If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Equation for simple harmonic oscillators

## Video transcript

all right so we saw that you could represent the motion of a simple harmonic oscillator on a horizontal position graph and it looked kind of cool it looked something like this and the amplitude of that motion the maximum displacement from equilibrium on this graph was just represented by the maximum displacement from equilibrium it looked like this and the period which was the time it took for this entire process to reset capital T is the period this is the time it takes reset was the time it takes to reset which would be from peak to peak or from trough to trough or from any point to any analogous point on that cycle this was the period t and so with a graph that's a sine or a cosine you could represent any motion you want so if it if you had some oscillator that had a larger amplitude you can imagine just stretching this thing vertically the period would stay the same but you could stretch out the amplitude or if you had something with a larger period you can imagine stretching it out horizontally and leaving the amplitude the same or stretch it both ways to represent any oscillator you want which is kind of cool however a lot of times you also need the equation in other words you might want to know what equation would describe this graph right here what equation would represent this graph here first of all what do I even mean by like the equation for this graph what I mean is that this graphs representing the horizontal position X which is how far the mass has been displaced this way from equilibrium as a function of time so we want a function that will be all right what is the value of the position of this mass as a function of time so what would this equation be going to be a function in other words you're going to feed this function any time you want and the function is going to give you it's going to spit out a value for the position and that should represent whatever position this graph is representing where the mass is at because the graph should agree with what this function is going to tell us and this function would tell us where the mass is at any given moment so what would this look like now we saw like this is a sine or a cosine right so this is either a sine or cosine that's the first choice do we want to pick sine or cosine and what I always do is I just look at the beginning and I say all right at T equals zero this one's starting in a maximum so I want to use cosine because cosine starts at a maximum and by starting at a maximum I mean think about it for cosine of zero if you remember your trig functions cosine of zero is equal to one and so because this is as big as cosine ever gets sine and cosine can only ever get as big as one this thing is starting at a maximum so cosine starts at a maximum at T equals zero this function here starts at a maximum at T equals zero I'm going to want to use cosine but I'm gonna have to add a few elements in here just cosine alone isn't going to do it for me because cosine only gets as big as one this thing has to get as big as a whatever a is this thing has to get that big so in other words my simple harmonic oscillators aren't always going to have an amplitude of one so I need something I need some variable here that will represent what the amplitude is for that given simple harmonic oscillator let me make this less abstract let me just say I'll say we happen to pull this thing back 20 centimeters or 0.2 meters so let's say our amplitude for a particular simple harmonic oscillator happen to be 0.2 meters that would mean that this here I could represent this here with 0.2 meters this doesn't even make it to one so if I just left this as cosine that would say this thing is going to get as big as one at some point in time and that's a lie this thing only gets as big as point two so it's easy though you might realize if you're clever we'll just multiply the front of this thing by the amplitude whatever the amplitude is multiply it because then one times amplitude means that this X only gets as big as the amplitude which is exactly what I want I want this thing to be as big as whatever the amplitude is of the motion and there's one more piece you can't you might be like alright we're done I'm just going to stick cosine of T in here that's not going to work we do want this to be a function of time right we want to be able to plug in a time and have this function spit out what is the value of the position of the graph and that would represent where is it so is it at point two is it at point one is it a point oh four five or something like that that's what this functions supposed to do but just plugging in T here just having T alone isn't going to be good because that would mean look at a cosine of zero we know cosine is one when does cosine get back to one that's going to be when the inside the argument in here is 2pi so we're going to be using radians you could use degrees if you wanted to but most physicists and professors and teachers are going to be using radians for this case so cosine of two pi would again be one because that's when if you remember your unit circle that's when this function for cosine has gone around one whole time it gets back where it started right so if something rotates through an angle of 2 pi you've reset the whole thing and that process has reset but that would mean this function resets every 2 pi seconds right because at T equals 0 the function was 1 and then at T equals 2 pi the functions 1 again that would mean the period for the cosine of T is 2 pi but our period isn't necessarily 2 pi right unless you've got a really special case the period is whatever the period is let's say it happened to be let's say our period happened to be like 6 seconds for this particular case so if this was 6 seconds we would not want a function that resets after 2 pi seconds we need a function the resets after 4 this case 6 seconds so how do we do that well we have to not just have T in here we saw that if we just have T the period is always 2 pi because that's when cosine of T resets how would we do this well we're going to be clever and if you're really clever you realize all right I'm just going to add a little variable in here I'm just going to add a little variable boom Omega and then multiply that by T and then I can tune this Omega however I want right if I can make Omega big or small I can make the period of this function whatever I want and if you're curious it might be like wait a minute Omega we've used that before and you'd be right Omega we have used before that was the angular velocity and remember angular velocity was Delta Theta over delta T the amount of change in angle over the amount of change in time which you might think isn't relevant here because this message is going back and forth this mass isn't actually rotating in a circle however you can represent repeating processes cyclic processes processes that go through a cycle on a unit circle so in other words let's say you start right here right so at T equals 0 you start we pulled this mass back and we let go so we start right there that would be right here on this unit circle and then it is through the equilibrium point that would be through a quarter of a cycle that means it would have made it to right here and then it makes it way over to this edge fully compresses this thing that would be over to here that would be through half a cycle and that would come back through me find another