simple harmonic motion going to be the topic of this lesson in my new General Physics playlist which when complete will cover a full year of University algebra based physics now in the last lesson we reviewed Springs as well as s and cosine functions in preparation for this one so if you haven't watched that one yet I highly recommend going back to that one before watching this one now in this one we're going to introduce equations for position velocity and acceleration of an object undergoing simple harmonic motion we'll talk about the frequency of oscillation we'll talk about the period of oscillation we'll talk about the amplitude of oscillation and we'll talk about the frequency Factor we'll talk about it both in the context of Springs as well as pendulums my name is Chad and welcome to Chad's prep where my goal is to take the stress out of learning science now if you're new to the channel we've got comprehensive playlists for General chemistry organic chemistry General Physics and high school chemistry and on Chads prep.com you'll find premium Master courses for the same that include study guides and a ton of practice you'll also find comprehensive prep courses for the DAT the MCAT and the oat in the last lesson we reviewed Springs and looked at hooks law and elastic potential energy conservation of mechanical energy and we took a look and said if we take an extend a spring outside of its equilibrium position and then release it so it will have a restoring Force pulling it back according to hooks law it'll pass through the equilibrium position and then reach a point where it is compressed by the same magnitude from which you extended it away from the equilibrium position on the other side and then it will snap back with the restoring Force taking you back through the equilibrium position and returning so and as as long as it's an ideal spring for which there's no internal friction and no non-conservative forces whatsoever it will just keep oscillating back and forth forever so obviously there's no such thing as a real ideal spring there's going to be friction or something somewhere so but for an ideal spring this theoretical construct it will just oscillate back and forth forever and the reason we also in the last lesson reviewed s functions and cosine functions so is these are repeating sinodal functions if you will wave functions that make a great model for modeling uh following the position uh of a spring in this case or of a pendulum swinging back and forth in this kind of repetitive motion so take a look at our typical cosine function we start there because it turns out that the function for position is one based on the cosine so it looks like this so xal a cosine Omega T and sometimes you'll see it written as X as a function of T and that's just math jargon so and probably a little more proper but we're just going to write it as xal a cosine Omega T and the only thing that X as a function of T tells us is that t is the variable here so we're going to follow the position as a function of time all right so take that X of T back off so there's our function for position so it follows a cosine function and here's just simply cosine of x or cosine of T if you will so and we see that it starts at a maximum so hit zero at both pi over 2 and then 3 pi over 2 but hits a minimum at a value of pi and then returns back to 2 its original value at 2 pi and so it repeats one full oscillation every 2 pi now in the last lesson we also learned that well if you put a factor in front of T here like if we put 2 T then all of a sudden that actually means you're going to fit two full oscillations over the course of 2 pi if it was 3T would' fit three full oscillations over the course of 2 pi and so in some way shape or form the period is related to this Factor we put right here and this is a w it's a Greek letter Omega and we call it the frequency factor and it turns out both the frequency and the period are going to be related to this term right here now on the other side we also learned that the S and cosine functions oscillate between positive one and negative 1 so but if you multiply by a term out here which in this case is our amplitude well if this part of the function oscillates between a maximum of one well then the entire function is going to have a maximum value of a * 1 or a so and on the other side it has a minimum value here for cosine Omega T of1 well then the entire position function would have a minimum value of a * 1 or simply negative a and that's why we call it a here for amplitude because it is the boundaries the minimum and maximum values that the function can take so from a maximum value of a and then a minimum value of negative a so when you see an equation for position of an object undergoing simple harmonic motion the coefficient out front you should definitely recognize that as your amplitude so and then we've got our frequency Factor here which we'll talk a little bit more about in a second now it turns out if you were taking a calculus based physics class this would be a little bit easier because it turns out we're going to derive the equations for velocity and acceleration using a little bit of calculus and I'm going to talk about that calculus but you're not on the hook for Calculus if you're taking an algebra based physics class so but I will derive those equations using a little bit of calculus and I will explain it cuz I hate just throwing equations out in the middle of nowhere and not tell anywhere they come from so but that's kind of the deal so if we take the d derivative of the position function with respect to time and if you recall we said that velocity was equal to the change in position I.E the displacement over the change in time well in calculus terms we'd say that velocity was equal to the change in position over the change in time so DX DT and so in this case if we took dxdt taking the derivative of this equation with respect to time that would get us an equation for the velocity that's ultimately what we're doing with a little bit of calculus and so in this case we're going to get V equals Well turns out the derivative of this equation a