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# AP Physics 1 review of Energy and Work

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- [Teacher] Not only are there many different kinds of energies, but both objects, and systems of objects, can have energy. Once they have that energy, they can transfer it to another system or object or that energy could transform to a different type of energy inside that system. When energy gets transferred, we call that work, and the amount of work that's done is the amount of energy that was transferred. You often hear people say, "Energy is conserved," which really just means that you can't create or destroy energy. You can simply transfer it between objects or systems. So, what are all the different types of energy? There's kinetic energy which is the energy due to something moving, and the formula's one-half the mass times the speed squared. There's gravitational potential energy which is the energy something has due to its height, and the formula's the mass times the magnitude of acceleration due to gravity times the height of the object. Height above what? Height above whatever you're choosing as the H equals zero reference line. Is that cheating? No, because all that really matters is the change in gravitational potential energy, not the actual value itself. There's also spring potential energy which has to do with a compressed or stretched spring, and the formula's one-half times the spring constant times x, which is not the length of the spring. X is the amount that the spring has been compressed or stretched. These three types of kinetic and potential energy constitute what we call mechanical energy. Mechanical energy's another word for the kinetic energy plus gravitational potential energy plus spring potential energy in a system, and it's important to know that mechanical energy does not include thermal energy. Thermal energy's the heat energy generated by dissipative forces like friction and air resistance, and you can find the amount of thermal energy generated by taking the size of the dissipated force times the distance through which that force was acting. The unit of energy is Joules and energy is not a vector. But maybe the most important thing to remember about energy is if there's no external work done on a system, then there's no change in the energy of that system. In other words, if there's no external work done on a system, the initial energy of that system will equal the final energy of that system, which is the way you solve many conservation of energy problems. So, what's an example problem involving energy look like? Let's say a box started with an initial speed and slides from one platform up to another platform. We'll assume that frictional forces and air resistance are negligible. And for the system that's consisting of the mass and the Earth, what's happening to the total mechanical energy in this system? So, you've got to pay special attention to what is in your system. Since my system includes the mass, which is going to be moving, my system's gonna have kinetic energy, and since my system has two objects that are interacting gravitationally, the mass and the Earth, my system's also going to have gravitational potential energy. So, when I asked about the total mechanical energy of the system, that's really just code for the total kinetic and potential energy of the system. So, as this mass slides up to a higher point on the ramp, the gravitational potential energy increases, but the mass is gonna slow down, so the kinetic energy's gonna decrease. However, since the Earth and the mass are in our system, and there's no dissipative forces, there's no external work done on our system. Yes, the Earth is doing work on the box, but the Earth is part of our system so it can't do external work, and that means energy just gets transferred from one form to another within our system, and the total mechanical energy, here, is gonna remain the same for the entire trip. Now, what if we asked this same question but we consider a system that consists only of the box. In that case, our system has a box that's moving, so it'll have kinetic energy. But, our system no longer includes two objects interacting gravitationally so our system will have no gravitational potential energy. What happens to the total mechanical energy in this case? Well, the only energy that I've got in my system, now, is kinetic energy and since that kinetic energy decreased, the total mechanical energy of the box, as a system, decreases. How does it decrease? It decreases because now the Earth is outside of our system and the work that it is doing on the box is external work and it's taking away energy from the box. What does work mean? In physics, work is the amount of energy transferred from one system, or object, to another. In other words, if a person lifted a box and gave it 10 Joules of gravitational potential energy, we'd say that person did positive 10 Joules of work on the box since that person gave the box 10 Joules of energy. But since the box took 10 Joules of energy from that person, we'd say that the box did negative 10 Joules of work on the person since the box took 10 Joules of energy. So, you can find the work done if you can determine the amount of energy that was transferred. But, there's an alternative formula to find the work done. If something's having work done to it, there's got to be a force on that object, and that object has to be displaced. So, if you take the force on the object times the displacement of the object, and multiply by the cosine of the angle between the force and the displacement, you'll also get the work done. In other words, one way to find the work done is by finding the amount of energy that was transferred. But, another way to find the work done is by taking the magnitude of force exerted on an object times the displacement of the object and then times cosine of the angle between the displacement and the force. Since work is a transfer of energy, it also has units of Joules. And even though work is not a vector, it can be positive or negative. If the force on an object has a component in the direction of motion, that force will do positive work on the object and give the object energy. If the force on the object has a component in the opposite direction of the motion, the work done by that force would be negative and it would take away the object's energy. And, if the force on an object is perpendicular to the motion of the object, that force does zero work on the object. It neither gives the object energy nor takes away the object's energy. So, what's an example problem involving work look like? Let's say a box of mass M slides down a frictionless ramp of height, H, and angle two-theta, as seen in this diagram, here, and a separate box of mass