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# Intuition about simple harmonic oscillators

## Video transcript

all right we should talk about oscillators and what an oscillator is is an object or variable that can move back and forth or increase and decrease go up and down left and right over and over and over so for instance a mass on a spring here is an oscillator if we pull this mass back it's going to oscillate back and forth and that's what we mean by an oscillator or another common example is a pendulum and a pendulum is just a mass connected to a string and you pull the mass back and then it swings back and forth so you've got something going back and forth that's an oscillator these are the two most common types masses on Springs pendulum but there's many other examples and all those examples share one common feature of why they're an oscillator so you could ask why do these things oscillate in the first place and it's because they all share this common fact that they all have a restoring force and a restoring force like the name suggests tries to restore this system but restore it to what restore the system to the equilibrium position so every oscillator has an equilibrium position and that would be the point at which there's no net force on the object that's oscillating so for instance for this mass if this mass on the spring was sitting at the equilibrium position the net force on that mass would be zero because that's what we mean by the equilibrium position in other words if you set the mass there it would just stay there because there's no net force on it however if I pull this mass to the right the springs like aa now I'm going to try to restore this mass back to the equilibrium position the spring would pull to the left if I push this mass to the left the springs like ah we're moving this thing back to the equilibrium position we're trying to push it back there so if I push left the spring pushes right and if I pull the mass right the spring pulls left it tries to restore always tries to restore the mass back to the equilibrium position same for the pendulum if I pull the pendulum to the right gravity is the restoring force tries to bring it back to the left but if I pull the mass to the left gravity tries to pull it back to the right always trying to restore this mass back to the equilibrium position that's what we mean by a restoring force now there's lots of oscillators but only some of those oscillators are really special and we give those a special name we call the simple harmonic oscillators and you might be thinking that's a pretty dumb name because that that doesn't sound very simple but there's something called a simple harmonic oscillator so what makes simple harmonic oscillator so special is that even though all oscillators have a restoring force simple harmonic oscillators have a restoring force that's proportional to the amount of displacement so what that means is if I pull this mass to the right there'll be a restoring force but if it's proportional to the displacement if I pull this mass back twice as much I'd get twice the restoring force and if I pulled it back three times as much I'd get three times the restoring force same down here if I pulled this pendulum back with two times the angle I get 2 times the restoring force if that's the case then you've got what we call a simple harmonic oscillator and you still might not be impressed you might be like who cares if the restoring force is proportional to the displacement why should I care about that you should care about that because these satisfy some very special rules that I'll show you throughout this video and it turns out even though this doesn't sound very simple they are much simpler than the alternative of non simple harmonic oscillators so these are what we typically study in introductory physics classes and it turns out a mass on a spring is a simple harmonic oscillator and a pendulum also for small oscillations here you have to make a caveat you have to say only for small angles but for those small angles the pendulum is a simple harmonic oscillator as well now in this video we're just going to look at the mass on the spring to make it simple we can look at the pendulum later so I'm going to get rid of the pendulum so we could focus on this mass on a spring now you might not be convinced you might be like how do we know this mass on a spring is really a simple harmonic oscillator we could prove it because the force that's providing the restoring force in this case is the spring so the spring is the restoring force in this case and we know the formula for the force from a spring let's give them by Hookes law and Hookes law says that the spring force the force provided by the spring is going to be negative the spring constant times X the spring displacement so X is going to be positive if the spring has been displaced to the right because the spring is going to get longer so this would be a positive X amount and if you compress the spring the length of the spring gets smaller that's going to count as a negative x value but think about it if I compress the spring to the left my X is going to be negative and that negative combines with this negative to be a positive so I'd get a positive force that means the spring is there's a force to the right and that makes sense restoring it means it opposes what you do if you push the mass to the left the spring is going to push to the right if we did it the other way if we pulled the mass to the right now that would be a positive x value if I have a positive x value in here combine that with the negative I get a negative spring force that means the spring would be pulling to the left it's restoring this mass back to the equilibrium position and that's exactly what an oscillator does and look it up here this spring force this restoring force is proportional to the displacement so X is the displacement this is a force that's proportional to the displacement and that's the definition that was what we meant by simple harmonic oscillator so that's why masses on Springs are going to be simple harmonic oscillators because the restoring force is proportional to the displacement now to be completely honest it has to be negatively proportional to the displacement if you just had F equals KX with no negative then if you displaced it to the right the force would be to the right which would displace it more to the right which would create a larger force to the right this would be a runaway solution this thing would blow up that wouldn't be good so it's really forces that have a negative proportionality to the displacement that way it's going to restore back to the equilibrium position and if this is proportional you get a simple harmonic oscillator and so we should talk about this what what the heck do we mean by simple like what is simple about this it turns out what simple is that these types of oscillators are going to be described by sine and cosine functions so simple harmonic oscillators will be described by sine and cosine and that should make sense because think about sine and cosine what do those look like sine and cosine look like this so here's what sine looks like it's a function that oscillates back and forth and cosine looks like this it starts up here so it's also a function that oscillates back and forth and so these are simple turns out those are very simple functions that oscillate back and forth and because of that we like those in physics we love things that are described by sine and cosine it turns out they're pretty easy to deal with mathematically maybe you don't feel that way but they're much easier than the alternatives of other things that could oscillate so that's what simple harmonic oscillators mean but let's try to get some intuition what is really going on for this mass on a spring so let's imagine we pull the mass back right so the mass if the masters continues to sit at the equilibrium position it's a pretty boring problem because the net force are there would be zero and it would just continue to sit there so let's say we pull the mass back we pull it back by a certain amount so if you pull