**Concept:**

LetÕs introduce, this are some definitions. This is a really important page because thereÕs a lot of really key stuff you need to know from here, and itÕs the foundation for anything else. So, when an object, this is really simple, when an object moves in a repetitive way, if something moves in the same way over and over again, we call this periodic motion, and the best example of periodic motion is, whatÕs an example of periodic motion? Spring, right? ThatÕs boring. So the best example of periodic motion is sex, alright? So periodic motion aka oscillations. ThereÕs a lot of perverting stuff in this chapter as you can see, so oscillations, that could be pervert too. The classic and boring example of this, is the mass connected to a spring, this is the mass spring system, which is what 90% this chapter is about. In fact on this and at the end just like Oh, what if itÕs this way, and then thereÕs like a small adjustment but most of the other things would be the same, okay? So, the mass spring system, how does this works? So, I like to draw a line with three little dashes, we'll see more stuff later, the middle is my X equals 0, and we are going to call this equilibrium point. This is also where the spring starts, right? This is where the spring has no compression, so IÕm not going to draw a spring here, because itÕs going to be cut, but imagine thereÕs a spring here attached to a block, so thereÕs sort of a little block here, and thereÕs a spring here, okay? What I am going to do is IÕm going to get this block and pull it all the way over here, and then release, when I release, where does it go? To the left, because itÕs always going back to 0, right? So, weÕre going to pull this over here, and IÕm going to release it from here, so this is my initial position, X initial, initial position or initial compression, okay? Your initial position is going to be your amplitude, okay? So, what IÕm going to do is kind of go back and forth it between this definitions in the back, cool? So, the amplitude is your maximum displacement, itÕs the magnitude of X because if youÕre in the left, then itÕs negative 5, itÕs just 5 or it doesn't matter, right? So, it is always the initial pull. One thing thatÕs potential tricky about in the chapter, itÕs theyÕre not going tell you the amplitude of this 5, theyÕre going to use pull 5 and you release, whatÕs the amplitude? 5, okay? So, you just have to notice like little motions, right? So, thatÕs your initial pull, we give the big A, so X right there would be positive A and X over here would be negative A. Something you should actually know, is that if pull I 5 and I release, itÕs going to go 5, 0, 5, 0, 5, 0, forever, okay?

