Let Harmonic and the Science Behind It

a few weeks back i visited my friend simone who let me tie a bowling ball to a tree at her workshop in order to demonstrate the conservation of energy all without smashing my face and we're gonna come back to this most excellent demo because by the end of today's lesson you and i are going to figure out exactly how high that tree branch is just by watching how long it took the bowling ball to swing hello i'm diana cowan and welcome to lesson 16 of my class today's lesson is simple harmonic motion things that go back and forth i'm gonna ask you and you can think in your head or say out loud to your computer screen whether you think these things are simple harmonic motion okay the bowling ball on a rope swinging back and forth

yes that one was easy simple harmonic motion is the whole idea of this lesson and we started with that example so yes how about a spring vibrating like this hanging spiral decorations at a party moving up and down yes how about the ball in ping pong no yes the ball does go back and forth but it's not simple harmonic motion so this game maybe isn't that fair yet because i have to tell you more about what simple harmonic motion really means and where it shows up and how it's useful so to start modeling simple harmonic motion in a useful way let's use a tool that we already have conservation of energy okay the bowling ball swings in a gravitational field and it has the most gravitational potential energy here at the end where

it's at its highest point and then it's all converted to kinetic energy at the bottom when it's moving the fastest and then back to gravitational potential energy we also know that the ball is going to go to the same height on either side because it doesn't have a rocket attached to it and of course we're assuming that there's no air resistance so what can we say about the time it takes for each swing well we can measure it looking at the video at simone's workshop on my editing software i measured the time for one full return trip and it took 4.125 seconds and it's the same every time it better be or everything i'm going to teach you in this lesson is wrong so it is this time the return trip has a name we call it the period of oscillation or

just period now i said that the period is always the same but it's obviously not the same period for every single pendulum you know that if you've ever seen a foucault pendulum if you've never heard of these a foucault pendulum is this massive pendulum that swings back and forth and it changes its path the path turns as the earth rotates over the course of the day and i got to visit the biggest foucault pendulum in the world at the oregon convention center a few years back i was so fascinated that the cleaning staff let me stay after hours to just stare at it and film it so what do we need to change then about a pendulum to change the period how about the length of the string how about the mass of the cow

or the mass of earth or how about the size of my spherical cowl it turns out it's only two things that affect the period and those are the length of the string and the acceleration of gravity g that's nuts the mass of your cow has no effect so if i swung this little cow on the fuco pendulum versus the big ball they have it would be the same period this is crazy it's something that galileo first discovered in the 16th century the story goes that he was really bored in church and he would watch a giant chandelier on a big chain swinging back and forth and back and forth and he would think about its period of motion that's the kind of thing that i do during public presentations too so he wondered then and i'm going to ask you now

why does changing the mass and the displacement height of the object not change the period while changing the length of the string does hmm to answer that question is to get simple harmonic motion so it's going to help us to answer this question to relate the pendulum to another system the spring and i'm going gonna mathematically show you that the bowling ball pendulum or the cow pendulum when you drop it from short distances is mathematically identical to the spring cool okay so let's do this say we start with the bowling ball it's where it's hanging from on the tree and here's the rest position so my bowling ball is here and it is out at some angle theta from its rest position so when it's up here it feels a restoring force that pushes

it back towards its rest position in this direction so this force is my restoring force m g sine theta and i got the mg sine theta if you remember our free body diagrams because my gravitational force is downward with mg i get the component restoring back to its original position now we get to use one of my favorite little physicists mathematician tricks so grab your calculator and make sure that your calculator is in radians and not degrees because it does not work in degrees now plug in a small angle like 0.1 radians which is approximately 6 degrees and then when i hit the sign button i get 0.0998 which is approximately my original angle value 0.1 so i got my angle value back almost and this is true for all small angles

for all angles up to about 15 degrees in fact the small angle approximation is accurate to within one percent and this makes math really easy for physicists small angle approximation and that means that the restoring force on the ball is actually proportional to mg theta if my sine theta is approximately theta and that is for small angles and then i can do another cool trick using geometry if my string has a length l and i pulled up a distance x so we're going to say that this distance here is x then theta equals x over l and you might remember that from geometry on a circle where you take the arc of a circle that arc x equals theta times the radius of the circle in this case the radius is l i could either

