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# Introduction to harmonic motion

## Video transcript

let's see if we can use what we know about Springs now to to get a little intuition about how the spring moves over time and hopefully we'll learn a little bit about a harmonic motion we'll actually even step into the world of differential equations a little bit and don't get daunted when we get there or just close your eyes when it happens anyway so I've drawn a spring like I've done in the last couple of videos and zero this this point in the x-axis that's where the the Springs natural resting state is and in this example I have a mass of mass M attached to the spring and I've stretched the string I've essentially pulled it so the mass is now sitting at point a so what's going to happen to this well as we know the the force the rest rative force of the spring the rest rate of force of the spring is equal to minus some constant times the x position the x position starting at a so initially the spring is going to pull back this way right the springs are going to pull back this way it's going to get faster and faster and faster and faster and we learned that at this point has a lot of potential energy at this point when it kind of gets back to trusting state we'll have a lot of a lot of have a lot of velocity and a lot of kinetic energy but very little potential energy but then its momentum is going to keep it going and it's going to compress the spring all the way until all of that all of that kinetic energy is turned back into potential energy then the process will start over again so let's see if we can just get an intuition for what X will look like as a function of time so our goal is to figure out X of T X as a function of time that's going to be our goal on this video and probably the next few so let's let's just get an intuition for what's what's happening here so let me try to graph X as a function of time so time is the independent variable and I'll start at time is equal to zero so this is the time axis and let me draw the x axis this might be a little unusual for you for me to draw the x axis in the vertical but that's because x is the dependent variable in this situation so that's the x axis very unusually or we could say X of T just you know this is you know X is a function of time X of T and this state that I've drawn here this this is what at time equals zero right so this is at zero let me switch colors at time equals zero what is the x position of the mass well the x position is a right so if I draw this this is a actually let me draw a line there now I might come in useful this is a and then this is going to be let me try try to make it relatively that is negative a that's minus a so at time T equals zero where it where is it well it's at a so this is where the graph is right now let's actually let's do something interesting let's say let's define the period so the period I'll do with a capital T let's say the period is how long it takes for this mass to go from this position it's going to accelerate accelerate accelerate accelerate be really going really fast at this point all kinetic energy and then start slowing down slowing down slowing down slowing down and then do that whole process all the way back let's say T is the amount of time it takes to do that whole process right so at time zero today and then we also know that at time T at time T this is time T it'll also be a day right we're just I'm just trying to graph some points that I know of this function and just see if I can get some intuition of what this what this function might be analytically so if it takes T seconds to go there and back it takes T over two seconds to get here right the same amount of time it takes to get here was also the same amount of time it takes to get back so at T over 2 at T over 2 what's going to be the x position well T over 2 the the block is going to be here it will have compressed all the way over here so T over 2 it'll have been here and then at the points in between it will be at this at x equals 0 right I'll be there and there hopefully that makes sense so now we know these points but let's think about what the actual function looks like will it just be a straight line down than a straight line up and then the straight line down and then a straight line up that would imply if think about it if you have a straight line down that whole time that means that you would have a constant rate of change of of your x value or another way of thinking about that is that you would have a constant velocity right well do we have a constant velocity this entire time well no we know that at this point right here you have a very high velocity right you have a very high velocity we know at this point you have a very low velocity so you're you're accelerating this entire time and you're actually the more you think about it you're actually accelerating well you're actually accelerating at a at a decreasing rate but you're accelerating the entire time and then you're accelerating and then you're decelerating this entire time so your actual rate of change of X is not constant so you wouldn't have a zigzag pattern right and it'll keep going here and then you'll have a point here so what's happening when you start off you're going very slow your change of X is very slow and then you start accelerating and then once you get to this point right here you start decelerating you start decelerating until at this point your velocity is almost your velocity is exactly zero so your rate of change or your slope is going to be zero and then you're going to start accelerating back your velocities going to get faster faster faster it's going to be really fast at this point and then you'll start decelerating at that point so at this point what is this point corresponds to you're back at a so at this point your velocity is now zero again so the rate of change of X is zero and now you're going to start accelerating your slope increases increases increases this is the point of highest kinetic energy right here then your velocity starts slowing down and notice here your slope at these points is zero so that means you have no kinetic energy at those points and just keeps on going on and on and on and on and on so what does this look like well I haven't proven it to you but but out of all the functions that I have in my repertoire this this looks an awful lot like a trick metric function and if I had to pick one I would pick cosine Y because cosine when cosine of zero I'll write it down here cosine of zero is equal to one right so when T equals zero this function is equal to a so this function probably looks something like a cosine of and I'll just use the variable Omega T probably looks something like that this function and we'll learn in a second that it looks exactly like that but I want to prove it to you so don't just take my word for it so let's just figure out how we can figure out what w is and it's probably a function of the mass of this object and and also probably a function of the spring constant but I'm not sure so let's see what we can figure out well now I am about to embark into a little bit of into a little bit of calculus actually a decent bit of calculus and we'll actually even touch on differential equations this might be the first differential equation you see in your life so it's a it's a momentous occasion but let's just move forward close your eyes if you don't wanna be confused or go watch the the calculus videos especially the at least so you know what a derivative is so let's write this seemingly simple equation or let's rewrite it in ways that we know so what's another way what what's the definition of force force is mass times acceleration right so we can rewrite Hookes law as let me switch colors okay I choose which colors mass times acceleration is equal to minus the spring constant times the position right and I'll actually write the position as a function of T just so you remember we're so used to X being the independent variable that if I didn't write that function of T it might get confusing you're like oh I thought X's is the independent variable no we're actually going to do it because in this and this function that we want to figure out we want to know what happens as a function of time so actually this is also maybe a good review of parametric equations this is where we get into calculus what is acceleration what is acceleration if I have if I call my position X my position is equal to X as a function of T right I put in some time and it tells me what my x value is that's my position what's my velocity well my velocity is the derivative of this right my velocity at any given point is going to be the derivative of this function the rate of change of this function with respect to T so I would take the rate of change with respect to T X of T and I could write that as you know DX DT and then what's acceleration well acceleration is just the rate of change of velocity right so it would be taking the derivative of this or another way of viewing it it's like taking the second derivative of the position function right so in this situation acceleration is equal to we could write it as there's you know I'm just showing you all the different notations X prime prime of T second derivative of T of X with respect to T or these are just notational d squared X over DT squared so that's the second grade oh it looks like I'm running out of time so I'll see you in the next video remember what I just wrote