SHM (Simple Harmonic Motion) - A-level Physics

shm or simple harmonic motion describes how things move when they oscillate let's say that we have somebody on a swing and here they are at equilibrium that's where they would be if they weren't swinging and that's the midpoint isn't it because once they start swinging the swing should be symmetrical about that point what if this person wants to start swinging and they get pulled up to this position here and they're ready to be let go when they're at equilibrium do they feel any force trying to bring them back towards equilibrium no they don't because they are at equilibrium already but when they're up here they do feel a force it's trying to pull them back towards equilibrium is that always going to be the case when they're the other side is the force still going to be towards the left well no because we know that gravity their weight is going to try and pull them back towards equilibrium again it might seem a little bit obvious but it's actually very important that you know the conditions needed for shm and the first condition is this acceleration therefore forced to act in the opposite direction to displacement if we say that to the right is positive as the left is negative when they're displaced to the right the displacement is positive which way is the force and acceleration pointing negative when we're the other side the displacement is negative but the force is pointing in the positive direction so this is correct acceleration act in the opposite direction to displacement if i lifted them higher would they experience a greater force or a smaller force to begin with well they should experience a greater force shouldn't they as they come back towards the center here as they're in equilibrium we said that there's no acceleration back towards the center because they're already at the center so what we find is that acceleration and therefore force 2 is proportional to displacement so those are the two conditions needed for shm if number two was true but number one wasn't and that would mean that acceleration would be pointing in the same direction as displacement and the personality will be gone before you know it so let's make an equation for this then like we said acceleration is proportional to displacement we're going to give displacement the letter x trying to make a curly x instead of a multiply sign but we know that satisfies condition number two but we know it needs to be going in the opposite direction as well so we put the minus in front as well to turn this into an equation with an equal sign we need to introduce a constant and actually for shm that ends up being this the minus is there which is going to be equal to 2 pi f squared x f being frequency 2 pi f being angular frequency or angular velocity also equal to minus omega squared x now when we talk about frequency we are talking about complete oscillations per second so if we're starting at the top then that means going over to the other side and coming back again to the same position that's one complete oscillation same thing if we started at the middle to do one complete oscillation we'd need to go to the left to the right and back again not just to the left and back again so frequency that's complete oscillations per second x is displacement i should say from equilibrium i'm going to write equilibrium like that eq little m makes it a little easier and also we need to talk about time period that's for one oscillation how do we find the time period from frequency well they're reciprocals of each other one divided by the other one amplitude we're going to give that the symbol a that is the maximum displacement from equilibrium in other words you can't get any further away from equilibrium unless you have another external force involved more on that later so when it comes to any shm it could be somebody on a swing or it could be a pendulum similar to this or it could be a mass on a spring we need to understand what's happening to the displacement force acceleration and speed velocity at any given time so i'm going to draw myself a little grid here so looking at our swing here i'm going to give you the three positions here left center that's equilibrium and right i wonder if you can tell me what the displacement the velocity the acceleration and the forces at each of these points whether they're maximum zero and then to the left or the right but instead of left and right i want you to use negative and positive so 10 seconds pause the video if you have to see if you can figure those out so displacement nice and easy first when it's at the left we know it's at max and that's negative at the center it's zero at the right it's max but in the positive direction let's go for speed when it's at the left here what is the speed of that point well it's actually zero same at the right we get the max speed when we're going through the center whether it's going to the right or the left well as it's going from left to right the velocity is going to be positive and vice versa so we're going to say positive or negative depending on which way it's going what about the acceleration when do we have no acceleration we have no acceleration at the center no acceleration towards the center because it's already there when it's at the left which way is the acceleration pulling to the right and it's at a max there look at this displacement and acceleration going the opposite way we know that's got to be true and here as well that's at a max but it's going to be negative whatever the acceleration does the force is going to do the same so we know it's going to be the same as the acceleration there well done if you got those right so what about if we drew a graph to show what's going on with displacement velocity and acceleration at any point this side is positive this side is negative this is the midpoint this is equilibrium let's start off with displacement now with the swing we're going to start at a maximum so we're going to say that starts there it follows this shape here that is well it's supposed to be a sine wave so this is displacement now if i drew a dotted line joining the peaks together what am i showing i'm showing the maximum displacement and that's my amplitude so you should be able to see that already that the displacement at any point is going to be a fraction of the amplitude and we started at maximum whoopsie so that's going to be cos two pi ft as per usual if anything changes sinusoidally then it's either going to be cos two pi ft or times sine two pi ft so what we're saying is that displacement is going to be some fraction of a the maximum displacement where we started them from what about acceleration well we know that from our conditions the acceleration is always going to be in the opposite direction i'm not drawing this to scale so please don't get the idea that acceleration is as big as the displacement but i'm just drawing it to show you the shape so it's always going to be doing the opposite isn't it slightly trickier one to do is velocity well let's have a think about it here we start at the right hand side so that's our swing and it comes through to equilibrium like that so at this point here at equilibrium the swing is going from right to left so the velocity is actually negative so we put a cross down there here is going to be zero because we're at the top we know that speed is zero at the top and then we can just follow our pattern through so we can see that all three of these things are 90 degrees out of phase or pi over two radians out of phase with each other displacement is 90 degrees out of phase with velocity velocity is 90 degrees out of phase with the acceleration displacement and acceleration 180 