Jonathan Osbourne

**PhD., University of Maryland**

Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

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**Pendulum motion** basically depicts the motion of a mass hanging from a string that moves back and forth. The variables in pendulum motion are the mass, the length of the string, and the location, which is measured by an angle. Forces acting on the mass in pendulum motion are tension and gravity. The force that is perpendicular to tension is expressed in the formula *F = m x g x sine of the angle*.

So let's talk about pendulums. So what is a pendulum? Well a pendulum is just a mass hanging from a string that moves back and forth so in order for us to describe this motion we need to give ourselves some parameters so we've got a length of this string we've got a mass m that's hanging from the string and then the way that we're going to measure the location of this mass is by an angle and it's standard to measure the angle off the vertical alright so here I've got the pendulum sitting there and what forces are acting on it? Well I draw free body diagram we've got the string acting on it and that puts a tension on it t and we have the weight acting on it. Now notice that this cannot be equilibrium because this tension force and this weight force can't cancel each other out. Now the thing about strings is that this tension is going to be whatever it has to be so there's no motion in this direction. There can't be any motion in that direction because the string has to keep its length it doesn't get to get longer or shorter it does like this. Alright so what that means is that it's the part of the weight that is perpendicular to the tension that's going to generate acceleration. Alright, well this angle wow this angle right here is theta that means this angle is theta that means this angle is theta because of vertical angles from Geometry. Alright now this angle is theta this line is mg long so that means that this part the part then is opposite the angle must be mg sine theta. Alright so we've got a force mg sine theta that's pulling to try and make the angle smaller alright now that's then kind of complicated but we can make a nice simplification when theta is small.

Alright if theta is small, then it turns out that sine theta is approximately equal to theta now this only works in radians you've got to be in radians but that's actually what we're going to want to be in anyway so let's look at why this is true real quick. So over here in the unit circle I've got a small angle theta. Now where's sine theta? Sine theta is right here it is opposite theta alright where's theta? Well theta is that angle that since we're in radians that means that this distance right here is also theta so that's theta and this is sine theta. Now as theta gets smaller you can see that sine theta will get closer and closer and closer to theta because I've just get like over here and they'll be the same length so that's why that's true it's called the small angle approximation. Alright so we've got sine theta about theta and that means that the force is equal to mg theta its proportional to theta. Now this reminds us of something, remember that theta is telling us the location of the mass here so that means that we've got a force that's proportional to location but that's Hooke's law so that means that we've got simple harmonic motion so x is not theta because theta is not a length and x has to be a length but x will be the length times theta so that means that my spring constant will be mg over l alright? So that means that I'm going to have a period of motion which is 2 pi times the square root of l over g, now I could solve this for l if I want to and write it l equals gt squared over 4 pi squared alright we can actually use this to measure the length of a pendulum so let's look we got a pendulum right here now what I want to do is I want to displace this pendulum a little bit and then set it into motion so let's watch.

Alright so this is the amplitude that angle right there which is small fairly small you know if I let this go there it goes in simple harmonic motion. Now remember what simple harmonic motion meant, it meant that the period was supposed to be totally independent of the amplitude so let's just check this out so I want to go a little bit and I want to calculate the period so that means that I need to count the seconds until this mass gets back to the same point that's just difficult for me to do that for just one so let's do it for 2 or 3 whole oscillations. Alright so here we go I'm going to let it go and then I'm going to count seconds I'm going to try to do this 1, 2, 3, 4 alright so its about 4 seconds alright let's see what happens if we've got a little bit of a bigger amplitude 1, 2, 3, 4 alright almost the same, little bit longer period it takes a little bit longer because when the angle is larger it actually makes this approximation not true anymore and it makes it so that the period is a little bit longer than what's given by this formula, I was probably not very accurate in counting seconds either but we'll see alright so let's try to measure the length of this pendulum from my period so I'm again going to measure 2 periods alright 1, 2, 3, 4 alright so little bit more than 4 seconds is 2 periods alright? So let's take it as 2t equals 4 let's say point 2 seconds alright?

So that means that the period is 2.1 seconds alright so now I need to plug this into here and carry out all the calculations. Alright but I'm tired I don't want do that alright I just want to estimate so one thing that's neat about this formula is the inner SI units g is 9.8 meters per second squared alright? Pi squared, pi squares is almost 10 it's actually very very very close to 9.8 so these guys cancels out it's wonderful wonderful wonderful fact they didn't cancel exactly but I mean jeez my period is not exact either you know I go- got it so what's happening is the length is the period squared over 4 and this always will work you know approximately for any length pendulum as long as we don't move it too far alright as long as its small amplitude oscillations alright and this is going to give us the period sorry the length in what? Well we did this in SI units so that means that the period got to be in seconds and then that's going to tell me a length in meters alright well t, alright again I'm tired I don't I don't want to square 2.1 let's just square 2, alright 2 squared is 4, 4 divided by 4 well I can do that is 1.

So that tells me that this pendulum has a length of about 1 meter it's a little bit longer than that but I know that because the period was a little bit bigger than 2 seconds alright that's a really nice easy way that you can measure the length of very long things up to the top of the ceiling you know I've actually done that in a restaurant once where they're going to hang our lamp down no because they didn't like that very much but you just move it a tiny bit, the bigger it is the less accurate it is so you don't want to swing the lamp all over the place you just kind of go a little bit and it will go back and forth period get that seconds squared over 4 that's the length of the pendulum and that's pendulums.