two-M slides down another frictionless ramp of height, H, and angle theta, as seen in this diagram, here. And, we want to know how work done on the object by the Earth compares for each case? The easiest way to find the work done here is by finding the change in energy. The box will gain an amount of kinetic energy equal to the amount of potential energy that it loses. So, the work done by the Earth is just gonna equal positive m-g-h. Both heights are the same. So, the H's are equivalent, but one box has twice the mass. So, the work done by gravity on the mass, two-M, is gonna be twice as great as the work done on the mass one-M. What's the work energy principle mean? The work energy principle states that the total work, or the net work done on an object, is gonna be equal to the change in kinetic energy of that object. So, if you add up all the work done by all forces on an object, that's got to be equal to the change in the kinetic energy of that object. In other words, one-half m-v-final, squared, minus one-half m-v-initial, squared. So, this is a really handy way to find how the speed of an object changes if you can determine the net work on an object. In other words, if there's multiple forces on an object, and you can find the work done by each of those forces, you can determine how much kinetic energy that object gained or lost. So, what's an example of the work energy principle? Let's say a four kilogram box started with a velocity of six meters per second to the left. Some net amount of work is done on that box, and it's now moving with a velocity of four meters per second to the right. We want to know what was the amount of new work done on the box? Without even solving it, we can say since this object's slowed down, energy was taken from it, so the amount of net work had to be negative which means it's either B or D. To figure out which one exactly, we could use the work energy principle which says that the net work done is equal to the change in kinetic energy. So, if we take the final kinetic energy, which is one-half times four kilograms times the final speed squared, and we subtract the initial kinetic energy, one-half times four kilograms times the initial speed, squared, six meters per second, we get negative 40 Joules of net work. If you get a force versus position graph, the area under that graph will represent the work done. So, when you see F versus x, you should think area equals work. But, be careful, area above the x-axis is gonna count as positive work done, and area underneath the x-axis is gonna count as negative work done, and make sure the x-axis really is position. If you get a force versus time graph, the area's impulse, not work. So, what would an example of work as area look like? Let's say a box started at x equals zero with a velocity of five meters per second to the right and a net horizontal force on the box is given by the graph below. We want to know, at what position other than x equals zero, will the box, again, have a velocity of five meters per second to the right? Well, since the box will end with the same speed that it began with, the change in kinetic energy is gonna equal zero. But, that means the net work would also equal zero, since the net work is equal to the change in kinetic energy. So, if the box starts at x equals zero, how far do we have to go in order for us to have no net work done. Between zero and three meters, the work done is gonna be negative, and the area of this triangle is gonna be one-half the base times the height which is one-half times three meters time negative six Newtons which is negative nine Joules of work done, and the area under this triangle, between three and five seconds, would again be one-half base times height, which is one-half times two meters times height of four Newtons which is positive four Joules of work done. So, by the time that the box has made it to five meters, there's been a total amount of work done of negative nine plus four, which is negative five Joules of work. But, we want no net work done. So, we're gonna have to keep going until this positive area contribution is gonna equal the negative area. In other words, if I can make it so that all of this negative area is equal to all of the positive area, my net work's gonna equal zero. My negative area is negative nine. My positive area, so far, is positive four. If I continue on to the six meter mark, I've pick up another positive four Joules of work since the height of this rectangle is four and the width is one meter, which means we're almost there. Four plus four is eight. I'd only need to pick up one more Joule, so I can't go all the way to seven meters. I'd only need to go one more fourth of a meter to pick up one more Joule so that one plus four plus four is equal to negative nine. So, the net work would equal zero somewhere between x equals six and x equals seven which would ensure that the change in kinetic energy is zero and we would end with the same speed that we began with. What does power mean? In physics power is the amount of work done per time, which can also be thought of as the amount of energy transferred per time. In other words, the amount of Joules per second that are transferred, and the name given to a Joule per second is a Watt. So, you can solve for the power by finding the work divided by the time or the change in energy divided by the time. And you can increase the amount of power by increasing the work done or decreasing the amount of time it takes for that work to be done. And just like energy and work, power is not a vector. So, what's an example problem involving power look like? Let's say a box of mass M slid all the way down a frictionless ramp of height H and angle two-theta as seen in this diagram, and a separate mass M slides all the way down a frictionless ramp of height H and angle theta as seen in that diagram, and we want to know how the average power developed by the force of gravity on the boxes compares for each incline? So, we use the formula for power, power's the work done per time. The work done on these boxes is gonna equal the change in kinetic energy of these boxes which would equal the change in potential energy of the boxes, but the mass of the boxes are the same, the gravitational acceleration is the same, and the height they fall from is the same. So, the work done on the boxes are equal, but the time it takes for these boxes to slide down the ramp is not equal. The mass on the steeper ramp will reach the bottom faster which means it has a higher rate of power being done compared to the mass on the less steep ramp. So, even though the same amount of work is being done, the rate at which that work is being done is greater for the steeper ramp compared to the more shallow ramp.
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