To Master Physics First Master the Harmonic Oscillator

of all the systems you're likely to meet in your first physics class far in way the most important is the simple harmonic oscillator in other words the basic setup of block attached to spring no other system so thoroughly permeates virtually every corner of physics from classical mechanics to Quantum Mechanics to Quantum field Theory it's setup that you're guaranteed to meet again and again throughout your physics education and so it's good idea to learn it well from the get-go but why is this seemingly innocuous problem so ubiquitous in so many areas of physics that's what want to explain in this video we'll start by reviewing the basic setup and after that we'll get into why it's so important we've got block of mass sitting on frictionless table it's hooked up to spring which is itself anchored to wall on the other end if you give the block little kick or if you pull it out of ways and then let it go it'll oscillate back and forth from side to side like or cosine function we call these sinusoidal oscillations simple harmonic motion here's how we understand all using Newton's Laws whenever we move the block we stretch or compress the spring which therefore tries to pull or push the block back toward its equilibrium position the equilibrium is the position where the spring is happy and relaxed it's neither compressed nor stretched and so it exerts no force on the mass say we stretch the spring away from equilibrium by distance how hard the spring pulls the block back toward equilibrium will be proportional to as well as how stiff the spring is which is measured by another parameter called the spring constant so the force is minus KX and the fal ma equation for the block is ma = minus KX we can make this look little neater by moving the to the right hand side and then it's also convenient to define new symbol for the fraction that shows up there call it Omega squar where Omega is the square root of it'll be clear what that parameter means in second meanwhile if you've learned some calculus then you know that the acceleration is nothing but the second derivative of the position of with respect to time and so we can express the fal ma equation as the second derivative of with respect to Time = - Omega 2 * what that means is that to solve for the trajectory of the block we just have to ask ourselves what kind of function of when you take its rate of change twice in row gives us back the same function times this negative number minus Omega 2 that's exactly the property of and cosine and so the general solution of this equation is of = * cosine Omega + * sin Oma where and are numbers that depend on what the block is doing at equals 0 how far do we pull it out and or how fast do we kick it go through those derivatives and all this material in more detail in the notes which you can get at the link in the description so we indeed find that the block sinusoidally oscillates back and forth around its equilibrium position in simple harmonic motion the rate at which it oscillates is determined by Omega which we therefore call the natural frequency of the system you should check that it indeed has units of one over seconds as appropriate for frequency and notice that you don't get to pick Omega it's fixed by the stiffness of the spring and the mass of the block and however you set the block moving it'll always oscillate at the same rate unless you do something like kick it so hard that it breaks the spring or crashes into the wall here's an animation made that you can play with to see how this works just drag the sliders to choose the initial position and velocity that you want for the block and then press start to see what its position versus time graph is going to look like I'll put link to that too in the description and you can play around with it for yourself there's one more thing we need to review before we can step back and understand why this simple system is so prevalent and that's the potential energy of the spring as the block slides back and forth it's constantly speeding up or slowing down and so its kinetic energy = 12 mv^2 is always changing but if we add on the potential energy stored in the spring = 12 kx2 then the total energy is constant and we can check that just by taking the rate of change of in the first term we bring down that power of two to get * and then because of the chain rule we need to multiply that by the rate of change of which is the acceleration likewise in the second term we again bring down the two and get KX and then that gets multiplied by the rate of change of which is the velocity now if we pull out this common factor of then the thing in parenthesis vanishes because of ma so the total energy is indeed constant notice that the force = - KX is related to the potential energy = 12 kx^ 2 by = minus du by DX in other words the force is equal to minus the slope of the potential energy curve and that's the general relationship between force and potential energy notice that it means that in equilibrium where the force is equal to zero slope of the potential must vanish in fact you can think about the potential energy curve as if it were frictionless hill with particle sliding along it in the case of the harmonic oscillator the hill is parabola the equilibrium point is at the bottom of this well if you set particle at rest there at equal 0 it'll just happily sit there forever if you set it down away from the equilibrium however it will rock back and forth about the bottom of the well in the same way that the block oscillates back and forth around its equilibrium position thinking about the potential energy curve like this as hill is very powerful way of quickly developing your intuition for how particle in general potential will behave without ever doing the hard work of trying to solve the fals ma equation in an earlier video told you about how we can apply that strategy to learn ton about otherwise hard physics problems just by sketching picture of the potential like understanding the kinds of shapes planet orbiting star can follow I'll link that up in the corner if you haven't seen it even with very complicated potential where we have no hope of writing down simple solution for the trajectory we can still qualitatively understand what the particle will do just by picturing the potential like hill which brings us to the reason why the harmonic oscillator is so incredibly important and far-reaching consider particle