consider x as this arc here or as the line straight across it doesn't really matter because of the small angle approximation those distances are about the same so theta equals x over l so i'm plugging that in here i get force equals mg times x over l now let's hang out with this equation for a second it's pretty neat because it says that the restoring force is proportional to the displacement x f equals mg x over l the restoring force also depends on the ball's mass m g the gravitational acceleration and l the length of the string now since m g and l are numbers we know they're constants that means that the restoring force is linear meaning that we graph it as a line just like a spring if you remember

from the spring lesson our graph looked something like this was my displacement and this is my force it was linear so both the pendulum and the spring involve a linear restoring force proportional to this displacement x the mathematical models for the pendulum and the spring are starting to look very similar particularly when the pendulum is at small angles now back to our original question we want to find out what the period depends on so let's bring in newton's second law so my force equals m a so i'm going to say this mg over l x equals m a so i've got m on both sides those cancel so for the pendulum the masses cancel and the acceleration of the ball at any point in its motion a is going to be g over l times x this is a key step because we've shown

that the bowling ball's acceleration does not depend on mass the m is gone it just depends on gravity and the length of the pendulum and of course the displacement which is a stone's throw away from showing that for the bowling balls period now a side note before we relate this to the period notice that for the spring acceleration equation the mass doesn't cancel so check it out my spring acceleration equation over here is f equals minus kx now if i set that equal to m a i don't have m's on both sides so i can't cancel the mass so if i set that equal to m a my acceleration is minus k over m times x which means the spring's acceleration is actually proportional to k over m times the displacement so i've

got this model now for the swinging bowling ball's acceleration which is linearly proportional to its displacement time for the big mama let's relate the bowling ball's acceleration to its period to do that we have to use a little calculus it's not scary calculus but we do need to talk about derivatives for a hot second and if you know calculus you know how hard it's going to be for me to try to explain what a derivative is in a couple sentences but here it is the derivative is a function that tells you how fast another function is changing so it tells you the slope of another function that's the derivative in two sentences and that's basically what all of calculus is i mean there's the integral too but

derivative is a big part of it so how does the derivative apply here well the acceleration it turns out is the second derivative of position acceleration is the derivative of velocity meaning it describes how fast an object's velocity is changing we write that like this the derivative of velocity with respect to time how velocity changes change in velocity over change in time meaning it describes how fast an object's velocity is changing which you know and velocity is actually the derivative of position dx dt which means it describes how fast the position is changing as you know or as i like to say acceleration describes the rate of change of position we could also write that as acceleration equals d squared x dt

and interestingly there are two very common functions that are their own second derivative like this so where we would take the derivative of something twice and get back the same function almost the cosine function and the sine it turns out cosine is the function that tells you how fast sine is changing derivative of sine is cosine and vice versa the negative sign is the function that tells you how fast cosine is changing derivative of cosine is negative sine the negative is the reason that i said almost so then the rate of change of sine is negative sine and the rate of change of cosine is negative say that three times fast i want to write all that out okay so d

squared dx squared of cosine of x is minus cosine x and d squared dx squared of sine of x is minus sine of x so my pendulum oscillating can be described using one of these sinusoidal functions oh this is like a really big moment i'm gonna tell you why it's big because sine and cosine are these wavy functions like this it looks like a sine or a cosine and that is the motion we're describing in this whole lesson it's a big deal okay back to the math so i'm going to pick cosine to describe my position with some angular frequency omega because oh by the way we've we've only really used angular frequency described going in a circle but it works the same for a cycle this is a cycle it just describes cycles per time in

this case and i'm picking cosine because for the cosine function when time equals zero cosine is at the maximum and that's true of my bowling ball too at time equals zero the displacement is also at its maximum then it swings and goes on so i'm going to say x equals cosine of omega t so then the first derivative of this is going to give me its velocity at any time my velocity equals dx the change in my position over change in time and that's going to be negative omega sine omega t and then the second derivative of x the acceleration equals negative omega squared cosine omega t which gives me my position function back i get my cosine omega t back with this negative omega squared in front of it so

then i can plug this in and rewrite this as negative omega squared x and now let's go back to algebra so acceleration equals negative omega squared x let's go back and equate that to our other acceleration formula which you may have forgotten by now but it was a equals g over l times x so we've got negative omega squared and i'm going to give it a negative because the restoring force is in the negative x direction so it's minus g over l times x and my x's cancel displacement doesn't matter and these negatives cancel so i get omega squared equals g over l and now i know the relationship between angular frequency and period which is what we're looking for period equals 2 pi over my angular frequency omega so now i

