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Potential Energy, Conservation of Energy

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Recall Last Time Learned that work is force through a distance For a constant force, can take it out of the integral Here, d is the distance over which the (constant) force acts, and is the angle between the force and the direction of movement. Doing work on an object gives it kinetic energy

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Non-Constant Force—Springs There are times when the force is not constant. An example is a spring, for which the force depends on the distance the spring is stretched or compressed. The force can be written where k is the spring constant. Why negative? Because the force is in the opposite direction to the displacement. Now, since the force is not constant, calculating the work is trickier: It is the area under the force curve.

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Gravitational Potential Energy When you drop an object, it accelerates toward the ground and may be moving rather fast by the time it hits. Therefore, it has a lot of kinetic energy. Where did that energy come from? We can consider that it had the energy by virtue of the fact that it was held at some height above the ground. This is potential energy. It is stored in the object, waiting to be released. An object sitting on the ground, however, has no potential energy. We give it potential energy by doing work on it, i.e. by lifting it to some higher height, against the force of gravity. Imagine an applied force F app = mg that balances the force of gravity, and we lift the object from height y i to y f. We then do work But the object is not moving at the end of our lift, so the energy is potential energy, stored in the object by virtue of its location. We use the symbol U for potential energy, and identify (assumes y i = 0 ) Gravitational potential energy

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Spring Potential Energy Gravitational potential energy is not the only kind. A spring has potential energy by virtue of the compression or stretching of the spring. We just saw that the work done by the spring is The work done by an applied force compressing the spring is just the negative of this, and it is this work that is stored in the object, i.e. the spring potential energy In short, assuming x i = 0, the spring potential energy is Spring potential energy

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Relationship Between K and U Placing a system in a certain configuration (like compressing or stretching a spring, or lifting an object to some height) can store energy. When the system is “released,” that energy is available to do work on the object, i.e. make it move and give it energy of motion (kinetic energy). By the same token, a moving object, with kinetic energy, can move into a configuration in which the energy of motion is stored in the object, as when a ball is thrown upward. The ball moves upward and eventually comes to a stop. At that moment, high above the ground, the kinetic energy is gone, and is converted into potential energy. Here is a similar example with a spring. Animation

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Conservative Forces Both gravity and a spring are examples of conservative forces. There are two properties that conservative forces have. 1.The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle. 2.The work done by a conservative force on a particle moving through a closed path is zero. A B Work done in going from A to B is the same for any and all paths h h

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Non-Conservative Forces h h Not all forces are conservative. In fact, whenever friction is important in a physical situation (i.e. nearly always), the friction spoils this neat setup. Take the example we just gave. If friction is acting along the ramps, then the longer ramp will have more friction, so the ball will be moving more slowly after going along the longer ramp.

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Getting the Force from the Energy We have the work done by the component of a conservative force in the x direction is So the change in potential energy is An infinitesimal change in potential energy results from an infinitesimal increment dx, i.e. So we have the differential relationship Likewise in the other directions: In the case of gravity: In the case of a spring:

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The Potential Energy Function For a spring, the potential energy function looks like this: Then the force is the negative of the slope of this function. This kind of curve is called a potential well (looks like a bowl), and it indicates a stable equilibrium. Unstable Equilibrium

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Conservation of Energy One of the great laws of physics is the conservation of energy. What this law says is that energy can be neither created nor destroyed. It can only be changed from one form to another. A corollary to this law is that for an isolated system the energy is constant. What do we mean by an isolated system? It means the system does not interact with anything outside the system. For example, a falling apple, together with the Earth, can make an isolated system (but we have to ignore the Earth’s interaction with the Sun, the Moon, etc). Of course, no system is truly isolated. In a non-isolated system, the energy does not have to be constant. Energy can come in from outside, or go out from within. An example is mass and a spring, but with friction. Now energy can leave the “system” through friction with the table (which takes the form of heat). We can account for that if we add a term int. Conservation of energy states that where we have ignored other loss of energy due to transfer out of the system.

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Case of Conservative Forces When we ignore friction or other non-conservative forces, we have or Example, dropping a ball from rest. Initial K i =0, and initial height is h. So total energy is just mgh. While falling, the potential energy at position y is U = mgy, and kinetic energy is ½mv 2, so the total energy is just the sum of these. At the floor, y = 0, so U f = 0 and K f = ½mv f 2. The final energy equals the initial energy, so And we can solve for v f to get This is the same result we got when using forces.

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Case of Friction (non-conservative) In many cases, the friction force is constant (remember that kinetic friction depends on the normal force, which is just the weight of an object in the simplest case). Consider a block sliding on a rough surface. If it slides a distance d = x, the force of friction f k through that distance gives the form of energy we called internal energy (heat): Since then For this problem, U i = U f = 0, so you can see that the final kinetic energy will be lower than the non-friction case.

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Example, crate sliding down ramp Initial energy is mgy. Crate slides 1 m against friction force, so E int = f k d = 5 Nm. Final energy is all kinetic, ½ mv f 2. Putting it all together, f k = 5 N m = 3 kg Find v f.

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Power A useful concept is the amount of energy used per unit time, which is called power: Now, since, we have. But we can write which assumes that F is constant. The units of power are watts, 1 W = 1 J/s

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