moving in any potential of here's random example we would like to once again solve the fal ma equation to determine the trajectory of but outside of handful of potential energy functions described in textbook examples finding simple solution to this equation for complicated potential like this one will be hopeless task so how can we make progress well the first thing you should do when someone hands you potential energy curve is to identify its stable equilibrium points an equilibrium is again point where the force on the particle vanishes and so the slope of the potential will be zero there stable equilibrium point is one that's at the bottom of well as opposed to the top of hill we can of course write down the exact solution to the fal ma equation for particle that set down at rest at an equilibrium point the force vanishes and so the particle will just happily sit there forever on the other hand if you give the particle at stable equilibrium little tap it'll oscillate back and forth around the bottom of the potential just like the block on spring oscillated around the bottom of its parabolic well you'll see this behavior all the time in your daily life if you pay attention when you rock gently in chair you're oscillating around stable equilibrium when you drop an ice cube in drink it Bobs up and down around its equilibrium height your coat hanging on the hook is swaying slightly around its equilibrium axis and the basic reason is that when you're nearby stable equilibrium point of almost any potential the bottom of the hill looks just like the parabola of the simple harmonic oscillator and this isn't some rough qualitative analogy we can make it mathematically precise we can expand any potential of in powers of around stable equilibrium point using tailor series like you might have learned about in your math classes of = of 0+ Prime of 0 * + 12 Prime of 0 * x^2 and so on here's what's going on in this formula first of all if you're sitting right at xal 0 then when you evaluate the potential you're obviously going to get of then if you take tiny step away from there so that is small but nonzero number the potential is still going to be close to of 0 but now it'll be shifted slightly Away by the slope Prime of 0 times the displac in other words the change in is the rise over run times the Run that's already good approximation to most functions near given point but it's actually not that useful to us in this particular case because we've put our origin at an equilibrium point where of 0 and Prime of 0 both vanish to get better approximation we have to include the terms with higher powers of in the tailor series by including more and more of these terms we get better and better approximation to our function and when we add up all the infinite number of terms we reproduce the exact function but we don't need an infinite number of terms to get good approximation to our potential near the stable equilibrium point we just need the first nonzero one the quadratic term of = 12 * Prime of 0 x^2 but that's the potential energy of spring with spring constant equal Prime of 0 and that's the point almost any potential energy function however complicated reduces to the simple harmonic oscillator potential in the neighborhood of its stable equilibrium points this is why the simple harmonic oscillator is so prevalent systems tend to settle into stable equilibrium and small disturbances make them oscillate around it so the first thing we should do with any potential energy function is find its stable equilibrium points and then ask what happens when we perturb slightly away from them particle that's released there will oscillate around the equilibrium in simple harmonic motion with natural frequency < k/ that is Omega equal the < TK of Prime of 0 / to understand the physics farther away from the equilibrium points is usually much harder the bigger gets the more important the higher power Corrections in the tailor series like cubed and 4th become but we can often get approximate solutions by treating those as perturbations of our harmonic oscillator solution all this may or may not sound unfamiliar but if you studied the simple pendulum before you've seen it in action though maybe not using this language if pendulum is inclined at an angle Theta where the rod is of length then the mass will sit at height = lus cosine Theta above the bottom and so its potential energy is = MGH write for the Arc Length coordinate that the particle traces out so that Theta equal / then we can write the potential energy as of = mgl * 1 - cine of / here's what it looks like the stable equilibrium point is course in the middle at equal 0 where the pendulum hangs straight down at rest and we can check by taking the derivative of the potential energy the derivative of the cosine gives us minus and then we get 1 / from the chain rule making Prime of = mg * the of / and if we plug in equal 0 we indeed get of 0 equal 0 the second derivative is meanwhile NG / * cosine of / if we plug in = 0 there we get Prime of 0 = mg / note that it's positive that's how we know we're at stable equilibrium as opposed to the top of the arc where the slope of the potential also vanishes but the second derivative is negative that's an unstable equilibrium because if you take tiny step away from it the pendulum will swing far away so our tailor expansion around equilibrium is of = 12 Prime of 0 * s^ 2 and if we plug in Prime = mg / we get simple harmonic oscillator with that spring constant the frequency of oscillations around the equilibrium < TK will therefore be Omega equal theare < TK of which you might recognize as the familiar formula for the frequency of pendulum tailor expanding around the equilibrium in this context is often called the small angle approximation because it amounts to expanding the fal ma equation for the pendulum around Theta equal 0 in that neighborhood the system reduces to simple harmonic oscillator and so the pendulum Sally oscillates around its equilibrium position you can go through all these details in the notes wrote up which you can get at the link in the description you can also play with the animations made to go along with this video which think are really helpful for building up your intuition make sure you've seen that earlier video shared about how much you can learn about system by thinking about the potential as if it were hill please like subscribe and leave any questions in the comments and thank you so much for watching