Physicists have a difficult time describing what the quantity we call energy really represents. The scientist Romer once described energy as a abstraction, a mental book keeping device that we have invented. Energy is a number, a value we calculate about an object that has many useful properties. It helped us to describe the motion, location, or even the temperature of an object. There are many different forms of energy, all of which we measure in joules. Let's take a moment to introduce two new forms of potential energy. By potential energy, we mean energy that is somehow stored within the configuration of a system that can be used to do work. There are many different kinds of energy an object can have and each are related to a particular force. If a brick is lifted from the surface of the earth to a height of two meters, that object gains what we call gravitational potential energy. This means that that brick now has the ability to do work on another object. If we dropped this brick on a nail, it could apply a force over a distance and drive it into the ground. We calculate this gravitational potential energy by multiplying the gravitational force on this object, in this case mass multiplied by gravitational acceleration, by the height of the object above some reference point. The higher the object is above that point the greater its gravitational potential energy. >> There are two things that I'm solving for in this question. One if the gravitational potential energy when it reaches the second crest which is a height of 30 meters and the other thing I'm going to solve for is the speed of the roller coaster at that same position. Here I have a rough sketch. At the very start, at the fifteen meter mark, it does not have a speed, it's starting from rest. Unless they tell you otherwise, you're assuming you're starting from rest here. Which means the only type of energy I have is stored energy, gravitational potential energy. How about at the 30 mark? Well, here's the thing. It hasn't used up all of its stored energy, and I know that it's still in motion, because they asked me for the speed at that point. So I have potential energy as well as kinetic energy. What I need to do first for part a, is to solve for the gravitational potential energy. At the 30 meter mark. Well that equals MGH, So our potential energy is 1000, which is the mass of the roller coaster, g, I'm going to use ten meters per second squared, of course you can use 9.28 if you'd like, multiplied by 30 meters. When I solve for this, I end up with 300,000 joules, or 3x10 to the 5th power joules. Now I can go ahead and solve for the speed at the same point. In order to do that, I'm going to use conservation of energy. Conservation of energy tells me that the sum of all of my initial energy is going to equal to the sum of all of my final energy. Initially, all I had was potential energy. At the final point I have, still, some stored gravitational potential energy, as well as kinetic energy. So let's go ahead and plug in variables here. I have mgh initial equaling mgh final plus one-half mvf squared. You can do this a different way. You can just go ahead and solve for final velocity here. Had you wanted to to solve for kinetic energy, and then use that answer, set it equal to one-half mv squared, that would have been fine as well. The initial height was 50 and the final height that we're looking at is 30. Again, when you're solving for this finals feet don't forget that it's squared, so your going to need to take the square root to get to your final answer. And that final answer for me is a speed of 20 meters per second. >> So in looking at a problem like this, it is a box sliding down an incline plane, but there's a spring involved. This is a good example of a time that you can not use forces and kinematic equations to find your answer. Because as a spring compresses or un-compresses, the force changes. The more you push on a spring to compress it's equilibrium, the more it pushes back. So you get a change in acceleration because of a changing force. And kinematic equations will not get you to where you need to be. Instead, energy is a great answer to a problem like this. Same thing with roller coaster problems or pendulums. All of those require energy rather than kinematics. So, taking a look at this, what we have is an incline plane. Where there's a box up at the top, compressed against a spring. It tells me that that box is a height of 1.4 meters, Above the ground. And it wants to know how fast will that box be going when it's fallen half way, and then it wants to know what the max speed of the box will be and that is going to happen at the very end of the ramp. Because at the very end it's lost all of it's gravitational potential energy and all of it's spring energy which it lost long ago, to have all kinetic energy at the bottom of this ramp. And so, let's solve for the first portion. It asks what is the speed when it reaches that halfway point, well I always start off with my fundamental principal, sum of energy before equals the sum of energy after. This tells the grader and tells your physics teacher or your professor that you know the fundamental principal here is that energy is conserved. Well, how many types of energy do I have at the beginning of the problem up here? The blocks at rest, there is no kinetic energy, but it is compressed against a spring so there is some spring initial potential energy, which I represent with a capital U. There is also some gravitational potential energy. I'm going to set my zero for gravitational energy at the ground, which is a pretty common place to set it, but you can place it wherever you'd like. In the beginning of the problem then were going to have the potential energy for gravity, initial. When it has fallen half way down the ramp, somewhere around here, perhaps that's a little off, but we'll pretend that that's about half way. It still has some gravitational potential energy. We'll call that final, feel free to use whatever subscripts you find helpful. But it will also have some kinetic energy, because it will be moving. It will no longer have spring potential. It left the spring long ago. Let's just assume that the spring is relatively short. Which is usually