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Video transcript

so we saw that for a mass oscillating on a spring there's a certain amplitude and that's the maximum displacement from equilibrium but there's also a certain period and that's the time it takes for this process to reset in other words the time it takes for this mass to go through an entire cycle but what do these things depend on we know the definitions of them but what do they depend on well for the amplitude it's kind of obvious the person pulling the mass back whoever or whatever is displacing this mass is the thing determining the amplitude so if you pull the mass back far you've given this oscillator a large amplitude and if you only pull it back a little bit you've given it a small amplitude but it's a little less obvious in terms of the period what does the period depend on who or what determines the period maybe it depends on the amplitude so let's should check if I asked you if I asked you if I pulled this back farther right if I increase the amplitude farther will that change the period of this motion so think about it some of you might say yes it should increase the period because look now it has farther to travel right instead of just traveling through this amount well that looked horrible stead of just traveling through this amount back and forth it's got to travel through this amount back and forth since it has farther to travel the period should increase but some of you might also say wait a minute if we pull this mass farther we know Hookes law says that the force is proportional the force from the spring proportional to the amount that the spring is stretched so if I pull this mass back farther there's going to be a larger force that's going to cause this mass to have a larger velocity when it gets to a larger speed when it gets to the equilibrium position so it's going to be moving faster than it would have so since it moves faster maybe it takes less time for this to go through a cycle but it turns out those two effects offset exactly in other words the fact that this mass has farther to travel and the fact that it will now be traveling faster offset perfectly and it doesn't affect the period at all this is kind of crazy but something you need to remember the amplitude changes in the amplitude do not affect the period at all so pull this mass back a little bit just a little bit of an amplitude it'll oscillate with a certain period let's say three seconds just to make it not abstract let's say we pull it back much farther it should oscillate still with three seconds so it has farther to travel but it's going to be traveling faster and the amplitude does not affect the period for a mass oscillating on a spring this is kind of crazy but it's true and it's important to remember this amplitude does not affect the period in other words if you were to look at this on a graph so say you graph this put this thing on a graph if we increase the amplitude what would happen to this graph we'll just stretch this way right we'd have a bigger amplitude but you can do that and there would not necessarily be any stretch this way if you leave everything else the same and all you do is change the amplitude the period would remain the same the period this way would not change so changes in amplitude do not affect the period so what does affect the period I'd be like all right so the amplitude doesn't affect it what does affect the period well let me just give you the formula for it so the formula for the period of a mass on a spring is that the period here is going to be equal to this for the period of a mass on a spring turns out it's equal to 2pi times the square root of the mass that's connected to the spring divided by the spring constant that is the same spring constant that you have in Hookes law so it's that spring constant there it's also the one you see in the energy formula for a spring same spring constant all the way this is the formula for the period of a mass on a spring now I'm not going to derive this because the derivations typically involve calculus if you know some calculus you want to see how this is derived check out the videos we've got on simple harmonic motion with calculus using calculus and you can see how this equation comes about it's pretty cool but for now I'm just going to quote it and we're going to sort of just take a tour of this equation so the 2pi that's just a constant out front and then you've got mass here and that should make sense why why does increasing the mass increase the period look at that's what this says if we increase them ass we would increase the period because we'd have a larger numerator over here that makes sense because a larger mass means that this thing has more inertia right increase the mass this mass is going to be more sluggish to movement more difficult to whip around if it's a small mass you can whip it around really easily if it's a large mass very massive it's going to be difficult to change its direction over and over so it's going to be harder to move because of that it's going to take longer to go through an entire cycle because this spring is going to find it more difficult to pull this mass and then slow it down and then speed it back up because it's more massive it's got more inertia that's why it increases the period that's why it takes longer so increasing the period means it takes longer for this thing to go through a cycle and that makes sense in terms of the mass how about this K value that should make sense to if we increase the K value look at increasing the K would give us more spring force for the same amount of stretch so if we increase the K value this force from the spring is going to be bigger so it can pull harder and push harder on this mass and so if you exert a larger force on a mass you can move it around more quickly and so larger force means you can make this mass go through a cycle more quickly and that's why increasing this K gives you a smaller period because if you can whip this mass around more quickly it takes less time for to go through a cycle and the period is going to be less that confuses people sometimes taking more time means it's going to have a larger period sometimes people think if this mass gets moved around faster you should have a bigger period but that's the opposite if you move this mass around faster it's going to take less time to move around and the period is going to decrease if you increase that K value so this is what the period of a mass on a spring depends on note it does not depend on amplitude so this is important no amplitude up here change the amplitude doesn't matter those effects offset only depends on the mass and the spring constant again I didn't derive this if you're curious watch those videos that do derive it where we use calculus to show this something else that's important to note this equation works if the mass is hanging vertically so if you have this mass hanging from the ceiling right something like this and this mass oscillates vertically up and down this equation would still give you the period of a mass on a spring you'd plug in the mass that you had on the spring here you'd plug in the spring constant of the spring there this would still give you the period of the mass on a spring in other words it does not depend on the gravitational constant so little G doesn't show up in here little G would cause this thing to hang downward at a lower equilibrium point but it does not affect the period of this mass on a spring which is good news this formula works for horizontal masses works for vertical masses gives you the period in both cases so recapping the period of a mass on a spring does not depend on the amplitude you can change the amplitude but it will not affect how long it takes this mass to go through a whole cycle and that's true for horizontal masses on a spring and vertical masses on a spring the period also does not depend on the gravitational acceleration so if you took this mass on a spring to Mars or the moon hung it vertically let it oscillate if it's the same mass in the same spring it would have the same period doesn't depend on what the acceleration due to gravity is but the period is affected by the mass on a spring bigger mass means you would get more period because there's more inertia and it's also affected by the spring constant bigger spring constant means you'd have less period because the force from the spring would be larger
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