is just a constant it pulls out in front of taking the derivative and it's just going to still be a constant out in the front but it's really just taking the derivative of cosine function well it turns out the derivative of cosine is negative sign which is why we're going to have a negative sign way out front as well and we're going to have the sign function so and turns out the derivative of cosine X is negative sinx well we don't just have a simple X or in this case T we have omega T and when you've got an inner function and an outer function you got to multiply by the derivative now the inner function and the inner function is Omega T well the in I'm sorry the integral the the derivative of Omega T is simply Omega and so we're going to multiply by Omega out here as well and this is the function for the velocity of an object undergoing simple harmonic motion so negative a Omega sin Omega T so and then acceleration works the same way we said oh acceleration equals change in velocity over change in time and in calculus terms that would be DV DT so the derivative velocity with respect to time and so if we take the derivative of the Velocity equation we can then come out with an equation for acceleration and so in this case now it's negative a Omega that's going to pull out front so a Omega and it turns out the derivative of sign function is the the cosine function not the negative cosine just the plain old cosine function so we're going to end up with cosine Omega T but just like with the sign we did above or the cosine we did above when we took the derivative you also then got to multiply by the derivative of the inner function Omega T which is simply Omega and so we're going to multiply the whole thing by another term of Omega which is why we end up with Omega squar all right so going back now we've got three different equations one for position one for velocity one for acceleration so we want to take a look at what amplitude means in this context because the way we've defined amplitude it's only actually the amplitude of the position equation it is the coefficient out in front of the sign or cosine well if we see for velocity the amplitude is now out in front of the sign function it's not just simply a it's a Omega and so in this case what that tells us is that the maximum velocity I.E the amplitude for velocity is actually equal to a Omega that's the maximum value and the minimum would be negative a Omega and the negative just refers to Direction in that case same thing for acceleration now so the coefficient if you will out in front of the s or cosine function in this case cosine function is now a Omega or negative a Omega squ and again the negative just refers to Direction and so we can see that the max acceleration I.E the amplitude of the acceleration equation is that a Omega squar whether it's positive positive or negative is again just going to be uh dependent upon Direction so if we see our spring oscillating back and forth if you recall there's no Force for acceleration when you pass the the equilibrium position according to hooks law but it reaches a maximum at the extremes when you have the maximum displacement and that's when you're going to reach this acceleration maximum value of a Omega squ and on one you'd make it a Omega squ and the other negative a Omega squ depending on Direction in this one it's pointing to the right and so you'd probably make it positive by custom so over here the force and acceleration be pointed back to the left for the restoring Force and you typically make it negative by custom in such cases so but these are our three equations here again for position velocity and acceleration and we've got one more thing to talk about and that's that frequency Factor right here so and that frequency factor is related to the frequency but it's not the same thing as the frequency and if you recall we talked about having say cosine of x well I'm going to make a cosine of T this time and then cosine of 2T and then cosine of 3T and talk about what the difference was and again if you have cosine of T you get one full oscillation in over the distance or time in this case of 2 pi well if you have cosine of 2T we learn that oh you'd actually get one full oscillation in by the time you reach pi and then another full oscillation in by the time you reach 2 pi and you'd get two full oscillations in over the course of 2 pi cosine of 3T you get three full oscillations in over the course of 2 pi it's really just telling you how many full oscillations you complete over the course of 2 pi so if you want to find the period uh you know what we're going to put this over here if you want to find that period of oscillation it's really equal to 2 pi over Omega you just take 2 pi and divide it by the number so if it's just one 2 pi divided by one and you complete one full oscillation over the course of 2 pi so cosine of 2T again you'll finish two full oscillations over 2 pi which means you're only going to finish one full oscillation over a a time or distance of Pi depending on your units you're looking at and so in this case 2 pi/ 2 gets you that Pi so in the case of three full oscillations over 2 pi so in that case well you're going to take 2 pi over 3 and you'd figure out that a full oscillation would complete in 2/3 Pi if you will so that's how that works that's how we find the period now frequency on the other hand is just the inverse of the period 1 / T and if if you rearrange this a little bit you can see that uh your frequency factor is equal to 2 pi times your frequency and we'll come back to that in a second so if you look at frequency frequency is nothing more than just saying how frequently something is happening so if I start tapping my arm five times per second that's a frequency of five we typically measure that in hertz so five Hertz is just five Taps per second and it's five of whatever you want to talk about per second so and that's what a Hertz is so 5 Hertz equal 5 per second we often write that as seconds to the minus one so in this case we're talking about the number of oscillations per second and it doesn't technically have to be per second so