We're going to see that vertical systems are very similar to horizontal, and the only difference has to do with the equilibrium position. But rather than tell you, I just want to show you using this example. So let's take a look. We've got a 0.5 meter spring, and it's hanging from the ceiling. So we've got that distance 0.5 meters, and that's going to be the original length of the spring. Then it stretches down by some distance after I attach a mass to it. Now, why does it stretch down and stop? Well, we've got an extra force to consider because now we have the object's weight that pulls it down, whereas, originally, we didn't have that before in horizontal systems. So as the thing is pulling the spring down, the restoring force gets higher and higher upwards. And so what ends up happening is that that restoring force ends up cancelling out with the gravity, and so it reaches a new equilibrium position. So in horizontal mass spring systems, we said that x equals 0 is where like no forces were acting on it. But in vertical mass spring systems, the equilibrium is where these forces will cancel out. That's the important part. And so we call that distance, that hanging distance until it reaches equilibrium delta l. So in this case, it is delta l equals to 0.2. So again, that delta l really represents the spring system's hanging deformation or hanging distance. It's the amount that you need to stretch the spring so that you reach equilibrium. And that equilibrium condition is where these two forces balance out. Well, the upward force is going to be k times delta l, and the downward force is going to be mg. So that means at equilibrium, we've got k δ l = m g . So let's look at the second part of this problem.

After all of this happens, we're going to pull the spring system an additional 0.3 meters downward. So now, just like in horizontal spring systems, your additional push or pull was the amplitude. So that means that once you pull this thing downwards 0.3, now this thing is just going to go up and down between these two amplitudes. So this is going to be the positive and that's going to be the negative amplitude. So it's just going to oscillate up and down like that. And so it's important to remember that this amplitude is the additional push or pull, and it's not the delta l. So it's not the natural hanging distance or deformation; it's the additional push or pull. So in problems, what you'll see is that you're going to attach a mass, and it's going to stretch by some distance. What that represents is delta l, and then you're going to pull it down an additional something, and that is going to be the amplitude. So don't confuse those two. Alright. So now basically we have everything we need to solve the problem. This first part is asking us for the force constant, so it's asking us for k. So we're just going to use this new equilibrium equation that we have. So we've got k δ L = m g . Now I know what mg and delta L are. I just have to figure out what this k constant is. So I've got k, once I just move this to the other side, I've got k = 5 ⋅ 10 for gravity, and then the delta L is what? Well, the delta L we said, that the natural stretching distance was 0.2 meters. So that means I got a k constant of 250 newtons per meter, and that's it. So what about the second part here? The second part is now asking us at its maximum height, so it's oscillating. At maximum height, how far is that ceiling, from the block? Great. So let's check it out. So we've got this thing here, and so I'm going to represent this whole entire line here as this motion. The original distance, the original length of the spring was that black dot, and then you hang it down, and it reaches some equilibrium position. And then you're going to pull it down an additional 0.3 meters. So that's the amplitude, and it's just going to oscillate between these two points. Okay. So we're being asked for basically this distance right here. What is the distance between the ceiling and its maximum height? That's where it reaches that maximum amplitude. And so that distance here is going to be the letter d. Well, we're told that this thing has an original length of 0.5 meters, and the equilibrium position is when it stretches an additional 2. So that means that the length of equilibrium, I'll call this Leq, is equal to 0.7 meters. So now what happens is it goes 0.3 down and then 0.3 up. So at this bottom part here, this length here is 1.0, And at the top point right here, it's 0.7 minus 0.3. And so we say that at maximum height, the distance d away from the ceiling is equal to 0.4 meters, and that's it. Alright, guys. That's it for vertical oscillations.