a. When the displacement of a mass on a spring is ½A, what fraction of the energy is kinetic energy and what fraction is potential energy?

Video transcript

Everyone in this problem, we're asked to express the kinetic and potential energy in terms of the total energy for a horizontal spring. In simple harmonic motion, we're told to consider the displacement of the mass is C divided by five or C is the amplitude. We're given four answer choices here and they each have different ratios for the potential in kinetic energy in terms of that total energy. E, so let's start with the total energy. OK. The total energy E we call it E T is gonna be the sum of the potential energy plus the kinetic energy. OK. So we have U plus K. Now the potential energy we have a horizontal spring. So we don't have to worry about any gravitational potential energy, but we do need to worry about the spring potential energy. And so U is gonna be equal to one half K X squared. OK. And then we're adding one half M V squared are kinetic energy. Now, what we wanna do is pick a particular time point where we can find the total energy. OK. Now let's think about at max displacement at the max displacement of our spring, OK. X is gonna be equal to the amplitude A which we know is being called C and we know that the speed is gonna be equal to zero. OK? We get the maximum displacement, that speed is gonna be zero because that block or whatever is on that spring is gonna be stopping momentarily before turning around and heading in the opposite direction. So our total kinetic energy therefore is gonna be equal to one half K C squared. We have our displacement of C, our velocity is zero. So the second term goes to zero and that's gonna be our total energy. All right, now, we know energy is conserved. So again, that's our total energy throughout this entire situation. Now, we wanna look at when the displacement is C divided by five. OK. So why in the displacement is C divided by five? What happens? Well, our potential energy you is given again by one half K X squared when the displacement is C divided by five, that means X is C divided by five. So we have one half K multiplied by C divided by five squared. This tells us that our potential energy is equal to one half K multiplied by C squared divided by 25. And so if we want to write this as a fraction of the total energy or in terms of the total energy, what can we do? Well, let's look at the ratio of you to the total energy. This is gonna be equal to one half multiplied by K C squared, divided by 25 all divided by one half K C squared. Well, the one half K C squared is gonna divide out and all we're left with is one divided by 25. So what this tells us is that potential energy eu when the displacement is C over five is gonna be equal to the total energy E T divided by 25. Maybe we just multiplied both sides of our equation by that total energy. And now we have the potential energy expressed in terms of the total energy at this particular point, which is what we want. If we compare those to our answer choices. OK. We can already see that the correct answer should be A. OK. Option B has the potential energy as 24 divided by multiplied by E option C has E divided by five and option D has 4/5 E so none of those match what we have. We expect that the correct answer is gonna be a but let's move on to our kinetic energy and just double check that we haven't made a mistake. So moving to our kinetic energy K, OK. Well, recall that the total energy E T that we wrote at the beginning is equal to the potential energy plus the kinetic energy. Nice. Now we know that the potential energy is equal to one divided by 25 multiplied by that total energy. So we have E T is equal to E T divided by plus K. So solving for K K is gonna be E T minus E T divided by which gives us 24 E T divided by 25. OK? The two of them must add up to that total. And so K must be the total minus the potential. And that is the kinetic energy now expressed in terms of the total energy. If we look at our answer choices once again, OK, we thought that it was gonna be answer choice. A and we can see that that matches for that kinetic energy as well. So the correct answer choice is a here. We found that the potential energy could be expressed as the total energy E divided by 25. And the kinetic energy could be expressed as 24 divided by 25 multiplied by that total energy. E thanks everyone for watching. I hope this video helped see you in the next one.