degrees at a phase with each other which makes sense given our conditions earlier on if you really want to get mathsy with it the gradient of the displacement function actually gives you the velocity the derivatives of each other but you don't need to go into the derivatives and all that jazz but it is worth knowing the gradient of the displacement line gives you the velocity at that point here we have a negative gradient so that means that we have a negative velocity here we have zero gradient so we have a zero velocity here we have a positive gradient so we have a positive velocity that's how it works so let's just go back to our equation for acceleration real quick we had acceleration equals minus 2 pi f squared x what's going to be the max acceleration then when do we have max acceleration is it at equilibrium no because we know we've got no acceleration at the point it's going to be at the maximum or rather at the amplitude so we can say that we have the max acceleration when x is a when the displacement is the maximum displacement from equilibrium the amplitude velocity is a bit of a trickier one now you don't need to know the derivation for this but the velocity at any point is going to be plus or minus because it can be going to the left or it could be going to the right 2 pi f then the square root of the amplitude squared minus the displacement squared so let's say that the amplitude was 5 the displacement that we're looking at is 3 gonna be 25 take away 9 that gives us 16. square root of that is going to be 4. that's how we find that out when do we get maximum speed we get maximum speed at equilibrium because at the maximum at the amplitude we get no velocity no speed for a split second and then maximum speed as it comes through the equilibrium again so that's when x equals zero what does that give us plus or minus two pi f square root of just a squared in other words a now this is very useful when it comes to pendulums or swings pendulum let's go with pendulums useful with pendulums and given the height difference but on that note if we are talking about a pendulum and there it is equilibrium and then there it is at amplitude there's my height difference but that isn't my displacement where is my displacement this is my displacement here that's my x now you might see a bit of a problem with this in reality this is going to be a curved path because it is kind of like circular motion but all of this shm stuff is a good approximation for a pendulum that's displaced less than 10 degrees so if that's 10 degrees there then it's all good if it's more than that then we say that the model starts to break down a little bit that's why whenever you do an experiment with this you don't want to be putting your pendulum out there you just want to be bring it out just a tiny bit and you're going to find out that actually displacing it a little bit compared to a lot is as it turns out more accurate so the question is is how do you find the amplitude if you're given the height difference instead you have to go for energy what do you know that at this point it doesn't have any kinetic energy but it does have gravitational potential energy mgh when it comes back to equilibrium all of that has been turned into kinetic energy so that's half m v max squared m's cancel and you can find out your v max once you find your v max you can find your amplitude if you have the frequency one more small thing what is the tension going to be in the string or whatever that's holding this up well if you haven't seen my video on tension versus weight then it's worth having a look at that but we say that tension would be equals to mg in order to merely hold the mass on the end just there but because it's actually making it go in circular motion at a point as well we need to add on to there mv squared over r and yes that v there is going to be the v max now what would the graphs for energy look like for a pendulum or a mass on a spring or anything undergoing shm so what we can do is draw just simple axes energy on the y-axis and then we have displacement on the x-axis and we have positive and negative then also we're going to have amplitude as well so maximum displacement negative direction and then the same the other side so let's have a think about kinetic energy first so where is kinetic energy going to be greatest think about a swing think about a pendulum or a mass on a spring it's going faster when it goes through equilibrium so i know that it's going to cross up there now when an object reaches amplitude we know that for a split second it stops so i know that it's going to do that as well so joining these up we end up with this shape here so that's kinetic energy what about potential energy well when a pendulum or swing is at equilibrium it's not going to go anywhere so it has no potential energy so it's going to be the opposite and then at amplitude again it's going to be the opposite to kinetic energy so this is what we have for potential energy potential energy increases as we get further away from equilibrium so potential energy and that can be any kind of potential energy really with a pendulum it's gravitational potential energy with a spring it's elastic potential energy there can be others as well so what about total energy well we know that in any system total energy is equal to potential energy plus kinetic energy so if we add all of these up at every point we end up with just a straight line so in theory total energy should stay the same but that's only in theory that's only if there's no forces resulting in energy being taken out of the system energy being taken out of the oscillations therefore amplitude stays the same in reality we can have damping forces or actually driving forces as well which can reduce and increase amplitude respectively if you want to know more about that have a look at the next video so how can we find out what this total energy is equal to so we could find that out from potential energy maybe because at amplitude doesn't matter which one potential energy is equal to the total energy so if we could find out the potential energy then we've automatically got the total energy in the system so if it's a pendulum we could use eg mgh to calculate the potential energy and therefore the total energy but more often than not we will use instead of potential energy we'll use kinetic energy so at equilibrium it's the kinetic energy that is equal to the total energy we can see that because there's no potential energy so all of the energy is kinetic energy we saw earlier that v max is equal to two pi f a or omega a i've just switched from k e to e k excuse me same thing this is equal to half m v squared so therefore we could say that e k max is equal to half m v max squared so then all we would have to do is stick this into here and we can find out the maximum kinetic energy at this point here and once we find that well that's the total energy and we've also just proven why these lines are curves because we can see that kinetic energy is proportional to velocity squared and so because velocity depends on displacement that means that energy is proportional to displacement squared quite often in questions you'll be asked what is the total energy of the system and you'll be like i don't have an equation for total energy but just remember that if you can find out v max of the system you can find out the maximum kinetic energy and that's the same as the total energy of the system so i hope that helps if it did please leave a like if you have any questions or suggestions put them in a comment down below see you next time

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Length: 16min 24sec (984 seconds)

Published: Tue Dec 22 2020