it back this far and then we let go so since we let go of the mass we've released it at rest so it started at rest that means the speed initially over here is zero so it starts off with zero speed but this spring has been stretched and so this spring is going to try to restore right the spring is always trying to restore the mass back to the equilibrium position so the spring is pulling the mass to the left speeding it up speeds the mass up till it gets to the equilibrium position and then the spring realizes oh crud I messed up I wanted to get the mass here but I pulled it so much this mass has a huge speed to the left now and masses don't just stop on their own they need some force to do that so this mass has inertia and according to Newton's first law it's going to try to keep moving so even though the spring got the mass back to the equilibrium position that was its goal it got it back there with this huge speed and the mass continues straight through the equilibrium position and the spring starts getting compressed and the springs like oh now I've got to start pushing this thing to the right I want to get the mass back to the equilibrium position so now the spring is pushing to the right slowing the mass down until it stops it but the spring is compressed so it's going to keep pushing to the right now it's pushing in the direction the mass is moving so now it's got it going back to the equilibrium position again which is good but again same mistake the spring gets this mass back to the equilibrium position with a huge speed to the right and now the springs I go great I did it again I got this mass back where I wanted it but this mass had a huge speed and it's got inertia and so this mass is gonna keep moving to the right past the equilibrium and this is why the oscillation happens it's a constant fight between inertia of the mass wanting to keep moving because it's got mass and it's got velocity and the restoring force that is desperately trying to get this mass back to the equilibrium position I can never quite figure it out because it keeps overshooting each other and this oscillation happens over and over and over so just knowing this story lets you say some really important things about the oscillation one of them is that at these end points at these points of maximum compression or extension the speed is zero so this this mass is moving the slowest ie it's not moving at all at these maximum points of compression or extension because that's where the spring has stopped the mass and started bringing it back in the other direction whereas in the middle at the equilibrium position you get the most speed so this is where the mass is moving fastest when the spring has got it back to the equilibrium position and the spring at that point realizes I'll crap this mass is going really fast and the mass is coming at it or going away from it too fast for the spring to stop it immediately so at the equilibrium point this mass has the most speed during the oscillation so we could also ask where will the magnitude of the restoring force be biggest and where will it be least during this oscillation and we've got a formula for that look at this spring force is the restoring force so we just ask where will the spring force be biggest that's going to be where this X is biggest or smallest so if we wanted to know where the magnitude of this F is largest we could just ask where will the magnitude of the X be largest if we don't care about which way the force is we just want to know where we'll get a really big force we just try to figure out where will I get the biggest X X is displacement so the x value at the equilibrium position is zero so there's no displacement of the spring right here that's what it means to be the equilibrium position this is the natural length of the spring that's the length that the spring wants to be if the spring has that shape right there it doesn't push or pull but if you've displaced it this way or the other way this would be positive displacement and this would be negative displacement now the spring is going to exert a force so where will the force be greatest is to where the spring has been impressed or stretched the most so at these points here at the points of maximum extension or compression you're going to have the greatest amount of force so greatest magnitude of force because the spring is really stretched it's going to pull with a great amount of force back toward the equilibrium position we can say which way it points right the spring is going to be pulling to the left so it's gonna be a great spring force to the left technically that'd be a negative force so I mean if you're taking signs into account you could say that's the least force because it's really negative but if you're just worrying about magnitude that would be a great magnitude of force and then also over here at the maximum compression this spring is really pushing the mass to the right you'd get a great amount of force this way because your X even though it's very negative at that point it's going to give you a large amount of force and so here you would also have a great amount of magnitude of force which can be confusing because looking at these end points you have the least speed but the greatest force sometimes that freaks people out there like how can you have a great force and your speed be so small well that's the point where the spring has stopped the mass and started pulling it in the other direction so even though the speed is zero the force is greatest so be careful force does not have to be proportional to the speed the force has to be proportional to the acceleration right because we know net force we could say that the net force is equal to MA so wherever you have the largest amount of force you'll have the largest amount of acceleration so we could also say at these end points you'll have not only the greatest magnitude of the force but the greatest magnitude of acceleration as well because where you're pulling or pushing on something with the greatest amount of force you're going to get the greatest amount of acceleration according to Newton's second law so at these end points the force is greatest the acceleration is also greatest the magnitude of the acceleration is also greatest even though the speed is zero at those points so those are the points where you get the greatest force and greatest acceleration where will you get the least amount of magnitude of force and magnitude of acceleration we'll look it up here the least force will happen where you get the least possible displacement and the least possible right here in the middle this equilibrium position is when x equals zero that's when the spring is not pushing or pulling when it's at this point here so when the mass is passing through the equilibrium position there is zero force right that's the point where the mass got back there and the spring was like whoo I'm glad I got it back to the equilibrium position and then the spring quickly realized oh no this mass I got it back there but the mass was moving really fast so it shot straight through that point but right at that moment the spring had this glorious moment where it thought it had done it and it stopped exerting any force because at that point the X is zero and if X is zero we know from up here the force is zero so this would be the least possible force and I guess I should say it's actually zero force it's not just the least there is zero force exerted at this point and if there's zero force by the same argument we could say that there's zero acceleration at that point hopefully that gives you some intuition about why oscillators do what they do and where you might find the largest speed or force at any given point so recapping objects with a restoring force that's negatively proportional to the displacement will be a simple harmonic oscillator and for all simple harmonic oscillators at the equilibrium position you'll get the greatest speed but zero restoring force and zero acceleration whereas at the points of maximum displacement you'll get the maximum magnitude of restoring force and acceleration but the least possible speed