can find the period of my pendulum and i get that it equals t equals 2 pi square root of l over g and there it is and by getting to this equation my friends we have shown what we set out to do today we've shown that the only way to change the period of our pendulum is to change the length of the string period equals 2 pi square root of g over where l is link the string because i can't change 2 i can't change pi and unfortunately i can't change g unless i go to the moon or something and actually now we have all the tools now to do the thing that i promised you at the very beginning of this lesson we can figure out how high that tree branch is on simone's tree just by using the

period so if we assume that at rest the bowling ball is just above the ground which it's not it's about a meter up but we can use that too so i'm going to draw the tree here's my branch it's a weird tree here's my bowling ball if i assume that it's just above the ground then l is going to be the height of my branch but it's it wasn't it was about a meter off the ground so then the period of my swing is this t equals 2 pi square root of l over g and i know the period because we measured it to be about 4 seconds and i know obviously g is about 10 meters per second squared so 4 seconds equals 2 pi square root of the length i'm looking for over 10 meters per second squared do all this math and you get l the length

of my string equals 4 meters well that's eerie we started with four seconds for our period and we got four meters for the length of our string oh yeah the height of the tree branch is that plus one meter so all in all the tree branch is five meters high done amazing just by knowing the period but back to this eerie business so four seconds and four meters that's pretty intriguing and in fact if you plugged one meter into this equation two pi square root of g over l you would get really close to two seconds using 9.81 meters per second squared for g you actually get 2.006 seconds which means it takes almost precisely one second for a pendulum to swing from one side to the other and you can try this at home

cut yourself a meter long string and attach something pretty dense to it like some coins or some washers and then use a clock to time it and you'll get that one swing takes approximately one second to go across and then another second to swing back what is going on is it the stars landing in the seventh house of the age of aquarius no this is actually not a coincidence in the late 18th century during the french revolution when the french academy of sciences were trying to define the standard meter for the entire country they strongly considered using a pendulum set by the second if it swings to one side in a second it's a meter long and people did that informally but for various reasons when it came to the

formal definition they decided instead to define the meter as one ten millionth of the distance from the north pole to the equator measured along the meridian that passes through paris i wonder why they did that now of course the meter is defined in terms of the speed of light so none of those standards are used anymore anyways it's still a cool thing to try and now we've also just described how a grandfather clock works the pendulum of big ben is actually made of concentric tubes of steel and zinc that shrink and expand to oppose each other when the temperatures change so that the pendulum length stays constant which as you know now is very important and great now you have all the tools to understand

simple harmonic motion and you can answer these questions is a vibrating string on a guitar plucked simple harmonic motion yes how about a bouncing ball no the shape of a bouncing ball's motion doesn't look like this which is our simple harmonic motion sine and cosine it looks like this so it is harmonic motion but it's not simple this is a cool place to stop and describe what simple harmonic motion means so if you play music you've heard of harmonics and you know that you've got the main frequency of your note and you've got these higher frequencies above it those are harmonics and they add to your note and make it sound different with a simple harmonic motion what we're talking about is just one

frequency one simple sine wave that's the motion of our object so how about then a marble rolling back and forth inside of a semi-spherical bowl yes how about a door stop if you hit it and it vibrates yes that's just like the spring what about the newton's cradle nope the motion of one of the individual balls is not simply harmonic it is motion that looks something like this it has part of the swing like this part of the swing but then it stops so it goes like and then stops for a while and then it does the other part of the sign then stops and so forth it's like that but other examples of simple harmonic motion are the motion of a bobbing duck as ripples in the pond go by the motion

of air as you get a bottle to whistle as you blow across it a swing a tuning fork vibrating or anything that you twang like a piece of spaghetti or your motion falling back and forth through a tube that goes directly through the center of the earth or car suspension and of course the pendulum of a grandfather clock you can also build a simple clock using other systems that oscillate steadily like a mass on a spring so the period of a spring is proportional to and this is another useful period to remember so the period of a spring is proportional to i'm just going to give it to you p of a spring is proportional to the square root of the mass divided by the spring constant so square root of m over k times two pi now let's work a

