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Vertical springs and energy conservation

Video transcript

- [Instructor] Let's say you've got a mass connected to a spring and the mass is sitting on a frictionless surface. If the mass is sitting at a point where the spring is just at the spring's natural length, the mass isn't going to go anywhere because when the spring is at its natural length, it is content with its place in the universe. It neither pushes nor pulls. It has no spring energy. This is like me most days in the summer. So we call this point where the spring neither pushes nor pulls, the spring's natural length. For a mass on a horizontal spring, this is gonna be the equilibrium position. What we mean by equilibrium position is the point where the net force on the mass is zero. So for a mass on a horizontal spring, the equilibrium position is at the point where the spring is at its natural length because the spring wouldn't be pushing to the right or the left. If you just put the mass there at that point, it would just stay there forever at rest. That would be boring. So let's say we pull the mass to the right at distance d. If we do this, we give this spring spring potential energy. If we release the mass from rest while the spring has spring potential energy, the spring's gonna pull the mass back to the left and that mass is gonna move through the equilibrium position with the sum speed. We can figure out what that speed is just by using conservation of energy and it's not that hard. The potential energy, the spring with start with would be 1/2 k, the spring constant, times d, the amount of spring has been stretched, squared. There would be no kinetic energy to start because we release the mass from rest. As this mass flies to the left, it would start gaining kinetic energy, the spring energy would start turning into kinetic energy. When the mass gets to the equilibrium position, the d would be zero. So at that point there would be no spring energy and all of the spring energy would have turned into kinetic energy and you get this simple relationship that says all the spring energy equals all the kinetic energy at the equilibrium position. So if we solve for v, we can get that the v of this mass at the equilibrium position would be the square root of k over m times d squared. You could call this d out because d squared and square root. but this is the idea. This is the speed you would get of the mass passing through the equilibrium position. Let me ask you this. What if it was a vertical spring? And this mass is sitting here. We come find a vertical spring with a mass hanging on it. We're like hey, I wanna pull this down, the distance d. If I pull it down to distance d, when this mass reaches the equilibrium position again, will it also be going root k over m times d squared? Or, would it be going at some different speed because now it's hanging vertically? Well it turns the spring constant's the same and you pull it down from the point where the mass is hanging. This exact same procedure is gonna hold over here and you can find the speed in the exact same way and that should be surprising. That should not be obvious. Because when the mass is hanging over here, you don't just have a spring force. You got a gravitational force. You don't just have a spring energy and kinetic energy. Think about it. This mass is moving up and down. You got changes and gravitational potential energy. So why don't we have to taken into account the gravitational potential energy when we're doing conservation of energy in this equation? Well that's what I wanna prove to you in the rest of this video. If all you wanted was the result, if all you, if you're good, if you're like, "Man, all right, I can do the same thing above cases, "don't even tell me anything else." You're good. But I suggest you watch the rest of it because knowing why you can ignore the m g h in this calculation gives you better insight into what we really mean by this d and this h and this v as well as what we really mean by the equilibrium position and that will conceptually aid you if you get a problem that's more challenging. So let's prove this and figure out why we can get away with ignoring this m g h right here. So let me get rid of all these. Let's start fresh. Let's just say we have a spring hanging from the ceiling of spring constant k. Let's say this spring is light. If it was heavy, it might pull itself down by its own weight so I'm gonna assume this is very light spring and it's hanging right here. There's no mass connected to it initially so it's just hanging at its natural length. It's neither pulling up nor pushing down as you see it right here because it's at its natural length but we connect the mass m to it. When we do that, we lower the mass with our hand. We don't just let it fall and start oscillating. We first lower the mass. We connect it and lower it. We find the point where the mass is just gonna stay at rest. That would be the new equilibrium position. So this right here is essentially our new equilibrium position. In other words, that's the point where the net force on the mass would be zero. But this time that's not at the