So if itÕs the amplitude is 3 in one side, it's 3 to the other. So, it does in reality, right? Because thereÕs friction and thatÕs called a damping oscillation, we donÕt, itÕs a damping oscillation, but we donÕt have to do that because we are going to skip that but in reality it would eventually get to a 0, in reality it would look not like this, but like this, right? But we are going to sum the friction which makes everything much easier, okay? We are just going to jump back and front between A and negative A. The reason I call this negative A, is because A is a positive number and this becomes negative because A is to the right, it's all that matters. Positives to the right, negatives to the left, cool. LetÕs talk about, before we get into this, letÕs talk about the velocities here. These are the three important points here, the middle and the m's, okay? What do you think the velocity is here? Is velocity there? What do you think? No, right? So, a lot of this chapter, you just need to play this out in your head, like literally we are just going to have it like go and back forth, and itÕs weird but you are moving here, you release and it goes, stops, stops, okay? So, the velocity at the end is 0, and you have to know that, you donÕt need to memorize that, you can just kind of, What about the mu? do you think the velocity mu is 0? No, because it pass through, right? So thatÕs 0, let's move along, the velocity in the middle, I'm just going to say itÕs the maximum velocity, and it should make sense, right? If you have 0 at the m's, the highest has to be in the middle, okay? Look at this, check this out. When you are here you have been pull this, right? So you are getting faster, faster, faster, faster, faster, as soon you cross, what happens to your force? Flips, so now I'm going to slow down, does that make sense? So, this is the fastest youÕve ever been, is in the middle, because you are accelerating and then you start decelerating, okay? So, is like you are getting faster, faster, faster and then you start going slow, thatÕs the middle right there, maximum, alright. Period, period is the time for a complete cycle, it's the time for complete revolution, it's the time it takes to do a full rotation cycle of something. A thing you need that you need to realize here, remember, period is a little kind of counterintuitive, is not the time to go from here to here, thatÕs what a lot of people initially think period is, is the time to go back, okay? So, period is the whole thing. Now, let me make this even a little bit more in depth, but to let you understand something better. The whole thing is the period, so what fraction of the period do you thing this is? Half, What about this? A fourth. What about this? An eighth. Right, so thatÕs not an eighth, that stops there. You can only keep breaking this down until the half, okay? So, what I am going to do is just kind of do this, T over 4, T over 4, this is telling two things, one thatÕs the lowest you can cut, this in the middle is not T over 4 is some other number, okay? Why? Because thatÕs just not really works, it doesnÕt actually keep having, it doesn't matter, bla, bla, bla. So this reminds you of two things. One, this is the most you can cut, right? Think of this like this is the atom, you canÕt get any smaller than that and if you could then it would be the exact half of it. But this also tells you that if this is a quarter and a quarter, this whole thing is half and then you need one, two, three, four of these to do the full period, okay? So, if you just remember that every little sort of half of this, itÕs actually a quarter of the period, you are good, awesome. Frequency is the inverse of this thing, there is a thing you need to know called angular frequency, this is the most important variable in this chapter, but IÕll talk about this a lot, there will be a ton of this crap, and itÕs just the regular frequency multiplied by 2pi, okay? 2pi, pi if you remember has to do with circles, the unit circle, so pi is considered a rotational number or an angular number, so thatÕs why this is called angular frequency, because you have a frequency and you multiply by 2pi. Which, by the way, in other chapters this is called, in chapter nine where we will need rotation, we have the same exact thing, but instead of being call angular frequency, this is called angular velocity, right? ItÕs the same thing. I can write 2pi, F or I can write 2pi over T because F and T are inverse of each other. So, letÕs talk about the acceleration real quick. As you are moving left and right, right here, as the system moves, the only force doing work on it is the spring force, thereÕs gravity, a normal, gravity and normal donÕt do work, because remember, if you are at 90 degrees, you donÕt do work, right? So as you are moving left and right, gravity is up, gravity is down, m, g is up, so you don't move anything to it. The only force acting in this thing is the spring force, okay? Therefore, spring force is Fx equals negative K, x, thatÕs the only force that acts on you, so let me do this real quick, sum of all forces in the x axis equals m, ax, but the only force is the force of the spring, F is equals m, a, the only force is the force spring. The more we replace this force of spring with negative Kx, and then I am going to sum for ax. No, I am going to write it a little different than we have been seen, I am going to write it like this, a parenthesis X equals negative Kx over m. LetÕs put this on the box, this is another equation you can note. I am going to give you about fifteen equations today, it fucking sucks, right? Maybe like twelve. Most of them are going to be in the equation sheet, some of them arenÕt but you are going to remember them, because theyÕre going to make your life much easier, okay? aX equals Kx, m. ItÕs the first time that I write something like this here, you might remember this from math class, this X here is, this is function notation. The reason I am saying this is so that you donÕt think youÕre supposed to multiply by this X. This means a as the function of X, you plug, if you want to find a when x is 2, you write a 2 equals negative K 2 over m, this is the function, itÕs not multiplying, think of this as a just by itself or like this, aX equals negative Kx over m, okay? ItÕs functional notation, donÕt multiply by the X. Otherwise it would be silly, like why would you even write the X, it just fucking cancels, inaudible, right? You just cancel the two, so donÕt do that, okay? ItÕs just to say it's a function of X. Why are we doing this now? Because we are going to have an equation later for a as a function time, okay? lot of equations, don't get scared, most of this is like, oh, it's this equation, plug in numbers, done, okay? So, the reason why I'm deriving this is, one, you need to know, and two, you need to look at this stuff here. So look, K is a constant, mass is a constant, the only thing that is changing here is the X, okay? So, you can see that as X grows, my acceleration grows, okay? As X grows, my acceleration grows, what is the acceleration in the middle? Is it 0 or is it maximum? Why is it 0? ThereÕs no X, right?