however for SI units it's going to be per second but if we talked about you know how often does the sun rise well one time per day seven times per week 365 times per year or you know if you figured out how many seconds are in a day some very tiny fraction per second if you went that route so but it's just how frequently something is happening so if you notice if I said that you know the sun rises seven times per week okay that's a frequency of seven per week well then how long does it take for the sun to rise just once well 17th of a week and there's that inverse relationship again if the frequency was seven times per week well then the period would be 17th of a week if you will in return so just as if FAL 1 over T you can invert that and see that t = 1/ f and so again if the frequency is seven times per week then the period would be 17th of a week I.E one day all right so there's that relationship there if you know the frequency you know the period if you know the period you know the frequency but also you can see that the frequency factor and the frequency are directly proportional so and the constant of proportionality is 2 pi so if you take the frequency and multiply by 2 pi you get the frequency Factor or vice versa if you take the frequency factor and divide by 2 pi you get the frequency which makes sense because again frequency and period were inverse functions of each other so if the period equal 2 pi over Omega it makes sense that the frequency would equal Omega over 2 pi but a lot of similarities going on here it's easy to get these confused spend some time really making sure you understand how to get frequency period and the frequency factor from each other okay one more thing to talk about here we'll need a little bit of room all right in the last lesson we showed how F equal KX so for uh Hook's law for a spring with the restoring force and according to Newton's second law of motion that's also equal to Ma and we could set them equal to each other and come up with an expression for the acceleration and we could get K Over M * X now here's the deal when x equals zero IE you're at the equilibrium position well then the acceleration is zero no problem there but when you're at the extremes either Max compression or Max extension that's when x equals a in such case the amplitude so and if we take a look at that I'm going to substitute that in and so we can say that a maximum equals km K Over M * a well we just set a minute ago that the maximum of this you know sinusoidal or in this case cosine function is a Omega s or negative a Omega s so in this case I'm going to set that equal to a Omega s and all of a sudden we can see that oh the A's cancel the negatives cancel and I've got an expression that says that Omega squ is equal to K Over M and we've just arrived that Omega is actually going to equal the square root of K Over M so it is the ra the square root of the ratio of the spring constant value to the mass that's on the end of the spring so now we've got an expression for Omega and that's the same frequency factor that shows up here here and here as well as here in the velocity equation and here squared in the acceleration equation all right often we say whack as a way of remembering this lovely equation and we're going to compare that in a second to a uh a pendulum on a on a swing which we say wiggle as we'll see so I'll write that one out so wiggle so for a pendulum it turns out this Omega is going to equal the square root of the acceleration duee to gravity divided by the length of the pendulum we'll see where that comes from so next we're going to look at simple harmonic motion in the context of a pendulum ever so briefly so but it turns out a pendulum doesn't truly follow simple harmonic motion now if you do take this pendulum and pull it away from that perfectly vertical position which we'll call its equilibrium position here and then let it go it is going to oscillate back and forth but it turns out it doesn't truly follow simple harmonic motion however it approximately will follow it as long as the angle Theta here that you pull it away from the vertical is less than about 15° or so and then the more it is less the better an approximation uh it is to say that the the pendulum is undergoing simple harmonic motion now the the mass we have hanging on the end of the pendulum is called the Bob so it's hanging from you know some sort of wire or string or something like that of a given length which I'll write out here of a given length and then it's going to be away from the equilibrium position instead of being away from the equilibrium position by a linear distance x uh like in the case of the spring it's going to be an arc length that we'll call s in this case all right now we look at the restoring Force it's perpendicular right here so to the uh uh the length of the pendulum itself and the restoring force is equal to mg sin Theta and that's our problem is that as this thing gets shorter and shorter it turns out the force modulates by a factor of sin Theta not simply Theta and that's kind of our issue of why it's not truly simple harmonic motion with simple harmonic motion it would uh adopt something similar to what we saw with hooks law or at least a similar type of equation if we take a look there we could say that mg sin Theta can that be modeled after an equation like hooks law negative KX and the problem is so instead of being uh a force that a restoring Force that's proportional to a displacement from equilibrium it's proportional to the sign of a angular displacement instead and that's the problem well it turns out though that we have a little way to get around it and we commonly say that pendulums follow simple harmonic motion as long as again that angle Theta is small smaller than about 15 degrees or so and the reason this works is that if you take the sign of a lot of angles all the way up to say 15 degrees or so the sign of that angle and that angle itself are almost exactly the same value so the sign of like 10° and 10° and again if you put it in radians anyways uh almost exactly the same value and so as a result we can substitute this out for