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Energy in Simple Harmonic Motion

Patrick Ford
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Hey, guys. So now that we've seen the forces of mastering systems, we wanna look with the energy looks like for mastering systems. So at any point of simple harmonic motion, you've got this mass that's moving back and forth. And so it's exchanging energy, right? The energy that's associated with its motion is the kinetic energy and the other type of energy that might have has to do with the compression and stretching of the spring. So it's got some elastic potential energy Now, in all the cases that we considered, we've considered no friction. So there's no work done by non conservative forces and without allows us to do is use energy conservation. The mechanical energy is conserved. And so what we're gonna do is compare the energies at 22 special points of mass spring systems. The two special points are gonna be the amplitude and the equilibrium, so we've got minus and plus a and then we've got to the equilibrium in which X equals zero. So let's take a look what the energy looks like at all of these points. Well, we've got elastic potential energy here when X is equal to positive or negative amplitude. So that means the elastic potential energy is maximized here, one half k square. But the velocity of the specific point is equal to zero at the end points. And so there's no kinetic energy, so that contribution actually goes away. So there's no kinetic energy, which means that the Onley mechanical energy that we have is just due to the potential energy, theological energy. At that end point at the equilibrium here the situation is reversed because now what happens is when X is equal to zero. The elastic energy goes away. What does that make? It is the kinetic energy look like? Well, at the equilibrium position. We've got this object either flying to the left or to the right, and it's got its maximum velocity here. So that means that the kinetic energy will the elastic energy is equal to zero, and the kinetic energy is maximized. It's got one half V max squared, so that means that the total mechanical energy is on Lee, made up of the kinetic energy que zero Well, what about any other points and any other points? So, for instance, find Just label this point right here and I call that p so it X equals P. The elastic energy has to do with what the deformation looks like. So how much it stretched or compressed. But it also has some non zero velocity. It's also going either to the right or to the left, with some velocity. So it's got some kinetic energy. The elastic energy is just going to be one half times k x p squared, whereas the kinetic energy is gonna be one half M v p squared. And those were both not zero. So that means that the total mechanical energy is actually the sum of all of those energies. So now, if we've compared all of these energies that these different points and we remember that the total mechanical energy is conserved, it means that these things just get exchanged. They never get lost or destroyed. And so we can actually just plug in our expressions here for our energies. One half k squared. We've got one half m v max squared. That's equal to one half k times x p squared plus one half M v p squared. So this is the energy conservation equation. First Springs. It's really powerful. And what questions they're gonna ask you is ask you to relate all of these energies to each other, so we'll give you one and ask you for another on. So you need to know how to use this equation right here. So what we can do is we can use this equation and actually solve for the last cinematic quality that we need. That's the velocity is a function of position. So to do that, I'm just gonna cancel out some halves. I'm gonna manipulate some some numbers around, and then I get that the velocity is a function of position is equal to the square root of K over em times the square root of a squared minus x p squared. So that's basically it. Let's go ahead and take a look at an example. So in here in this example, we've got a 5 kg mass. We've got the k constant and we've got the amplitude, and we're supposed to find the maximum speed. So in this first part here, if I'm asked to find what the Max is, I'm just gonna write out my conservation of energy equation. Now, this is the conservation of energy equation. So I got all these things that are equal to each other. So I'm looking for specifically V max. So I'm looking for this guy right here. So let's take inventory of all my variables, I know what the mass is. I'm given what the amplitude and the K constant is. So I can actually just take a look at these two relationships in order to figure out what the max is. So the max, if I go ahead and solve for that, I'm gonna cancel out the half terms and they're gonna move the M to the other side and then take the square roots. So I'm gonna get K over em. And then I've got, uh, the amplitude that's squared. So yeah, the amplitude. So you might recognize this equation. V. Max is equal to a times omega, and that's because these two things are the same. So when I got the square root of K over over em, you might recognize that as that omega symbol. So if I've got all I need, I've got all of these variables I need. So I just take the square root of 30/5, which is Cavor M multiplied by the amplitude and when I get is a V max of 0.98. That's meters per second meters per second per second squared. So what is part V? Ask parties asking us for a speed at a specific position. So for part B now for me. So for part B, we've got X equals negative 0.2 m and we're supposed to find the velocity. So we're gonna use our velocity as a function of position equation. So we've got V when X is equal to minus 0.2 is going to be what? What we've got the square root of 30/5. That's K over em. And then we've got 04 squared minus 0.0 point two squared, and we've got this negative sign right here. So you go ahead and take the square root of that. What you're gonna find is that the velocity is equal to 0.85 m per second. So that is the velocity as a function or at that specific position. So now this last part here asks us to figure out what the total mechanical energy of the system is. So the total mechanical energy means we're gonna use that conservation of energy equation So that conservation of energy equation is this guy right here? So now which one is the one that we can use? Well, the answer is you can actually use any of them. So notice here how I have the k. I have a I also have m and V Max, so I can use all of those. So if I use this guy right here, So if I use the first terms, but I'm gonna use is one half the cake, Constant is 30. And then I got the amplitude, which is 0.4, and I square that what I get is a total mechanical energy of 2. jewels, right? So if I use those other two variables So if I go down here, I've got one half times five and then v max was 0. squared, so I got a total mechanical energy of 2.40 jewels, which is the same exact answer. So the idea is that you can actually use any one of these to relate the energy different positions. Alright, guys. So that's it for this one. Let's keep going