So look, if X is 0, a is 0, okay? And at the m's my acceleration would be maximum, thatÕs the most acceleration I can have, is at the ends, why? Because at the ends, what is the value of X at the end? If youÕre all at the end, what is the X value? The highest into the amplitude, right? So the X is the highest possible, a is the highest possible, does that make sense? So, itÕs always like this: minimum, maximum or maximum, minimum, right? Always opposite to each other, ok. What about the force? Where is the force at 0, in the middle or at the ends? ThereÕs sort of two ways you could know that. Where is the force 0, at the middle or at the ends? So, the force of spring is Kx, so where is the force 0? What was it? What? Good, thatÕs a third way, equilibrium forces cancels to forces to be 0 in the middle, another reason is because force spring, right here, is Kx, where is the 0? Fx is a 0. Another way is F equals m, a, right? If a is 0, F is 0, that make sense? We just kind of what you said, right? So right there, so this is maximum right here. So, if you notice, X, a and F are all sort of go together, theyÕre in sync, right? And v is the weird one, okay? And because this stuff moves like this, right? So, thereÕs this experiment that some guy did or whatever, inaudible you might have talked about this, imagine that I am doing this, right? And then thereÕs like a little needle here or a pen, this is will be weird, right? But if you are doing this, and then thereÕs like a paper thatÕs rolling up and you are writing on the paper as you are doing this, right? They would look like a sine function, alright? Because you are doing this shit and the paper would look like this. Because this thing is moving left and right like that, everything would look like a sine function, okay? The difference between this three, this four actually, is that X, a and F are all in sync, so they all look like that and the v is the other one, so itÕs going to look sort of like this, that was terrible but kind of the inaudible point, okay? So the idea is just theyÕre all sync up except v, v is like doing the other thing, when everyone else is at its maximum value v is at the minimum, does that make sense? Cool. You donÕt actually need to inaudible this diagram, I just want to make the point that they're a little bit different, This entire thing, you should be able to play this out in your head, okay? 0, velocity is the amplitude at the end, whatever, and then this is just because a and F are linked up to X, you get easily see by the equations, okay? Also, think about this, this is a resistive, a restoring force, so if youÕre to the left you're being pull to the right, so there's a force, right? And then you pull this way, so thereÕs a force as well and if thereÕs a force f equals m, a, there's an acceleration. Alright, enough for that. So, that was my attempt to build sort of mental picture and making that kind of playing back and forth in your head and this will make problem solving easier, so you donÕt have to memorize a hundred things, you can only memorize like twenty, right? Instead of a hundred and thatÕs too much. Alright. This is a concept, this is a purely conceptual point, I need you to just remember this sentence, because it might show up, okay? So a really big conceptual here. ThereÕs this thing called Simple Harmonic Motion, what is Simple Harmonic Motion? It doesnÕt matter, itÕs a type of motion, harmonic motion is repetitive basically, but itÕs a type of motion that is simple, which means you can use all kinds of simplifications, blah,blah, blah and use certain kinds of equations, for example, if I was asking you, does it keep going five and five, and not five, four, three and two, well, a simple harmonic motion itÕs five and five and everything is nice and beautiful and simple, simple harmonic motion is simple. as opposed to complicate harmonic motion? Which you donÕt want to see, right? So, when do you have simple harmonic motion? First of all, you are always going to have simple harmonic motion, because if you didnÕt, inaudible would be fucked up and you wouldn't be able to solve it, okay? So you are always going to have simple harmonic motion, but you still need to know what the condition for it is, this is kind of Bio-class, you have to remember that shit and that's it. The restoring force, I am going to call this Fr inaudible the space is tightening there, sorry about that, has to be directly proportional to the displacement from equilibrium, displacement form equilibrium, this entire phrase displacement from equilibrium is simply your X, thatÕs it. So, how would equation like this, what would the equation look like? What is the condition for simple harmonic motion? And thereÕs going to have a bunch of bullshit and one of them is going to say the inaudible, this one, okay? Whether you understand it or not, I like to simplify this by saying that the restoring force is directly