a pretty good assumption for these kind of problems. So remembering our equation for these types of energy, one half kx initial squared, that's the spring potential energy. Don't forget x represents how far the spring has been compressed from equilibrium, which is what it tells us, plus mgh initial, that's the initial gravitational potential energy. All those stored at the beginning. They can be used to give this thing, this block some kinetic energy. The gravitational potential energy at the end then looks very similar but has a different height final, plus one half m v final squared. Notice that kinetic energy can never be negative, it's either moving or it's not. The velocity being squared there will get rid of any negative signs, regardless of the fact here that kinetic energy is a scalar. It's a speed that that v represents that velocity, so don't sub in negative signs to begin with. Okay, let's scroll down to give ourselves some room here to calculate and plug in some numbers. Looking at my equation, I've got one-half. K represents the spring constant. How strong this spring is for us. That was 100 newtons per meter. How stiff the spring is is a good way to think of it in your head. Once the spring is made that doesn't change as long as you're not breaking the spring or stretching it beyond its elastic limit. .08 meters is how far it's been stretched. They gave that to be meters, which is awfully nice of them already, plus the mass of this object, 20. G is 10. And the height, originally, 1.4. On the other side, the mass has not changed, 20. G is 10. The height if it's fallen halfway, 0.7. Plus one half the mass is 20. We see at the end here is what we're looking for. Big equation, very easy to actually mix up your numbers so just make sure you're going slowly and deliberately. Combine all the terms on the left. Move all the other terms, for example, this to the left and then solve for V. I get 3.75 meters per second. That's the speed, not the velocity. When we're talking about energy, it's a gives you the speed. It doesn't give you the direction. We'd have to find that through other means, which we'll do in a later problem. So that gives me this speed at the mid point. Now, I also wanted to know what the maximum speed was. So again, looking at my problem, now I'm going to know when it's down here and the block has made it to the very bottom. At that location, its going to have lots of kinetic energy but no spring or gravitational potential energy. It's lost both, it's now what we call zero for gravitation. So, looking at that problem, we'll be acting a little bit faster. Sum of energy before equals sum of energy after is where we start. Again, I have spring potential initial. I'm going to set my initial point at the very top again. If you wanted to, you could make your initial point the middle, because the energy throughout the problem should be the same. The total energy should be the same. And had initial gravitational potential energy and at the end it will have all kinetic energy. So subbing in our expressions for that like before, And now I have a one half m in the final square. Now as a check, before we plug in numbers, I'd expect this speed to be higher than the speed we found earlier because now we're at the bottom. This should be the maximum speed. This is where it lost all of that gravitational potential energy, v that we did here should be bigger than 3.75. And if it isn't, we need to go back and make sure that our answers seem reasonable and what mistake we might have made. Subbing in our numbers, one half, 100, .08 is the displacement from equilibrium that spring has been stretched. Plus 20, ten for G. The height, the original height, above our zero, 1.4, one half. Again, the mass is 20 and I'm solving for vf squared. Don't forget that squared. We'll have to take the square root at the end and solve for b. Coming down, I get a final velocity, then, of 5.29 meters per second, which is more than the speed we got earlier, which is what I was expecting. >> So if you've solved the problem like this before using kinematics you know how long that can take. We did a problem like this in our kinematics module where we had to use the quadratic, and we solved it another way as well, which was even longer. Now let's go ahead and look at a way that's simpler. So, initially we have this rock that's thrown from the top of a 100 meter cliff. It's got an initial velocity. It's launched from a height and it reaches the sea below. We want to know what the final velocity is when it reaches that stopping point. So let's go ahead and solve for it. We're going to use conservation of energy. This means the sum of my initial energy equals the sum of my final energy. Let's look at what type of energy I have initially. Well I definitely have gravitational potential energy, it's at a height, so it has that stored energy. But it also has an initial speed, so I also have kinetic energy. How about at the final point? Well, if it's coming towards its very last point before reaching the ground, then that means it is using up all of its stored energy. Where is all of that stored energy going? It's going towards kinetic energy, which means it has it's maximum speed now. So now all I have left at the final point, is kinetic energy. When I'm solving for maximum speed, I tend to only have kinetic energy because I'm using all of my stored energy to get to that maximum speed. Let's go ahead and solve for this now. I have mgh initial, plus one half mv initial squared. This is my potential energy plus initial kinetic energy. All of this equaling one half and the final square. I can cancel the mass out over here. Or not, either is fine. I don't have the mass, so it's kind of nice to cancel it out. G is ten meters per second squared. The height initially is 100 meters, plus one half, my initial speed, of five, be sure to square that, equaling one half vf squared. Now, when I go ahead and solve for my final speed, I end up with 45 meters per second. That was quick and painless, and if you go back and look at the kinematics module, you'll find that if solve for your total, your resultant final velocity, not just the vertical component, but the resultant, you will also end up with 45 meters per second.
