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# Definition of amplitude and period

## Video transcript

alright so there's some terminology you got to get used to when dealing with simple harmonic oscillators because people and books and teachers and professors are going to throw these terms around like crazy and if you're not used to them it could all sound like mathematical witchcraft so the first term you got to know is that if you displace a mass from equilibrium and why wouldn't you do that that's how you get the thing to oscillate by displacing it from equilibrium the maximum magnitude of displacement so this amount right here whatever that distance is here is called the amplitude so we represent the amplitude with a capital A and it's called the amplitude and it's defined to be the maximum magnitude of displacement for that oscillator so if this mass only ever makes it this far away so from here to here and I'm drawing arrows but this is not a vector it's the magnitude right the magnitude of the displacement so it's the magnitude of a vector so it's always positive so we can draw that over here as we if we want to we can just say that this amplitude here this would also be the amplitude because we're just talking about the maximum magnitude of displacement so it's going to get displaced equally on either side of the equilibrium position and that maximum amount is called the amplitude so in other words if I pulled this mass back 20 centimeters then that means 20 centimeters would be the amplitude or if you wanted it in meters it'd be point 2 meters and then that means when it shoots through the equilibrium position it would also come over here and compress this spring by 20 centimeters on this side so it's always equal on both sides now there's another term that you got to get used to and that's the period so the period is represented with a capital T why is the period represented with a capital T when there's no T in the word period I'm not sure but capital T is kind of like time so T might stand for time maybe they thought that was a good idea because what the period means is the time required for an entire cycle so what does this mean an entire cycle what we mean is that you've got oscillations going on and so this process is repeating itself so in other words if you start the mass over here it's going to eventually make it over to this and over here right it goes over here compresses the spring then it's going to come back the time it takes us a little hard to see sorry let me draw that up here so the time it takes for it to go to here and then come all the way back after this happens the whole thing just repeats now it's back here the spring is going to pull it back to the left and then go to the right it's going to pull it back to the left push it back to the right so this process is repeating itself there's not something new happening it's just the same process over and over the time it takes to go through one entire cycle ie the time it takes to reset essentially once this entire system resets to the same position that's the period and so that's going to be the same whether I count as is from this point back to that point or if I imagine just starting my clock here from this point it's going to go over to here then it's going to come back here that would also be the period because it's the time it took to reset so the time it takes for this process to reset is what we call the period it'll be given in seconds and so for the sake of making this a little less abstract let's say for example the period of this mass on a spring was 6 seconds what would that mean it would mean that it took 6 seconds for the mass to go from this point and then all the way back to that point resetting itself now this is getting kind of messy and honestly for that reason people often draw what the simple harmonic oscillator looks like on a graph it turns out to be particularly elegant and useful to represent these ideas on a graph because look it if you just drew what's happening you'd be like all right the mass goes here and then there and then there and then they are you're drawing all over yourself and so that's kind of ugly looking it's better to represent this on a graph what would that look like so let me get rid of this it would look like this you'd have a graph of the horizontal position X so what does that mean that means this so we're essentially graphing what this is this is X the horizontal position has a function of time now already you might be upset you might be like wait a minute why did we stick the horizontal position on the vertical axis isn't that a dumb thing to do well perhaps but long ago physicists decided you know what time if x involved we're sticking that bad boy on the horizontal axis this is just designated this is just by default gonna go on the horizontal axis so if you have anything else you want to graph with it that's gotta go on the vertical axis and so unfortunately we're going to be graphing horizontal position on this vertical axis what that means is that this equilibrium position remember this is the point where the net force the restoring force that net restoring force is zero the only force on this mass in this case is the spring force which is given by Hookes law and that means this equilibrium position is going to be the point where x equals zero if I want my force to be zero I better have x equals zero so this equilibrium position right here this this line right here let me give it a special color this equilibrium position is essentially just this x equals zero line right these two lines are representing the same thing they represent x equals zero and if I go this way if I pull this mass to the right I'm essentially going up on this graph because I'm going toward positive horizontal positions and if I go to the left if I push this mass to the left I'm essentially going down toward negative horizontal positions on this graph so hopefully that doesn't freak you out too bad let me show you what this looks like if we do displace this mass let's say we pull it to the right so like we had over here right we have this mass we pull it to the right and if we start at 20 centimeters from the equilibrium position and let go what is that going to look like on this graph well it started to the right so if it starts to the right I'm going to start way over here at this point is my initial position that means I'm going to start up here let's start up here at x equals 20 centimeters so if we put that in meters technically SI units you should have meters for the default unit so this would be 0.2 0.2 meters and that's also the amplitude so remember this is the amplitude so this distance here is the amplitude and then what is the mass do well it shoots back toward equilibrium that's x equals 0 and then it oscillates it goes through that point it comes back so essentially what you're going to have on here goes toward equilibrium so it looks like this goes toward equilibrium boom hits equilibrium that's when it's at x equals 0 passing through this point right here then it's going to come back down it's eventually going to compress the spring and stop that's when your way over here and you've been stopped the mass has been stopped by the spring that's going to come back up in this process it's going to repeat it's going to go back through the equilibrium position and come back up which by up means over here back to this initial point that's one whole cycle look at that's gone through a whole cycle look I kind of made this a little too high let me let me make that a little better it should never go any higher than it started here so it's going to look something like that come back down and this whole process repeats over and over and over and if I was drawing this perfectly it'd be perfectly smooth but hopefully you get the idea and this is great because now we can draw the variables we talked about earlier like amplitude because the amplitude is the maximum magnitude of displacement from equilibrium that would equal point two meters that's what we represented on this graph here we could also represent the period remember the period was the time it takes to go through an entire cycle so if our math started here to go through an entire cycle it better get back to that point and have reset completely so that would be two here so on this graph this is the period so the time it took to do that is one whole period that would be the period t which if we recall what we said earlier we said that the period was six seconds so if it really is six seconds we could say that this here would be if we count this is time T equals zero this would be six seconds this would be three seconds that would be half of a period or half of a cycle this would be nine seconds this would be twelve seconds which would be two whole periods and make sense it's gone through two whole cycles once it gets back to that point now notice you didn't have to measure the period from peak to peak you could have measured it from sometimes people call these troughs or valleys so you can measure a trough to trough or Valley to Valley look at three seconds to nine seconds that's a time of six seconds it took six seconds to go from three seconds to nine seconds that's still one whole period or you can go from this point here I guess this would be like seven point five seconds all the way to what is this thirteen point five seconds that would also be one whole period just make sure you don't do this sometimes people are like Oh a period a repeat a whole cycle a are animal go from this equilibrium addition back to that equilibrium position that's not a whole cycle look at at this point the mass was going that way and this point the mass is going that way so it you can't you can't start your clock when the mass is going that way stop it when the mass is going the other way and say you've gone through a whole cycle because that hasn't fully reset if you're going to fully reset you got to go from mass heading to the left through equilibrium all the way back to mass heading to the left through equilibrium so you'd have to go from this equilibrium point all the way to that equilibrium point to have a full cycle a cycle would look like this whole process right there so recapping the amplitude of a simple harmonic oscillator is the maximum magnitude of displacement from the equilibrium position you can measure it that way or you can measure it this way you'll get the same amount and the period is the time it takes for an oscillator to complete one entire cycle which you can find on a graph by measuring the time it takes to go from peak to peak from Valley to Valley or from equilibrium position skip an equilibrium position and then get to the next equilibrium position