quick problem with that to get it and by the way i always get confused about which goes on the top or the bottom m or k but i think to myself what should it do if i have a higher coefficient or a higher spring constant it'll be a stiffer spring so it'll oscillate back and forth faster so the period is going to be shorter so that means k should be on the bottom and if i put a big mass on it then it's harder for it to accelerate so it's going to swing back and forth more slowly making the period longer so m should be on the top so this equation makes sense okay now for the final problem you can get a rough idea of the spring constant of the springs in your car by thinking about the frequency of

oscillation when you're on a bouncy road and thinking about how often you go up and down all right so it turns out one hertz is pretty typical oscillation frequency for mass-produced cars so if my frequency of my car oscillating is one hertz and i'm assuming a mass of about one ton or a thousand kilograms we can work this out pretty quickly because one hertz is one oscillation per second which conveniently also then means one second per oscillation so that means that my period is also then one second so then plug it all into this equation oh i should square this it's a big one so t of my car is going to equal 2 pi times the square root of my mass of 1000 kilograms over k and my t of my car is one second i do all this math

and i get that k equals newtons per meter that's huge but remember that there are four wheels in your car and the springs are in parallel so they add so really we get about 10 kilonewtons per meter 10 000 newtons per meter for each spring and that's also huge but it seems pretty reasonable now you can use this model of a spring to build a clock in space a clock based on a pendulum won't work in space because you don't have gravity to restore it you don't have that restoring force so yeah you could use a spring and one last thing i want to show you this is a really cool simulation of the pendulum's position as it swings back and forth so the position of say the bowling ball on the string follows the cosine function and i'm

going to overlay on its velocity the velocity as we know now follows a sine function so it's slightly offset compared to the position the equation shows us that but it also makes perfect sense if you think about it because at time equal to zero you're at the top of your swing where you have no velocity but that's also your maximum displacement at the bottom of the swing when you have your maximum kinetic energy you're at position zero but your velocity is maximum and then you swing to the other side here velocity and position are changing both sinusoidally but just out of phase and then here is the acceleration overlaid and it is even more beautiful because the acceleration is in phase with the position but exactly opposite

that's where the negative comes in when is your maximum acceleration well when you're displaced the most when is your minimum acceleration it's at the bottom think about that for a second there's no acceleration at the bottom because there's no more gravitational potential energy at the bottom to be converted into kinetic energy and also because all of your gravitational force is pointing down tangential to your path so there can be no restoring force so i want you to think about that about all of this next time you're in a playground and get on a swing as i still do as you swing through the lowest point think about the fact that there's actually no acceleration on you

at that point and then as you're moving up you can feel the gravitational potential energy accelerating you backwards until you get to the max height and you have no velocity for a split second and then you're accelerating backwards think about them apples swinging all right now is the part of the video where i tell you about how what we just learned is a simplified model because the pendulum is only approximately a simple harmonic oscillator when it's swinging at a small angle we had to use that small angle approximation but that doesn't mean that this was all some dumbed down useless concept in fact it's so incredibly useful because every type of oscillatory motion to first approximation is simple harmonic

motion like the bouncing of your car in a bumpy road there's maybe some other interesting oscillations added in but there's one main frequency one main period of oscillation like the vibration of a plucked guitar string there's one main frequency or a swing set which is basically a pendulum okay that is today's lesson on simple harmonic motion when they ask you today what you learned on youtube here are your two key takeaways one simple harmonic motion is all about a linear restoring force and two the period of a pendulum is two pi square root of l over g and the period of an oscillating spring is two pi square root of m over k and now here are all the problems that we did in today's lesson because as you know the best way to make sure you

understand these concepts is to practice applying them yourself and so go do all these practice problems again on your own and now finally a message from a special guest hey everyone kyle here and i have a bachelor of science in physics with an emphasis in astrophysics i'm also a member of the united states space force and i'm currently at the air force institute of technology studying operations research now i just want to let you guys know that in the united states space force we use physics all the time in a lot of really cool ways we use capillary in motion which is the behavior of two orbiting objects about each other we use concepts of time dilation when we send information from earth to our satellites and back down we use

newton's laws all the time in launching and recovering our rockets and we even use orbital mechanics because things in the vacuum of space behave a little bit differently than they would down here on earth so physics is really awesome i just want to let you know that if you apply yourself you work really hard have some fun along the way physics is gonna take you far so it's great that you're watching these videos keep at it