just simply Theta and all of a sudden now we can model this with something that's proportional to the angular displacement versus the linear displacement so and write this a little bit different uh if you take a look we're going to take a look at the Arc Length instead of a linear displacement and it turns out there's Rel relationship between the Arc Length the length of the pendulum and the angle Theta and that relationship is that L * Theta is equal to that arc length and what we really want in this equation is going to be that arc length and so in this case if we rearrange and solve for Theta and substitute it in we can get Theta equal s over L and so this case we'll take out that Theta actually you know I'll leave that in and just rewrite it so mg * s over L equals K and instead of looking this in a hook law context with a linear displacement we're going to look at it in the context of having an ark length displacement instead away from that equilibrium position and all of a sudden now we can come up with an expression for K and we can see that K is going to equal mg over L and if we go back to the frequency factor which was the square root of k m and substitute this in that's where this is ultimately going to come for so now Omega equals the sare < TK of K mg over L all over M and we see that the M's here are going to cancel and we're just left with the frequency Factor equaling theare root of G over L and that's where this comes from and so your frequency Factor uh for the spring is we use whack to remember that Omega equals k m whereas for the frequency factor in a pendulum it's wiggle so frequency Factor equals the of g l one thing to note here is the mass of the Bob on the end of the pendulum the frequency factor is not affected by that at all only by the acceleration due to gravity which isn't going to change unless you you know change planets or something like that uh but only on the length of the Bob in this case not on the mass I'm sorry the length of the pendulum not on the mass of the Bob uh instead cool besides that though and how we get this frequency Factor Factor so whether it's a spring or a pendulum you're going to get very similar looking equations for position velocity acceleration you're going to find frequencies and periods exactly the same way the only thing you're going to do different is how you get that frequency Factor uh from some of the the constants associated with the creation of that pendulum or that spring so either with the spring constant and the mass or from Gravity as well as the length of the pendulum let's do some problems all right so the next series of questions are all going to be related to this problem right here and a spring with a 2.0 kg Mass attached obeying simple harmonic motion follows the following equation of motion so x = 0.20 m time cosine of 4.0 per second time time all right we're going to see what is the amplitude of oscillation what is the period of oscillation what is the frequency of oscillation and three more questions when we get there now this might look a little bit different than you're used to seeing your position equation look and I highly recommend that maybe you rewrite it without the unit so it's a little easier to line up and stuff like that but putting units in there which is important uh maybe makes it look a little less like what you're used to seeing so let's rewrite this so x = 0.20 * cine 4.0t and it's a little easier to see what's going on here and if we match this up with our equation for position we can easily see that the amplitude which is the first question is equal to 0.20 and including it back here 0.20 meters so there's our first question so we can also see that our frequency Factor here Omega is equal to 4.0 and in this case 4.0 per second now that wasn't one of the questions it never actually said what is the frequency Factor but the next two questions were what is the period of oscillation and what is the frequency of O oscillation both of which are dependent upon that frequency factor and we can see here that the period is just equal to 2 pi divided by that frequency Factor so we'll start there and so in this case uh your period is going to equal 2 piun over 4.0 per second and we can see this is going to come out in units of seconds then so in this case 2 over 4 is 1/2 and so we're going to get pi over 2 seconds and so this thing is going to complete a full oscillation in Just pi/ 2 because it completes four full oscillations over the course of 2 pi seconds if you will all right we're also got to find the frequency here so and the frequency is just equal to 1 over T so if we already figured out that t was pi/ 2 seconds we can figure that 1 over pi over 2 seconds is going to equal 2 over Pi per second that's the frequency so there's the first three questions in pretty short order found so let's move on next question related to this equation says what is the equation for the velocity and what is the max velocity of the mass so now we want the equation for velocity here well again velocity is just equal to a Omega sin Omega T so in this case we get velocity equals negative a okay well the a here again was 0. 2 so and we're going to multiply this out actually because it's negative a * Omega we figured out Omega was 4.0 so. 2 * point I'm sorry Time 4 is going to be8 0.80 notice it's G to be meters per second as it should be for a velocity and then times s of Omega T So 4.0 per second time time whether we include the units or not all right so there's the velocity equation and then we wanted the max velocity and if you recall the max velocity is just the amplitude on the front the a * Omega which because I already multiplied them together is evident right from the equation and that Max velocity actually I'm going to write that over here Max velocity is just simply going to be 0.80 m per second it's the amplitude of that velocity equation which is just the coefficient out in front of the sign function all right finally next question says what is the equation for the acceleration and what is the max acceleration of the mass and so in this case we've got negative a Omega 2ar so in this case