spring's natural length. The way it was on the horizontal spring. This time the equilibrium position is this place, the distance a away from the spring's natural length because right now it's battling the force of gravity. In other words the spring force exerted upward k x minus the gravitational force which is m times g has got to equal zero in order for this mass to be an equilibrium. So we can actually figure out what this distance a would have to be in terms of given variables. Since at the equilibrium position, x, the distance the spring has been stretched has just gonna have to equal, m g divided by the spring constant k. This is what a is gonna equal. So the distance, the mass hangs down at the equilibrium position from the natural length of the spring is just gonna be m g over k. This is a in this diagram. This is gonna be key. So we're gonna hold on to this result right here. Well let's do this. Let's ask if we take this mass and pull it down an extra amount b from the new equilibrium position, well at that point the forces won't be equal. The spring's gonna be stretched extra. It's gonna be pulling up with more force in the force of gravity. So this mass is gonna accelerate upward. It's gonna reach this equilibrium position with some speed. It's gonna shoo pass that. Now the spring force is less than the force of gravity and so gravity wins in that case and it keeps going back and forth. We're gonna ask ourselves the same question we did before. If we pull this down, the distance b, what is the speed of the mass when it passes through the equilibrium position? Again we're gonna use conservation of energy to answer this. So we're gonna say that the initial energy in our system is gonna equal to final energy in our system. We're gonna choose two points. Let's choose initially this point down here. We release the mass from rest when it's pulled down at distance b below the new equilibrium position. Then our final point is gonna be right here at the equilibrium position because that's where we wanna know the speed of the mass. So let's try to figure out how much energy there is in the system initially if I pull this mass down and let it go. Well initially if I'm just letting this mass go, the mass is starting from rest. If the mass starts from rest, it's got no speed and if it's got no speed, it's got no kinetic energy. So there's no kinetic energy to start with if this mass is starting from rest but there is gonna be spring potential energy and there's gonna be a lot of spring potential energy. Because think about it. Not only is this mass stretching the spring passed the new equilibrium position by an amount b but the new equilibrium position itself is stretched from the spring's natural length a. This formula, when you have 1/2, the spring constant times the length that the spring has been stretched, that's the total amount the spring has been stretched. So the total amount the spring has been stretched from its natural length is gonna be a plus b. I've got to square that whole term. This is how much spring potential energy there's gonna be in the system initially. So how much gravitational potential energy we're gonna start with? Well, that's kind of hard to us because you can always choose where you want your h equals zero reference line to be. In other words, I'm gonna choose this lowest point here because that's often convenient when choose this to be the h equals zero reference line. We'll measure all heights from that point. This is allowed because it's only really differences in gravitational potential energy that matter so you can do this. You just have to be consistent with your choice. But with that choice where this is h equals zero, the height my mass has at the initial position is gonna be zero. So that means the gravitational potential energy which is given by m g h is also gonna be zero at that initial point. So in terms of initial energies, this is all I've got. This is my total initial energy, just the energy from the spring. Now we can set that equal to our final energies so we're gonna have any spring potential energy here in our final position. You might think no because the final position is at the new equilibrium position but remember this new equilibrium position still displays from the spring's natural length. So what I have, I'm gonna have 1/2 k times the amount the spring has been stretched from its natural position. At this new equilibrium position, the amount that has been stretched is just a. So I'm gonna write a here because that's how far the spring has stretched at this new equilibrium position. I have to square that because that's 1/2 k x squared. We're gonna have kinetic energy. This mass is gonna gain speed as it flies upward. It's gonna be moving with some speed when it gets up to that point. So the kinetic energy is gonna be 1/2 times the mass, times the speed the mass has at the equilibrium position squared. That's what we want to determine. But there's also gonna be gravitational potential energy. We said h is zero down here. So if the mass is not there, it's gonna have potential energy due to gravity. If it's b above this point, look at it, we pulled it down b so when it gets back