 ## Physics

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17. Periodic Motion

# Intro to Simple Harmonic Motion (Horizontal Springs)

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## Intro to Simple Harmonic Motion 3m
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Hey, guys. So for this video, I'm gonna introduce the motion of simple harmonic motion, the position, velocity and acceleration. And I'm also gonna give you guys some variables and definitions of useful for the rest of the chapter. So let's get to it. So the most common type of simple harmonic motion that will study in this chapter, or sometimes you'll see this oscillation is the mass spring system. So that's where you attach some fast to some spring. So if I pull this thing all the way back here, right, So I pull this on a spring and I let it go, The only force that's gonna be acting on that thing is the Spring Force, and that's equal to K X, Right? So I'm gonna pull this thing back. I'm gonna let it go, and basically this thing is gonna go all the way back to the equilibrium position. But it's gonna overshoot it and then land on the other side. So it's gonna go like something like that. And so basically, after that, it's just gonna oscillate between those two points forever, right? Those maximum displacements here, So the displacement is maximum on the left side, here and on the right side, right here. And so that the equilibrium position, We know that the X is equal to zero. So we call these two points on the outsides, the amplitude. So we've got plus a and minus a right here. And the amplitude is just the maximum displacement that this object has. It's always the initial push or pull that you apply against this object. So if you push it out to some distance and release it, that's the amplitude. So what is the velocity look like? When you do this, we're gonna pull this thing all the way back, and then you're just gonna let it go from rest. So that means that at this point, right here at the amplitude, the velocity is equal to zero. So then what happens is this thing starts speeding up faster and faster and faster goes through the equilibrium position. And then when it gets to the other side right here, it has to stop and then turn around. So that means that the velocity at this other amplitude is also equal to zero. And then what happens is as it's going back through here, the velocity is maximum here, and then it just keeps on doing that over and over again. So we have maximum velocity at the center, and so does this whole entire cycle over and over again. So the two variables that are related to that are the period and the frequency. The period, which is the letter T is the time that it takes for you to complete one complete cycle for you. Finish one complete cycle, and what's related to that variable is the frequency. So these things are just related by in verses of each other. So you see that the frequency is just one divided by T, So if you ever have one of them, you can get to the other one. So, for example, if I have a T that's equals, like two seconds on the frequency is just gonna be equal to one half of a cycle per second. And the units for that are gonna be in hurts now. Another variable that's related to the frequency. But not the same thing is the angular frequency. That's the letter. Omega. It's the units are rats, Radiance per second and all Omega is is just the frequency times two pi. We could also write it in a different way because the period and the frequency are in verses of each other. Okay, so what are the forces look like? A these three points. So when I pull it all the way back, the spring wants to push harder and harder, harder against me. So at this maximum displacement here that that means the force is also going to be maximum, Right? So take a look at K X if X is maximum that that means that F s also has to be maximum. And so we know that the maximum forces are gonna be here a the end points. And because of the equilibrium position, the X is equal to zero. Then that means that the force is equal to zero. Now, what about the acceleration? Will the acceleration and the force are related just by f equals m A. So if f is Max here, then that means the acceleration is Max here and here. And if f equals zero, then that means a is equal 20 So it looks like X f and A are all related to each other, and they're all in sync. So we've got X is all is zero F zero and a zero in the middle, whereas the weird one is this velocity here. So that's maximum when the other ones are zero. So let's take a look at an example here and see what we can find out.
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## Example 2m
Play a video:
Hey, guys, let's take a look at this. This example here. We've got a mass on a spring and it's pulled 1 m away from its equilibrium position. So we've got 1 m and we're just gonna draw a little box here, and then you release it from rest. But this is a mass spring system, right? So it's just gonna go and reach the other side where now the distance here, the displacement is gonna be negative 1 m, and that's just gonna go back and forth forever. So here in this first part were asked to calculate the amplitude. But we know that the amplitude is just the maximum displacement on either side. So that means that the amplitude of this is just 1 m. So the second question now were asked to find the period. So that is going to be that letter t. So what is the period look like? What we're told that the mass takes two seconds to reach the maximum displacement on the other side. So if you release it over here, then this time that it takes for you to go all the way over to the other side was equal to two seconds. Now the question is is that the whole period? No, it's not because we said that the period is equal to the time that it takes you to complete one whole entire cycle here. So this two seconds really only represents Ah, half period. This thing has to go all the way back to the other side for another two seconds. And that's gonna be another half period. Which means that these points in between here are actually quarter periods. Right, So this is a quarter, This is a quarter. These are all quarter periods, and this is the smallest sort of division that you could make. So you've got half periods and quarter periods, so we're told that it's two seconds to get to the other side, which means that the full period of this motion is going to be four seconds. Now we've got in this third part. Here is the angular frequency off the motion. So how do we relate angular frequency to the period? Well, we've got an equation appeared that could do that. So if we're looking for the angular frequency Omega we can use is either two pi times the frequency or we can use to pie divided by the period which we actually know. So I'm just gonna go ahead and use that I've got omega equals two pi divided by the full period of cycle, which is for and what you should get is 1 57. And that's gonna be radiance per second. So that's it for this one. Let's keep going with some more examples.
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Problem

A mass-spring system with an angular frequency ω = 8π rad/s oscillates back and forth. (a) Assuming it starts from rest, how much time passes before the mass has a speed of 0 again? (b) How many full cycles does the system complete in 60s?

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## Equations of Simple Harmonic Motion 7m
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Problem

A 4-kg mass on a spring is released 5 m away from equilibrium position and takes 1.5 s to reach its equilibrium position. (a) Find the spring’s force constant. (b) Find the object’s max speed.

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example

## Example 4m
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Problem

What is the equation for the position of a mass moving on the end of a spring which is stretched 8.8cm from equilibrium and then released from rest, and whose period is 0.66s? What will be the object’s position after 1.4s?

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## Example 3m
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