Springs and Hooke's law
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Spring potential energy example (mistake in math)
Welcome back. So let's do a potential energy problem with a compressed spring. So let's make this an interesting problem. Let's say I have a loop-d-loop. A loop-d-loop made out of ice. And I made it out of ice so that we don't have friction. Let me draw my loop-d-loop. There's the loop, there's the d-loop. All right. And let's say this loop-d-loop has a radius of 1 meter. Let's say this is-- this right here-- is 1 meter. So of course the loop-d-loop is 2 meters high. And let's say I have a spring here-- it's a compressed spring. Let's say this is the wall. This is my spring, it's compressed, so it's all tight like that. And let's say its spring constant, k, is, I don't know, 10. Attached to that compressed spring-- so I have a block of ice, because I need ice on ice, so I have no friction. This is my block of ice, shining. And let's say the block of ice is, I don't know, 4 kilograms. And we also know that we are on Earth, and that's important, because this problem might have been different if we were on another planet. And my question to you is how much do we have to compress the spring-- so, let's say that the spring's natural state was here, right, if we didn't push on it. And now it's here. So what is this distance? How much do I have to compress this spring, in order for when I let go of the spring, the block goes with enough speed and enough energy, that it's able to complete the loop-d-loop, and reach safely to the other end? So, how do we do this problem? Well, in order-- any loop-d-loop problem, the hard part is completing the high point of the loop-d-loop, right? The hard part is making sure you have enough velocity at this point, so that you don't fall down. Your velocity has to offset the downward acceleraton, in which case-- and here, is going to be the centripetal acceleration, right? So that's one thing to think about. And you might say, wow this is complicated, I have a spring here, it's going to accelerate the block. And then the block's going to get here, and then it's going to decelerate, decelerate. This is probably where it's going to be at its slowest, then it's going to accelerate back here. It's a super complicated problem. And in physics, whenever you have a super complicated problem, it's probably because you are approaching it in a super complicated way, but there might be a simple way to do it. And that's using energy-- potential and kinetic energy. And what we learned when we learned about potential and kinetic energy, is that the total energy in the system doesn't change. It just gets converted from one form to another. So it goes from potential energy to kinetic energy, or to heat. And we assume that there's no heat, because there's no friction. So let's do this problem. So what we want to know is, how much do I have to compress this spring? So what I'm essentially saying is, how much potential energy do I have to start off with-- with this compressed spring-- in order to make it up here? So what's the potential energy? Let's say I compress the spring x meters. And in the last video, how much potential energy would I then have? Well, we learned that the potential energy of a compressed spring-- and I'll call this the initial potential energy-- the initial potential energy, with an i-- is equal to 1/2 kx squared. And we know what k is. I told you that the spring constant for the spring is 10. So my initial potential energy is going to be 1/2 times 10, times x squared. So what are all of the energy components here? Well, obviously, at this point, the block's going to have to be moving, in order to not fall down. So it's going to have some velocity, v. It's going tangential to the loop-d-loop. And it also is going to have some potential energy still. And where is that potential energy coming from? Well, it's going to come because it's up in the air. It's above the surface of the loop-d-loop. So it's going to have some gravitational potential energy, right? So at this point, we're going to have some kinetic energy. We'll call that-- well, I'll just call that kinetic energy final-- because this is while we care about alpha, maybe here it might be the kinetic energy final, but I'll just define this as kinetic energy final. And then plus the potential energy final. And that of course, has to add up to 10x squared. And this, of course, now, this was kind of called the spring potential energy, and now this is gravitational potential energy. So what's the energy at this point? Well, what's kinetic energy? Kinetic energy final is going to have to be 1/2 times the mass times the velocity squared, right? And then what's the potential energy at this point? It's gravitational potential energy, so it's the mass times gravity times this height. Right? So I'll write that here. Potential energy final is going to be mass times gravity times the height, which also stands for Mass General Hospital, anyway. You can tell my wife's a doctor, so my brain just-- anyway. So let's figure out the kinetic energy at this point. So what does the velocity have to be? Well, we have to figure out what the centripetal acceleration is, and then, given that, we can figure out the velocity. Because we know that the centripetal acceleration-- and I'll change colors for variety-- centripetal acceleration has to be the velocity squared, over the radius, right? Or we could say-- and what is the centripetal acceleration at this point? Well it's just the acceleration of gravity, 9.8 meters per second squared. So 9.8 meters per second squared is equal to v squared over r. And what's the radius of this loop-d-loop? Well it's 1. So v squared over r is just going to be equal to v squared. So v squared equals 9.8-- we could take the square root, or we could just substitute the 9.8 straight into this equation, right? So the kinetic energy final is going to be equal to 1/2 times the mass times 4 times v squared times 9.8. And that equals-- let's just use g for 9.8, because I think that might keep it interesting. So this is just g, right? So it's 2 times g. So the kinetic energy final is equal to 2g-- and g is normally kilogram meters per second squared, but now it's energy, right? So it's going to be in joules. But it's 2g, right? And what is the potential energy at this point? Well, it's the mass, which is 4, times g times the height, which is 2. So it's equal to 8g. Right. So what's the total energy at this point? The kinetic energy is 2g, the potential energy is 8g, so the total energy at this point is 10g. 10g total energy. So if the total energy at this point is 10g, and we didn't lose any energy to friction and heat, and all of that. So then the total energy at this point has also got to equal 10g. And at this point we have no kinetic energy, because this block hasn't started moving yet. So all the energy is a potential energy. So this also has to equal 10g. And this g, I keep saying, is just 9.8. I just wanted to do that just so you see that it's a multiple of 9.8, just for you to think about. So what do we have here? [? I'll do ?] these numbers worked out well. So let's divide both sides by 10. You get x squared is equal to g, which is 9.8. So the x is going to be equal to the square root of g, which is going to be equal to what? Let's see-- if I take 9.8, take the square root of it, it's like 3.13. So x is 3.13. So we just did a fairly-- what seemed to be a difficult problem, but it wasn't so bad. We just said that, well the energy in the beginning has to be the energy at any point in this, assuming that none of the energy is lost to heat. And so we just figured out that if we compress this spring, with the spring constant of 10. If we compress it 3.3 meters-- 3.13 meters-- we will have created enough potential energy-- and in this case, the potential energy is 10 times 9.8, so roughly 98 joules. 98 joules of potential energy to carry this object all the way with enough velocity at the top of the loop-d-loop to complete it, and then come back down safely. And so if we wanted to think about it, what's the kinetic energy at this point? Well we figured out it was 2 times g, so it's like 19.6 joules. Right. And then at this point, it is 98 joules. Right? Did I do that right? Well, anyway I'm running out of time, so I hope I did do that last part right. But I'll see you in the next video.