Good morning! This is my review of Unit 7, Oscillations, for AP Physics 1. ♪ Flipping Physics ♪ This video is a free portion of my AP Physics 1 Ultimate Review Packet. If you are learning the AP Physics 1 curriculum this year, I definitely recommend you also get my full AP Physics 1 Ultimate Review Packet. Link is, of course, in the video description. Bobby, please tell me what periodic motion is. Uh, periodic motion is motion which is repeated in equal intervals of time. (Correct. And what is Simple Harmonic Motion?) Simple harmonic motion, which we often abbreviate as SHM, is periodic motion which results from a restoring force acting on an object where that force is proportional to the displacement of the object from equilibrium or rest position. Where equilibrium or rest position is the location where the net force acting on the object is zero, so the acceleration of the object is zero, so this is where the object can remain at rest. And a restoring force is always directed towards the equilibrium or rest position. Well done Bobby, thanks. Notice the motions of the hands of an analog clock are periodic motion, however, that periodic motion is not simple harmonic motion. However an ideal mass-spring system oscillating back and forth is in periodic motion and also simple harmonic motion. And remember an ideal mass-spring system has a massless and frictionless spring and the surface is also frictionless, yeah. Bo, what is the period of simple harmonic motion? The period of simple harmonic motion, capital T, is defined as the time it takes to go through one full cycle or oscillation. (Correct. And what is amplitude?) Amplitude of simple harmonic motion, capital A, is defined as the maximum distance from equilibrium position. {time with SHM video} Yes, thanks. Now, this mass-spring system has positions which I have arbitrarily numbered as {with video} 1, 2, and 3, and the mass-spring system passes through those positions in the following order: {with video} 1, 2, 3, 2, 1, 2, 3, and so on. Billy, what are four examples using these numbered positions the mass-spring system could go through during one full cycle? It could go through position 1, 2, 3, 2, 1, or position 2, 3, 2, 1, 2, or position 3, 2, 1, 2, 3. He said four examples, that is three. Oh, uh, it could also go 2, 1, 2, 3, 2. Yeah. Correct Billy. Now, each of you please tell me what we know about the force, acceleration, velocity, and displacement of the mass-spring system at one of the positions. Absolutely! At position 1 we know the velocity of the mass is momentarily zero because the direction of the velocity changes at position 1, so the velocity must momentarily be zero there. The displacement from equilibrium position of the mass at position 1 is equal to the amplitude, which is the maximum magnitude for the displacement of the mass from equilibrium position. Therefore, because the magnitude of the spring force equals the spring constant times the displacement from equilibrium position, the magnitude of the force from the spring on the mass must be at its maximum value at position 1. And because net force equals mass times acceleration and the only force acting on the mass in the x-direction is the spring force, the acceleration of the mass is also at its maximum value. And the force from the spring on the mass and the acceleration of the mass are directed to the left. But I thought Bobby said it would remain at rest at the equilibrium position, which is position 2, not position 1. Uh, I said it could remain at rest if it starts at rest at the equilibrium position because the acceleration at position 2 is zero. At position 1, and position 3 actually, the mass is instantaneously at rest because the velocity of the mass is changing directions. Oh, right. That makes sense. Thanks. You’re welcome. At position 2 we know the displacement from equilibrium position is zero, therefore, the spring force and the acceleration are zero. The magnitude of the velocity of the mass at position 2 is at a maximum. And the direction of the velocity of the mass at position 2 is either to the left or to the right. At position 3 we know the displacement from equilibrium position equals the negative of the amplitude. The magnitude of the spring force and acceleration are at their maxima. The spring force and acceleration are directed to the right. And the velocity of the mass at position 3 is zero. Perfect. Thanks. Now, the equation for the period of an ideal mass-spring system in simple harmonic motion is period equals 2 pi times the square root of the mass of the mass in the mass-spring system divided by the spring constant of the spring in the mass-spring system. Because mass is in the numerator of that equation, if the mass of a mass in a mass-spring system is increased, the period of the mass spring system increases. Because the spring constant is in the denominator of that equation, if the spring constant of a spring in a mass-spring system is increased, the period of the mass-spring system decreases. Because amplitude and gravitational field are not in that equation, if either the amplitude of the mass-spring system or the magnitude of the gravitational field the mass-spring system is oscillating in are changed, the period of the mass-spring system will not change. And realize the restoring force for a mass-spring system is the force from the spring acting on the mass. Billy, please do for a simple pendulum what I just did for a mass-spring system. Certainly! The equation for the period of a simple pendulum in simple harmonic motion is period equals 2 pi times the square root of L divided by the gravitational field strength the pendulum is in. Uh, A simple pendulum consists of a mass, or pendulum bob, hanging from a string and fixed at a pivot point at the top of the string. The “L” in the equation is the distance from the center of suspension to the center of mass of the pendulum bob and L is often referred to as the length of the pendulum. Because the length of the pendulum is in the numerator of that equation, if the length of the pendulum is increased, the period of the pendulum increases. Because the magnitude of the gravitational field the pendulum is in is in the denominator of that equation, if the gravitational field is increased, the period of the pendulum decreases. Also, a simple pendulum is considered to be in simple harmonic motion for small maximum angles and for AP Physics 1 that maximum angle can be as large as 15 degrees, however, some textbooks say the maximum angle needs to be less than 10 degrees, unfortunately, there is not universal agreement about that number. Technically it’s just that the larger the amplitude the larger the error in the simple harmonic motion equations and there