proportional to the displacement from equilibrium, itÕs a lot better than this entire sentence. This little Jesus fish right there, means directly proportional. So you need to remember that, alright? If I inaudible, to you at South Beach next weekend I am going to ask you what is the condition for simple harmonic motion? You better have it, alright? Cool. LetÕs do a problem. A mass on a spring is pulled 1 meter away from itÕs equilibrium position, then itÕs released from rest. So letÕs draw this. I like to draw a start here, just to sort to get the mental map going, right? You always want draw, like does shit to your brain, and remembers information better, weird, so we draw that little thing in there and this is 0, amplitude, amplitude. And it says here, you pull 1 meter away from the equilibrium position and you release. It doesnÕt say if you pull into the left or the right, it doesnÕt matter, so letÕs just say here is equilibrium, I am going to put 1 meter and I release. Part A, is this periodic motion? Yes or no? And look at the definition periodic motion in the first line of this page, is this periodic motion? Yes, right? As it keeps repeating left and right, itÕs always periodic motion, in this situation, inaudible motion. B, is it simple harmonic motion? Yes, why? Because mass spring system is simple harmonic motion, always, okay? And the longer answer is because of this is crap here, but it just always is, okay? C, ok so it says the mass takes 2 seconds to reach maximum displacement on the other side, letÕs draw this out, okay? Always draw. It takes 2 seconds to get maximum displacement on the other side, so itÕs to go from where to where that it takes 2 seconds? From A to negative A, awesome. So, to reach maximum displacement on the other side, okay? So let me just tell you, get something out of the way real quick, this chapter is easy, okay? But in physics thereÕs this thing, thereÕs a theory that I invented called, remember how is Conservation of Energy, Conservation Momentum? in physics they have this thing called Conservation of Difficulty, right? So this chapter is easy, so what are we going to do? Make up a bunch of bullshit to try to make it hard, right? The other stuff is hard, or some of the other stuff we did is hard, so that they just keep a bare minimum problems like inaudible this one thing right, here they're going to do all kind of bullshit to try to make it confusing, okay? But thatÕs what we are going to go through so inaudible all the shit down, what is the amplitude? X? Well, the letter is A, 1 meter, right? Why? Because amplitude is your initial pull and itÕs also your maximum displacement, right? So, amplitude is, you can think of it as your initial pull or you can think of it as your maximum displacement, X max, the absolute value of your X max. How much did you pull? 1 meter, right? WhatÕs the maximum can ever be on roof? 1 meter, thatÕs it. So, when the problem said you pulled 1 meter away, thatÕs your amplitude right away, okay? And you would be really sad if you got this wrong in a test, right? That you pulled 1 meter, whatÕs the amplitude? 1 meter, right? If you got that wrong, just should really reconsider reading and everything, right. Period. WhatÕs the period? So you have to be careful here, what is the definition of period in these problems? A, all the way back to A, so whatÕs period here? 4 seconds. So I hope you can see how you could, you know, if you do this quickly and you donÕt remember some basic shit, you might have thought that period was 2, right? So be careful, again itÕs easy but is one of those things that if you get it wrong, you better like inaudible yourself forever, right? And you should. So, like never get over. So, frequency is, whatÕs the definition of frequency? Or whatÕs an equation for frequency? Frequency is related to the period, inverse of the period. Awesome. 1 over T, 1 over 4, inaudible this in fraction, I mean inaudible is 0.25. Units of frequency? Hertz, thatÕs good hertz, alright, and F angular frequency, what variable is angular frequency? 2pi, F, the variable is omega or W, angular frequency is 2pi, F, so remember, angular frequency is just frequency with the 2pi in front of it, thatÕs why itÕs called angular, itÕs got 2pi. So you just multiply this and weÕre going to go 1.57. You guys remember what the units for omega was, like two chapters ago? The chapter of angular rotation, was called angular velocity? Regular velocity is meters per seconds, angular velocity was radians per seconds, itÕs the same thing. If you remember, awesome, If you donÕt just remember, just to memorize from now that omega is radians per seconds, thatÕs it, okay? We have to do it like very little here, you have to know almost nothing, itÕs just very, let me, you have to know very lot of stuff but thereÕs very basic meters per second, you have to know how to find A, how to find your T, and everything is just

