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# Harmonic motion part 3 (no calculus)

## Video transcript

welcome back and if you are covering your eyes because you didn't want to see calculus I think you can open your eyes again there shouldn't be any significant displays of calculus in this video but but just to review what we went over we just said okay if we have a spring and I drew it vertically this time but pretend like there's no gravity or maybe pretend like we're you know reviewing we're looking at at the top of a table because we we don't want to look at the effect of a spring and gravity we just want to look at a spring by itself so this could be in deep space or or something else but we're not thinking about gravity but I drew it vertically just so that we can get more intuition for this curve what we started off saying is if I have a spring and zero x equals zero is kind of the natural resting point of the spring if I just let this mass if I didn't pull on the spring at all but I have a mass attached to spring and if I were to stretch the spring to point a we used we said well what happens well it'll it starts with with very little velocity but there's a rest root of force it's going to be pulling it back towards this position so that force will accelerate the mass accelerate the mass accelerate the mass until it gets right here and then I'll have a lot of velocity here but the change in but then it'll start decelerating and then it'll decelerate decelerate decelerate it's velocity will stop and it'll come back up and if we draw drew this as a function of time this is what happens it it starts moving very slowly accelerates at this point at x equals zero it has its maximum speed so the rate of change of velocity or the rate of change of a position is fastest and we can see the slope is very fast right here and then and then we start slowing down again slowing down until we get back to this bottom a and then we keep going back and up and down up and down like that and we showed that actually the equation for the masses position as a function of time is X of T and we use a little bit of differential equations to prove it but this equation not that I recommend that you memorize anything but this is a pretty useful equation to memorize because you can use it to pretty much figure out anything about the position or the of the mass at any given time or the frequency of this oscillatory motion or anything else you even you the velocity if you know a little bit of calculus you can figure out the velocity anytime the object and that's that's pretty neat so so what can we do now well let's try to figure out the period of this of this of this Oslo oscillating system and just so you know I know I put the label harmonic motion all these this is simple harmonic motion simple harmonic motion is something that can be described by a trigonometric function like this and it just oscillates back and forth back and forth and so what we're doing is harmonic motion and now let's figure out what this period is remember we said that after T seconds it gets back to its original position and then after another T seconds it gets back to its original position let's figure out what this T is and and that's essentially its period right what's the period of a function especially its how long it takes to get back to your starting point or how long it it takes for the whole cycle to to happen once so what is this T so let me ask you a question what are all the points that if this is a cosine function right what are all of the points at which cosine is equal to 1 or this function would be equal to a right because whenever cosine is equal to 1 this whole function is equal to a it's these points well cosine is equal to 1 when so theta let's say when is cosine when is cosine of theta equal to 1 so at what angles is is this true well it's true with eita is equal to 0 all right cosine of 0 is 1 cosine of 2 pi is also 1 right we could just keep going around that unit circle you should watch the unit circle video if this makes no sense to you or the graphing trig functions it's also true at 4 pi really any multiple of 2 pi this is true right cosine of that of that angle is equal to 1 so the same thing is true this function X of T is equal to a at what points X of T is equal to a whenever this expression within the cosines within the whenever this expression is equal to 0 2 pi for pi etc and this first time that it cycles right from 0 to 2 pi from 0 to 2 T that'll be at 2 pi right so this whole expression will equal a when K and that's these points right that's when this function is equal to a and let it happen again over here someplace when this little internal expression is equal to 2 pi or really any multiple of 2 pi so we could say when so X of T is equal to a when square root of K over m times T is equal to 2 pi or another way of thinking about it is let's multiply both sides of this equation times the inverse of the square root of K over m and you get T is equal to 2 pi times the square root and it's going to be the inverse of this right of M over K and there we have the period of this function this is going to be equal to 2 pi times the square root of M over K so if you if someone tells you well I have a I have a spring that I'm going to pull from some you know I'm going to stretch it or compress it a little bit then I let go what is the period how long does it take for the spring to go back to its original position it'll keep doing that and so maybe we have no friction or no gravity or any air resistance or anything like that air resistance really is just a form of friction you could immediately if you memorize this formula although you should know where it comes from you should you could immediately say well I know how long the period is it's two pi times M over K that's how long is going to take the spring to get back to complete one cycle and then what's what about the frequency if you wanted to know cycles per second well that's just the inverse of the period right so if I want to know the frequency that equals 1 over the period right period is given in seconds per cycle so frequency is sike second sorry frequency is cycles per second so and this is seconds per cycle so frequency is just going to be one over this which is one over two pi times the square root of K over m that's the frequency but I've always had trouble memorizing this and this you always lose K over m M over K all of that all you have to really remember eyes is this and even that you might even have an intuition as to why it's true you can even go to the differential equations if you want to reprove it to yourself because if you have this you really can answer any question about the position of the mass at any time the velocity of the mass at any time by taking the derivative or the period or the frequency of the function as long as you know how to take the period and frequency of trig functions and you could watch my videos or watch my trick videos and to get a refresher on that one thing that's pretty interesting about this is notice that the frequency and the period right this is the period of the function that's how long it takes to do one cycle this is how many cycles it does in one second both of them are independent of a so it doesn't matter I could stretch it only a little bit like there and it'll take the same amount of time to go back and come back like that as it would if I stretch it a lot it would just do that if I stretch the if I stretched it just a little bit the function would look like this I'll make sure I do this right I'm not doing that right edit undo edit undo if I just do it a little bit the amplitude is going to be less but the function is going to essentially do the same thing just going to do that it's going to take the same amount of time to complete a cycle I'll just have a lower amplitude so that's that's interesting to me that how much I stretch it it's not going to make it take longer or less time to complete one cycle that's interesting and so if I just told you that I actually start having a object compressed right so in that case let's say my a is minus three I have a spring constant of let's say K is oh I don't know 10 and I have a mass of two kilograms then I can immediately tell you what the equation of the position as a function of time at any point is it's going to be X of T will equal I'm running out of space so X of T would equal this is basic substitution minus 3 cosine of 10 divided by 2 right K over m is 5 so square root of 5 t square root of 5 t I know that's hard to read but you get the point I just substitute it back but the important thing is to know is this this is I think the most important thing and then if given a trig function you have trouble remembering how to figure out the period of frequency although I just think about when does when does this when does this expression equal 1 and then and then you could figure out it's well 1 2 right when does it equal 1 or 1 is it equal zero and you can figure out its period if you don't have it you can memorize this formula for period and this formula for frequency but I think that might be a waste of your brain space anyway I'll see you in the next video