are different estimates of how much error is acceptable, right? Yeah. And, because amplitude is not in the period of a simple pendulum equation, if the amplitude of a simple pendulum is increased, the period of the simple pendulum will not change – as long as it stays below 15 degrees. And because the mass of a pendulum bob is not in the equation, if the mass of a pendulum bob in a simple pendulum is increased, the period of the pendulum remains the same. And the restoring force for a simple pendulum is the component of the force of gravity acting on the pendulum bob which is tangent to the direction of motion of the bob. Uh, is that equation for the period of a simple pendulum in simple harmonic motion on the equation sheet? Yes, it’s right there. Then what’s that weird curly letter under the square root radical? Oh, right. On the equation sheet they use a weird curly L for the length of the simple pendulum. Oh. Right. Thanks. You are welcome. Impressive Billy. Thanks. Realize the frequency, lowercase f, of simple harmonic motion is defined as the number of oscillations per second. The units for frequency typically are cycles per second, which are called hertz, capital H, lowercase z. And frequency and period are inverses of one another. In other words, frequency equals one over period. Alright, An equation which can describe the position of an object in simple harmonic motion is position equals amplitude times the cosine of the quantity 2 pi times frequency times time. And remember, when using this equation for simple harmonic motion, your calculator must be in radian mode. And it’s probably worth noting that capital T refers to the period of the system while lowercase t is a variable referring to any particular moment in time for the system. Okay, let’s create the graphs of position, velocity, and acceleration of a mass-spring system moving through positions 1, 2, 3, 2, and 1. Each of you please describe one of these graphs. Absolutely! The position graph is a cosine wave, which starts at position 1 at its maximum value which is the amplitude, then it curves as a cosine wave until it gets to position 2 where it is at the equilibrium position, then it continues as a cosine wave where it is at its largest negative distance from equilibrium position at position 3 which is the negative of the amplitude, then it continues as a cosine wave until it gets again, to position 2, where it is again at the equilibrium position, and then it continues as a cosine wave until it gets to position 1 at its maximum value which is the amplitude. I will also point out that, if we had instead used sine in the position equation, the wave would have been phase shifted by 90 degrees or pi over 2 radians. In other words, when you use cosine, the initial position of the object is at the amplitude, or position 1, and when you use sine, the initial position of the object is at the equilibrium position, or position 2. That’s really the only difference between using cosine and sine in the simple harmonic motion position equation. Okay. For the graph of velocity as a function of time that goes along with the x-position equation that uses cosine, Well, the slope of position as a function of time is velocity, and the slope of the position versus time curve at position 1 is zero, so the initial velocity on the velocity versus time graph at position 1 is zero. That makes sense because we already showed the velocity of the mass at position 1 is zero. From position 1 to position 2, the mass moves to the left, so the velocity of the mass is negative and below the time axis. As we discussed before, at position 2 the magnitude of the velocity is at its maximum value. That means the velocity is the most negative at position 2. Oh, and the slope of the position versus time graph has its largest magnitude at position 2. I see it now. The velocity versus time graph is a negative sine curve which starts at zero at position 1, goes to its maximum magnitude negative value at position 2, goes back to a velocity of zero at position 3, has its maximum velocity at position 2, and then back to zero velocity when it gets back to position 1. As we discussed before, the acceleration of the mass is left and negative at position 1, and it also has its largest magnitude at position 1. At position 2, the acceleration of the mass is zero. Right, the acceleration versus time graph is a negative cosine curve. That’s all it is. Thank you all. That was great. Now let’s talk about energy and simple harmonic motion. The total mechanical energy of an object in simple harmonic motion is the sum of the kinetic energy and the potential energy of the system. Due to conservation of mechanical energy, the total mechanical energy of an isolated system in simple harmonic motion is constant. That means that, when a system in simple harmonic motion has its maximum kinetic energy, its potential energy is at its minimum. And when a system in simple harmonic motion has its maximum potential energy, its kinetic energy is at its minimum. When looking at a horizontal, ideal mass-spring system specifically, the potential energy of the system is elastic potential energy, or one-half the spring constant times the displacement from equilibrium position squared. That means that the total mechanical energy of a mass-spring system when the displacement from equilibrium position of the mass is equal to its maximum value, which is the amplitude, the total mechanical energy equals one half spring constant times amplitude squared. And, you can see from the equation that, if the amplitude of motion of a system in simple harmonic motion is increased, the total mechanical energy of the system is increased. The total mechanical energy of a horizontal, ideal mass-spring system in terms of maximum speed is one-half mass times maximum speed squared, because we just replace the speed in the kinetic energy equation with maximum speed. Because both of these equations equal the total mechanical energy, we can set these two equations equal to one another to solve for the maximum speed of a horizontal, ideal mass-spring system. Total mechanical energy equals one-half spring constant times amplitude squared which also equals one-half times mass times maximum speed squared. The one-half’s drop out. Everybody brought one-half to the party! Sure. And we can rearrange that to get the maximum speed of an horizontal, ideal mass-spring system equals amplitude times the square root of the quantity spring constant divided bythe mass. And you can see from that equation that, if the amplitude of motion of a system in simple harmonic motion is increased, the maximum speed is also increased. Thank you very much for learning with me today, I enjoy learning with you.