https://youtu.be/oAXRxoGiIv0
PHYS 1101: Lecture Twenty-Three, Part Two
Okay. We’re on to our new material now. Let’s start with the big picture. My tree vision of where this all fits in. We’re now doing, we’re called springs and the simplest kind of motion that can result when an object is attached to a spring, which is called simple harmonic motion. We have a set of equations now–not a lot of new equations–but the result of these equations is very different motion. We have an equation for the force caused by a spring. So again, it’s always going to be an object attached to the spring that we care about and the force that that spring puts on that object is given by this equation, and I’ll define it carefully in this lecture.
In essence, x is the distance that this spring is stretched or compressed. If it’s stretched, x would be positive, if it’s compressed, x is negative. So I do have a sign that’s important to tell me the direction of the force. And you’ll see a little later in the lecture how that comes into play. K is called the spring constant. It’s just a property of the spring. The three equations I show below represent the x coordinate of this object as time goes on.
The next equation represents the velocity vector for this object as time goes on, versus time. And the last equation is the acceleration for this object as time goes on. Notice the acceleration is not a constant value anymore. It’s not just five meters per second2; it changes as this object oscillates back and forth and this equation properly tells you how it changes.
Okay. An important thing to emphasize are these quantities that are in front of these trig functions. For my x coordinate, in front of this cosine function I multiply, I have this factor of a. Cosine and sine only ever vary between plus and minus one. For different values here of my angle inside, I can have any number between -1 and 1, but nothing outside of that. It means when I multiply by a, my x position, it’s never larger than a or -a in the other direction. This is my maximum x value.
Similarly, here in green, let me highlight this, everything in front of the sine function here, a times Omega, that’s my maximum velocity. My velocity oscillates between minus a times Omega and plus a times Omega. So this is v max.
Similarly, down here, for everything in front of this trig function, represents my maximum acceleration value, a max. Okay. This collection of information about force and these kinematic equations are useful for describing a collection of problems that’s at the interface between the kinematic section describing motion and then Newton’s second laws, the information about force and the connection to acceleration. I circle this area here in purple. It’s just a special case, a particular kind of motion resulting from forces.
Okay. The motion we’re going to study now is in general called oscillatory motion. That’s where this, an object, rather than following a smooth trajectory or a straight line or even going around in a steady circle, it’s actually oscillating back and forth, right-left-right-left, perhaps up-down-up-down. The simplest kind of oscillatory motion is what’s called simple harmonic motion, and that’s the focus of this lecture. It’s mathematically the simplest kind to describe and it’ll give us a sense of what the implications are and the interesting physics aspects of this oscillatory motion. A lot of things in real life oscillate like this and it has very broad applicability.
It’s not always obvious because of two effects. One is that a lot of objects that are oscillating, our eye can’t follow it, move fast enough it’s actually oscillating at a very high rate. This is the case for example, with musical instruments, where if you pluck a guitar string you don’t see it but the string is actually vibrating and moving back and forth really fast. Cars going over speed bumps, you know if you have bad shocks, as the car hits the speed bump it starts bouncing up and down. Eventually it settles down as you go past the speed bump. If you’ve got good shocks, though, you know that it doesn’t go bounce up and down many times before it settles out.
So that’s another kind of oscillation that you might not notice so strongly because of what’s called damping of intentional designing so that this oscillation is not very extreme. But there are just lots of applications around us day-to-day life that involve an object oscillating back and forth.
Here are the key features that cause this type of motion. The first one is the idea that this object has what’s called an equilibrium position. Let’s think of these bullets here in the context of this meter stick. I want you to imagine hanging over the end of the table. You’re going to hold one end down and I’m going to consider plucking one end. By plucking, I mean with your finger you push this meter stick up and then you’re going to let it go and of course it’s going to spring back, and it’s going to go back and forth. It’s going to go oscillate fast, and eventually it’s going to settle back down to this nice steady straight position.
Okay. For this meter stick there is what’s called this equilibrium position. It naturally would be oriented straight hanging off this table. When you push against it, you can imagine you find a real resistant force that’s trying to oppose this pushing up that you’re doing before you let it go. That’s what’s called a restoring force. The molecules that are being deformed in this stick are resisting that and they’re applying a counter force to oppose that.
So as a result of having this equilibrium position of some external object disturbing it from equilibrium, you plucking this meter stick, the fact that then I have these internal forces inside this object that are trying to restore it to its equilibrium position, those forces that cause the thing to vibrate as it tries to then return to its equilibrium position. And the simplest form, as I say, is called simple harmonic motion, and in fact the best example of that is having a mass that’s just attached to a spring.
So just some object, here’s a wall, here’s an object, a mass, mass m, and it is attached to a spring. If I don’t touch the system, I leave it alone, this spring has a nice natural length that it wants to be at. If I pull this mass to the right, I’m going to stretch the spring, I’m going to feel the spring pulling back. If I let it go, then this force from the spring is going to pull that mass back as it goes back past this equilibrium position, its momentum is going to carry it through that and it’s going to start compressing the spring. A compressed spring puts the force in the direction opposite, to try to again push it back to the equilibrium position. What results is a motion that oscillates, it’s oscillatory motion about this equilibrium.
Okay. Let me show you with an animation here what I mean by that. This is an animation of a mass, this red object that’s attached at the end of the spring. In the horizontal direction, the only force on this object is going to be the pushing or pulling from the spring. This is very analogous to this meter stick, that meter stick above being pushed away from its equilibrium position. If you’ve, everybody’s held a spring and I think be able to think about the sensations that you’ve experienced when you stretch or compress a spring.
But first let me point out, as you take this spring and you pull it away from the equilibrium position, that the size of the force gets larger the more you stretch the spring. The arrow that’s drawn there is a force vector that’s representing that in this animation. If I’m at equilibrium where the spring is just naturally, its natural position, its natural length, there’s no force exerted on this object right-left. If I compress this spring, this spring will push to the right trying to restore the position back to equilibrium. The more I compress it, the larger that force is.
Back to equilibrium, the more I stretch the spring, the larger that force gets. If you think about it, you’ve probably had this same sensation when you play with a spring. You feel that the more you stretch it, you’ll feel it pulling back against your hand with the larger force, and the same with compression. Let’s offset this to a large position here so the spring is really pulling back and then let’s let this spring go, meaning I’m letting go of my force that I was exerting to hold the object there. Now the only force on it is this force due to the spring. This animation is showing you this oscillatory behavior of this object as it cycles back and forth.
At the same time, you’re seeing how the force of the spring changes. This is definitely not a constant force during this motion, agreed? The size of this force changes as the clock ticks on and the direction. So we need to study two aspects of this for this lecture. We first one to understand this equation that describes this force due to a spring. Then we want to look at the equation that describe the kinematics, the position versus time for this mass that’s oscillating back and forth, and then the velocity versus time, et cetera.
Okay. So here I’ve pasted into our lecture notes just a copy of that animation that you saw in real time.