plug inaudible, cool? I am going to do another one, and IÕll tell you a few more things and we're going to do a bunch of problems. So, the graph shows position over time. Every time you see a graph, the first thing youÕre going to look at is what is on the axis, what is the graph showing you. HereÕs position. ItÕs usually time over here, right? ThatÕs how we use it, graph things, where time is the X axis but this is the position here, well, this is in centimeter, here, so I might have to change that, cool. And this is 2, 2 is the highest value here and negative 2 is the lowest down here. One is this point right in the middle, this is a 2 over here, this is a 3 here, this is a 4 here, which means, If you look at that gaps here and between this means that 0 is right here, right? See 1, 2, 3, 4, so many gaps. Cool. The mass spring system has a K of 400, letÕs write that now, K equals 400. A, what is the amplitude? What is the amplitude here? 4? How do you get 4? But thatÕs time. So whatÕs the definition of magnitude? The maximum, the highest value of X you can have, right? As you are moving left and right, the maximum value of X you can have. By the way, if you have this graph, inaudible if you can picture this, you got this graph and you flip this way, thatÕs kind of what spring do, isn't it? you got this graph and you flip it like this, it would look like this, thatÕs exactly what springs do. And this is our maximum X right here, 2, 0, 2. So the amplitude is 2 centimeters, positive or negative? positive, amplitude is just by definition positive. Now, check it out, itÕs not 2 just because itÕs 2 here, itÕs 2 because itÕs 2 on X graph, if a gave you, this is really important, if I gave you a v, t graph, and this is a 2, this 2 is not your amplitude, this 2 is your maximum velocity, does that make sense? So here, the max equals 2, here x-max equals 2, and that is what we call amplitude, so go ahead and draw this here, letÕs put in a big parenthesis, just to let you know itÕs not just because itÕs the amplitude of that graph, you have to know what the graph is telling you, amplitude is the maximum value of X, which here you canÕt know, you cannot know what the amplitude of this thing is, you just know how fast is moving, okay? So, you just have to be careful with the graph is telling you. B, B says, where is the period? Period is the attempt to go for a point back to the same point, so what do you think the period is? 2 seconds. How do you do this? Take a point to all the way back to the same point in 2 seconds. You could have picked here, and it goes from 1 to 3, okay? By the way, this is always how you find the period of any wave, next chapter we are going to do is sound of waves or waves and sound, itÕs going to be exactly the same thing, you are going to have some waves like this, thatÕs how I find the period, you pick a point here, you pick a point here and thatÕs the gap between the two, 2 seconds. What if a gave you this chart, it works the same way as well here, okay? So, the period would also have been from here to here, so if this was a v, t graph and looked exactly like this, the period would also be 2, okay? So, this is a 2 here, the period would have been 2 as well, so in that sense that aspect of the two graphs is the same. How do I find frequency if I have period? 1 over T, so itÕs 2.5 hertz. How do I find angular frequency? Angular frequency is 2pi F, right? So 2pi times 0.5, so itÕs just pi or 3.14 radians per second, inaudible, cool? E, at what times does v max occur? Remember, weÕre here, where do you have the maximum velocity? So, think about this here, okay? Where? here we have the maximum velocity? In the middle? Right. So v max happens at X equals 0, okay? So, at what time does that happen? At what time does X is equals 0? Because look, the reason I wrote this by the way, you canÕt find v max directly from this graph, right? Why is that? Because this graph shows you X, but if you know that v max happens on X equals 0, you just have to look at the graph and say, where is X equals 0 in this graph? And X equals 0 cross right here, right? X equals 0, so this happens at, what is this number? 0.5, which is T equals 0.5, 1.5, 2.5, 3.5 centimeters. Now, what do you do with F? F is very similar; at what time does a max occur? Now, remember, you canÕt directly tell a max from this graph, so what you have to do is figure out that a max happens at X equals whatever, and then figure out, at what time does that happen? So, do that really quick. In what X positions do you have a maximum acceleration? If you donÕt remember, feel free to look back into that page, as long as you are eventually going to know this without inaudible. So a max is what X equals 2, good, not 0. I'm looking for a, so thatÕs cool, a is 2, right? A max happens at the amplitude, when X equals 2, positive or negative 2, right? So what times are those? 1, 2, 3, 4, 0, so it happens here, here, here, here and here, so 0, 1, 2, 3, 4 and if this thing kept going it would be 5, 6, 7 does everybody see why those are the numbers? You see how v max and a max are inverted 0, 0.5, 1. 1.5, cool.

A simple harmonic oscillator consists of a mass on a horizontal spring, oscillating with an amplitude A and negligible friction. What is the distance of the mass to the equilibrium position (x = 0) at the instant when the kinetic and potential energies are equal?

i. 0

ii. A

iii. A/√ 2

iv. A/2

Watch Solution

The displacement in simple harmonic motion is a maximum when

A. acceleration is zero.

B. potential energy is maximum.

C. kinetic energy is maximum.

D. potential energy is zero.

Watch Solution

A 250 g mass on a spring oscillates with the following equation:

x(t) = (2.8 cm) cos((3.4 s^{-1})t)

How long does it take the spring to travel a distance of 18.3 cm?

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