to the equilibrium position, if this is h equals zero, that's gonna be h equals b above where it started. So I have to put in m times g times the b value. This length right here. Since in moving up this mass gained m g times b a gravitational potential energy. So how do we make progress here? I wanna solve for the speed v but I've got this mess over here. Look, I've got a plus b squared so I better handle that first. So let's say we do the 1/2 k and then we square this out. Remember we do FOIL. So it's First, Outer, Inner, Last. I'm gonna get a quantity of a squared and then this cross term, I'm get plus two times a times b and then plus b squared. That's what happens if I square this whole term right here. These start looking really bad but don't despair. Something great, something wonderful is about to happen because I'm gonna set this equal to the right hand side. So if we multiply out on the left hand side, we're gonna get a 1/2 times k times a squared plus a 1/2 times k times this 2 a b term, plus I'm gonna get another one, 1/2 k times this b squared term. We can say that that's supposed to equal everything on the right hand side. So we can already see something that we can cancel out. I've got a 1/2 k a squared on each side. So if we subtract that from both sides, I can get rid of that. This 1/2 here is gonna cancel this two and I'm left with k a b on the left hand side plus 1/2 k b squared. But, what is k a? If you're clever and you look up here, you're like, wait a minute, I remember what k a was. K times a, if we just multiply both sides by k here, it's gotta equal m g because that's what just the statement of equilibrium that at the equilibrium position k times a has to equal m times g. So I can replace k times a over on this left hand side with m times g. M g might be like, why would I ever wanna do that? Why would I wanna replace k times a and I still multiply by this b? Why would I replace k a with m g? Because now that term is gonna cancel with the other m g b on the other side of this equation. One half m b squared plus m g b but I've got m g b on the left and m g b on the right now. That's why I replace this k a with m g. I can subtract that from both sides and magically I just get the exact same relationship we had for the horizontal spring measured from the new equilibrium position. This is important so let me restate this. You can either when solving these vertical spring problems, measure your spring displacement from the natural unstretched length of the spring like we did right here. We had to add a plus b because that was the distance from the natural unstretched length of the spring all the way to where the mass was. You can do that and include gravitational potential energy and get the right answer. But what we just saw is that these terms always cancel so the alternative is that you can just measure the spring extension, the spring displacement from the new equilibrium position. If you do that, you just leave off all mentioned of gravitational potential energy and you get the same answer. You can think of gravity simply as shifting the equilibrium position down a bit and then the mass and spring behaving just as I would on a horizontal surface as long as you only think about spring displacement from that new equilibrium position. So in other words, if you were given a problem, let me get rid of all these. If you're just given a problem and you were told a three kilogram mass is hanging from a vertical spring of spring constant 50 newtons per meter and this line here represents the equilibrium position, it's just hanging out right there at rest. You pull this mass down from the point where it was hanging at rest, 0.3 meter and let go. You wanna figure out what speed will it be going when it reaches equilibrium position. You can just do this. You can say, all right, down here it's got spring potential energy, 1/2 k is 50 newtons per meter times the amount the spring has been stretched but I'm just gonna consider stretches from the new equilibrium position. So I'm just gonna do 0.3. I'm not gonna worry about the fact that the spring has actually already been stretched to get to this equilibrium position. I'm just gonna put this stretching from equilibrium. I'm gonna set that equal to the kinetic energy the mass has at the equilibrium position. We know that m is three kilograms times v squared. I'm not gonna include information about the gravitational potential energy at all because I only measured the displacements from the new equilibrium position. So at this point, I can just solve for my v. If you solve that algebraically for v, you cancel out the twos. You divide both sides by three and take a square root. You get to the speed of the mass at equilibrium is gonna be 1.2 meters per second. So to recap, even though it seems initially like vertical springs would be much harder than horizontal springs because you've got gravitational forces and gravitational potential energy to worry about, if you measure the spring displacement from the new equilibrium position as opposed to the natural spring length, you can simply use conservation of energy without mention of gravitational potential energy at all.