it's more sluggish to accelerations, it's more difficult to move around and change its direction. That means it should take longer to complete a cycle. Maybe that means that the period should increase because the time would increase. But other people might say, wait a minute, if we increase the mass, that would increase the gravitational force. Right, gravity's going to be pulling down harder now on this mass, and gravity is the force that's gonna be restoring this mass back to equilibrium. Gravity's gonna be pulling down and if it pulls down with a greater force, you might think this mass is gonna swing with a greater speed and if it's got a greater speed, it'll complete this cycle in less time because it's moving faster, and since it takes less time, you might think that the period goes down, but these two effects exactly cancel. So, the fact that the mass is gonna have more inertia, with greater mass, that means it's harder to move, and the force is gonna increase due to the force of gravity getting larger. Those offset perfectly and this mass will not affect the period. So, it turns out, it's kinda weird, changing the mass on here does not affect the period at which this swings back and forth. So, imagine this. So, if you go get on a swing at the park, and you swing back and forth, and then a little kid, tiny kid, five year old comes on and swings back and forth, they should have the same period of motion as you do because the mass at the end here does not affect the period. So, that's a little weird but it's true and you should keep that in mind. Mass does not affect the period. So, what does affect the period? Well, I'm just gonna write the formula down for you. I'm not gonna derive this. The derivation requires calculus. It's an awesome derivation. If you know calculus, you should go check it out. But just in case you haven't seen calculus, I'm just gonna write this down, give you a little tour of this equation. Show you why it should make sense and hopefully give you a little intuition about why the variables are in here that they are. So, the first variable is L. L goes on top, the length of the string, and then the acceleration due to gravity, little g goes on the bottom. So, why is this the formula? Well, the two pi is just a constant, you get a square root. L is on top, that means if you increase the length of the string, you're gonna get a greater period. So, increasing the length should increase the period. Why is that? Well, think about this. A mass on a string rotating back and forth, if there's rotation, a quantity that's useful to think about is the moment of inertia. So, the moment of inertia of this mass on a string would be equal to, this is a point mass, rotating about an axis, so the axis of rotation is this point right here. And a point mass rotating around an axis is just given by mr squared. That would be the moment of inertia. But this r is the distance from the axis to the mass, so this is just mL squared. This is the moment of inertia. And look it, if we increase the length, we increase the moment of inertia. So, bigger L gives us bigger moment of inertia. What does that mean? Moment of inertia is a measure of how difficult it is to angularly accelerate something. So, it's a measure of how sluggish this mass is gonna be to changes in its angular velocity. So, bigger moment of inertia means it's gonna be harder to take this mass and whip it around back and forth and change its direction. So, since it's harder to move this mass around, it's gonna take longer to move it back and forth, that's why bigger length means bigger moment of inertia and bigger moment of inertia means it takes longer to move this thing back and forth, that's why the period gets bigger. Now, some people out there might object. If you're really clever, you might say, wait a minute, if this length increases, the thing causing this to angularly accelerate is the torque, and I know the formula for torque. The formula for torque looks like this. Torque is rf sine theta. And r is the distance from the axis to the point where the force is applied. So since gravity's supplying the torque, that r would also be this L. It'd go from the axis to the point where gravity's applied, so I'd have L times the force of gravity times sine theta. So, you might say, look, if the length increases, so would the amount of torque. So, I've got more torque trying to make this thing move around, I've also got more inertia, so it's harder to move around. Do those offset like so many of these other things offset? They don't. Look it, this torque will increase but it only increases with L, it's only proportional to L. This moment of inertia's proportional to L squared. So, if you double the length, you've quadrupled how difficult it is to move this mass around but you've only doubled the ability of this torque to move it around. That's means it's gonna take longer to go through a whole cycle and that period is gonna increase. This larger torque is not gonna compensate for the fact that this mass is harder to move as there's more inertia to the rotation of this mass. Alright, so that's why increasing the length, increases the period. But why does increasing g, the gravitational acceleration, decrease the period? Well, think about it, if I increase the gravitational acceleration, so I take this pendulum to some planet that's extremely dense or massive and it's pulling down with a huge force of gravity. So, bigger g means a bigger force of gravity, pulling downward on this mass, that gives me a larger restoring force. So, a larger force means it's gonna pull this mass more quickly, it's gonna have larger acceleration, that means it's gonna have a larger speed, it's gonna move back and forth faster, and if it moves faster, it takes less time to complete a cycle. That's why increasing the gravitational acceleration increases the force and it decreases the period. Essentially, if you're cool with torque, if you know about torque, you increased the force that increases the torque which would increase the angular acceleration and it would take less time for this thing to go back and forth, that's why the period goes down if you increase the gravitational acceleration. Now, if you're really clever, you'll be like, wait a minute. This is just like the formula for the mass on a spring. If you take the period from a mass on a spring, it was two pi, square root, something over something, and the term on top for the mass on a spring was the mass that was connected to the spring, and the term on the bottom, was the spring constant. And so you might say, wait, this is the same idea. Increasing the mass is just increasing the inertia of that system. That's why it's taking longer to go through a cycle. Just like over here. Increasing the length is increasing the inertia, at least the rotational inertia, the moment of inertia of that system, so it takes longer to go through a cycle. And you might say, increasing the k value, that's increasing the force on the system, and if you increase the force on the system, you make that system have a larger acceleration, greater speeds takes less time to go through a period, that's why this force constant k for the spring appears on the bottom, same as this g. Increasing the g, increases the force on the system which gives you a larger acceleration, greater speeds takes less time to go through a period. So, these formulas are very similar and they're completely analogous. There's an inertia term on top, a force term on the bottom, and they both affect the period in the same way. One more thing you should notice, amplitude does not affect the period of a mass on a spring, and the amplitude, this theta maximum will not affect the period of a pendulum either, as long as your amplitudes are small. So, we've gotta assume we're in this small amplitude region where this mass on a string is acting like a simple harmonic oscillator. And if that's true for small angles, the amplitude does not affect the period of a pendulum just like amplitude doesn't affect the period of a mass on a spring. Let me tell you about one last thing here. This simple pendulum only acts like a simple harmonic oscillator for small angles. And that means this period formula for the pendulum is only true for small angles. But how small does the angle have to be? So, to give you an idea, let's say your theta maximum, this amplitude for how far back you pull this pendulum to start it, is, let's say, less than 20 degrees. If you pull it back less than 20 degrees, the amount that this formula is gonna be off by compared to the true period of the pendulum, is gonna be less than one per cent. So, this formula gets you really close to the true actual value of the pendulum. I mean, it's really close to being a simple harmonic oscillator here. And let's say the theta maximum was less than 40 degrees, you're still only gonna be off by less than three per cent. So, the value you get from this equation is only off from the true value by three per cent. And let's say your theta maximum was less than 70 degrees, you get all the way up to 70 degrees, the error that this formula's gonna be is still less than 10 per cent. So, not nearly as good but still not bad. So, this formula gives you the period of the pendulum. It works really well for small angles. As that angle gets bigger, the value you get from this formula will deviate from the true value by more and more. So, recapping, for small angles, i.e. small amplitudes, you could treat a pendulum as a simple harmonic oscillator, and if the amplitude is small, you can find the period of a pendulum using two pi root, L over g, where L is the length of the string, and g is the acceleration due to gravity at the location where the pendulum is swinging.