dark color it would come back through the equilibrium point and that would be down here and then we get back to the initial point and that would be one whole cycle so you can see how we can represent cyclic processes on a unit circle and that's how this makes sense that might seem abstract but it's really useful because watch what we could do naively you might think all right how would we even define this well one cycle on a unit circle is 2 pi radians right if we're using radians then one cycle would be 2 pi because 2 pi is once around the circle and how long does that take well I know for a simple harmonic oscillator we defined the period to be the time it takes for one whole cycle so we'd have 2pi over the period and this is what you would plug in down here so it turns out this does work so even naively just using our ideas of angular velocity plugging in 2 pi over the period will give us a function that resets exactly when we want it to you might not be convinced and if you're if that doesn't make sense I don't blame you I might be confused too so let me show you what I mean in other words if we take this function instead of writing Omega we can just do this we can just be like all right forget this taken this Omega is the angular velocity sometimes it's called the angular frequency in this case so people use different terminologies you'll hear it as angular velocity or angular frequency if you take this angular velocity or angular frequency we just smack that right in here so we should put that in for Omega and then multiplied by T watch what happens this is beautiful so if we take this now it's going to work so we multiply by GT as our variable so little T is our variable 2 pi is a constant the period capital T is also a constant it'll be different for different harmonic oscillators but for a given harmonic oscillator capital T the period is a constant so watch what happens now at t equals 0 this hole inside becomes 0 so let's say I plug in T equals 0 we get to plug in little T whatever we want that is our variable so if I plug in little T equals zero cosine of zero gives me one but now what happens if I plug in T equals all right after one whole process right after one whole cycle it's gone through one whole period so if I plug in little T as capital T the period look what happens this capital T cancels with that capital T and you just get 2pi in here and the cosine of two pi is also one that means this thing goes through a cycle every capital T period that's what we wanted we didn't want something that always had to have 2pi as the period now we've got a function that we can plug in whatever our period is down here that way whenever this little T makes it to the period capital T this whole argument in here becomes two pi and the cosine resets itself and you get a graph or a function that will give you a graph that resets every period which is exactly what we wanted so in other words to make this less abstract let's take this thing here for this particular function here for this particular choice of amplitude and period we could say that the graph that's representing this so the function that would represent this here instead of amplitude we plug in point two so zero point two let me try to fit it in here zero point two and put the unit's down here meters times cosine remember we wanted cosine because it starts at a maximum and this graph started at a maximum if it started down here and went up I'd use sine because sine starts at zero this one started at a maximum and I have 2pi over the period I can't just leave that as period T that's a little bit vague I'd put in my actual period and we said that the actual period for this mass on a spring was six seconds and then little T a lot of times people get confused they're like all right what do i plug in for little T you don't typically like if you just want the function for the position as a function of time you leave little T as the variable that's the variable that you have sitting here right if I wanted to know what is the value of the position of this mass at nine seconds I would plug in nine seconds I would calculate this function with the nine seconds in there that would be the position at nine seconds or if I wanted the position at twelve point two five seconds I plug in 12.25 seconds for our little tea time calculate this function plug it into the calculator in other words and that would give me the position at twelve point two five seconds that's what this function can do for you that's how we can represent the motion of the simple harmonic oscillator and now you might be like dude I took a long time do they all take that long no once you get good at this it's real easy watch let me get rid of all that let's say you got this problem on a test or a quiz or whatever on homework and it was like hey make an equation that describes this simple harmonic oscillator it's easy first thing you do do I want to use sine or cosine so you might be like oh crud it doesn't start at a maximum and it doesn't even start at zero sine would start there it starts down here that's okay it starts at a minimum so we're still going to use cosine so we're going to say that X as a function of time is going to be what's well what's the amplitude the amplitude here is three meters so three meters is our amplitude because that's the maximum displacement from equilibrium so I'm going to have three meters out front and I'm going to do cosine because it starts at an extreme value either a maximum or a minimum value cosine of and then I need two pi over the period what is my period I look at my graph and I ask how long does it take to reset so start it down here to minimum when does it get back to a minimum I took four seconds so four seconds would be the periods would be two PI over four seconds and then little T what do I plug in four little T I don't this is the variable that sits there and waits for me to plug in whatever I want so that's my variable little T that X is a function of but I'm not done this would be a graph that starts up here and goes down like that this graph starts down here but that's easy just multiply by a negative sign out front and you've turned your cosine into negative cosine and negative cosine starts down here so note our amplitude is still three if the question asked what is the amplitude the amplitude is the magnitude of the displacement maximum displacement that's still positive three meters even though it started down here but you could just include an extra negative out front that essentially goes along with the cosine that would give you negative cosine and there you have it that would be your function keep in mind it's good to remember if you start up here you're going to want to use cosine if you start down here you're going to want to use negative cosine if you start right here you're going to want to use sine if you start here and go up that's going to be sine and if you start here and go down that's going to be negative sine that's what those functions look like so recapping you could use this equation to represent the motion of a simple harmonic oscillator which is always going to be plus or minus the amplitude times either sine or cosine of 2 pi over the period times the time this 2pi over the period is representing the angular frequency or angular velocity and you would choose positive cosine if you started at a max negative cosine if you started at a min positive sign if you start at 0 or equilibrium and go up negative sine if you started equilibrium and go down