we did a * Omega and it was 2 * 4 now it's going to be 2 * 4 * 4 and so in this case 2 * 16 well 16 * 2 is 32 so 16 * .2 is going to be 3.2 and notice now it's going to be me/ second per second which is meters per second per second or me/ second squared the proper SI unit for acceleration and then cosine Omega T So cosine 4.0 per second times time all right other half of that question was what is the max acceleration and the max acceleration again is just equal to the coefficient out in front of the acceler equation a Omega s which we already calculated to be 3.2 m per second squared we're good to go now one last part to these last two questions it's not actually in the question but I want to know where do we reach this Max acceleration where do we reach this maximum velocity well you might recall that uh Max velocity is reached on an oscillating spring when you pass through the equilibrium position where the max acceleration and the max force are both reached at the extremes either Max extension on one end or Max compression on the other hand in either case when you have maximum displacement all right last question related to this is what is the maximum value of the restoring force and where does this occur and again that Max restoring force is still going to happen at the extremes either when you have a maximum displacement either compression or extension but now we want the maximum value of that restoring force and so in this case uh it's not one of our traditional equations notice force is nowhere up here but you guys do know a couple different ways we can calculate this so we could calculate this with Newton's Second Law And if I want the maximum Force well then I want the maximum acceleration that'd be one way to go about it or we could try to use hooks law FAL KX and ultimately either one is going to get you the right answer uh but one's a little easier to do than the other now we're told the mass here is 2.0 kg and we just figured out the max acceleration was 3.20 m/s squared and so this is an easy one to use so 2.0 kg let's write that out time 3.2 m/s squared so in this case is going to get a 6.4 I'm erase hooks law and rewrite it if we need it 6.4 Newtons now could we have used hooks law to figure this out yes it just would have been a little more of a pain in the butt so if we do that up here let's say so FAL KX and the problem is is we don't have the spring constant wasn't given in this problem and it didn't need to be given because everything else you need actually is given and if you recall whack here tells us the relationship between the frequency Factor the spring constant and the mass and we know both the frequency factor and the mass so we can calculate out that spring constant so Omega equal theun k over M so or we could say that Omega 2 * m is going to equal K and so in this case Omega again was equal to 4.0 so squared is going to be 16 and then times the mass of two tells us that K is going to equal 16 * 2 is going to be 32 Newtons per meter for our spring constant and now we could use FAL KX if we knew the maximum displacement all right well how do we get that maximum displacement again that's just the amplitude of the position equation that maximum displacement is2 M and so if we took our spring constant FAL KX so multiplied it by that uh maximum displacement of 0 2 m we would once again still get 32 * 2 so 32 * 2 is 64 so 32 * 2 is still 6.4 Newtons we find out that the max force is once again the same answer we got lot easier to have gotten it using Newton Second Law though all right one more problem and this one's going to be more related to a pendulum so in this problem says if the length of a pendulum is increased by a factor of four and the mass of the Bob is doubled what will be the effect of the frequency of oscillation now I've asked this one on purpose and I've told you that I've changed the mass and you should be like Chad you told me mass doesn't matter and yes for a pendulum the mass does not matter your frequency Factor here only depends on the acceleration of gravity and on the length length of the pendulum not on its mass and you might recall well you know why is a pendulum swinging back and forth what's causing it to want to return back to equilibrium well it's gravity that is so and if you recall the acceleration du of gravity for any object on earth as long as we're ignoring air resistance is 9.8 m/s squared it doesn't care about the mass and if gravity's in charge of the restoring Force for pendulum then it shouldn't be too surprising that the mass of the Bob doesn't matter all right so if the length of the pendulum is increased by a factor of four four and the mass of the Bob is doubled so notice gravity wasn't changed so what will be the effect on the frequency of oscillation so a couple things you need to remember here is that the frequency factor is proportional to the frequency and here we can see that the relationship between uh frequency factor and the length is that and I put approximate let's get proportional in there it's proportional to the square OT of 1 over L and therefore that's going to be proportional to the frequency well what did we do to L well we can see that as L goes up the frequency and frequency Factor are going to go down there's an inverse relationship not inversely proportional because of the square root but there is an inverse relationship and so it shouldn't be too surprising that if we made the length of the pendulum longer it goes up now the school I used to work at we had a pendulum that was several stories tall and it moves so unbelievably slow and it's easy for me therefore to see that oh yeah as you make the pendulum longer and longer and longer it should make the frequency slower and slower slower if you will so or lower and lower and lower more proper so in this case if the length goes up by a factor of four then the frequency factor and frequency should go down not by a factor of four but by the square root of four so go down by a factor of two I.E they are cut in half and that is the answer to that last question if you have found this lesson helpful consider giving it a like happy study