Work and Potential Energy One general type of energy is potential energy, U. It is the energy that can be associated with the configuration (or arrangement) of a system of objects that exert forces on one another. Nov. 6, 2017 We can now more thoroughly explain the work done on an object as an object is tossed up in the air. The work transfers energy from the kinetic energy of the object to the gravitational potential energy of the object as it rises, hence the work is negative. As the object slows and begins its descent, the work transfers energy from the gravitational potential energy to the kinetic energy of the object. This gives us the relationship ΔU = W This also applies to our block spring system. If we were to apply a force and send the block moving rightward, the spring force acts to the left and thus does negative work on the block, transferring energy from the kinetic energy of the spring to the elastic potential energy of the spring.
Work and Potential Energy (cont'd) Conservative and Nonconservative Forces Here are the key elements in the two previous situations 1. The system consists of two or more objects 2. A force acts between an object in the system and the rest of the system. 3. When the system changes, the force does work (W 1 ) on the object, transferring energy between the kinetic energy, K, of the object and some other type of energy of the system. 4. When the change is reversed, the force reverses the energy transfer, doing work (W 2 ) in the process. In a situation where W 1 = W 2 is always true, the other type of energy is a potential energy and the force is a conservative force. The kinetic frictional force and drag force are nonconservative forces. When an object is slides across a frictional surface, work is done by the force slowing the block down by transferring kinetic energy to thermal energy. This energy transfer cannot be reversed. When only conservative forces act on an object, we can greatly simplify difficult problems involving motion of the object.
Path Independence of Conservative Forces The primary test for whether a force is conservative or nonconservative is: Let the force act on a particle that moves along any closed path, beginning at some initial point and returning to that position. The force is conservative ONLY IF the total energy it transfers to and from the particle during the round trip along this, and any other closed path, is zero. The net work done by a conservative force on a particle moving any closed path is zero. The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle.
Determining Potential Energy Values Consider an object that is part of a system in which a conservative force F acts. When that force does work, W, on the object, the change in potential energy, ΔU, associated with the system is the negative of the work done. We now get
Determining Potential Energy Values Gravitational Potential Energy Consider an object with mass m moving vertically up along the positive y axis. As the object moves from y 1 to y 2, the force F g does work on it. We now have Only changes in the GPE are physically meaningful. However, we would like to say that a certain GPE value, U, is associated with a system when the particle is at a certain height, y. If we take U i to be the GPE of the system when it is in a reference configuration in which the object is at a reference point, y i. We usually take U i = 0 when y i = 0.
Elastic Potential Energy Determining Potential Energy Values Consider a block spring system with the block on the end of a spring with spring constant k. As the block moves from x 1 to x 2, the spring force does work on the block. We associate a potential energy value, U, with the block at position x by letting the reference configuration correspond to the when the spring is in its relaxed length and the block is at x i = 0. Then the EPE U i = 0.
Conservation of Mechanical Energy The mechanical energy, E mec, of a system is the sum of its potential energy, U, and the kinetic energy, K, of the objects within it: When a conservative force does work on an object within a system, that force transfers energy between kinetic energy, K, of the object and potential energy, U, of the system. and therefore or which can be rewritten The sum of K and U for any state of a system The sum of K and U for any other state of a system
Example A roller coaster of mass 80.0 kg is moving with a speed of 20.0 m/s at position A as shown in the figure. Neglect friction. (a) What is the total mechanical energy of the roller coaster at point A? (b) What is the total mechanical energy of the roller coaster at point B? (c) What is the speed of the roller coaster at point B? (d) What is the speed of the roller coaster at point C? Ans: (a) 1.73 x 10 5 J (b) 1.73 x 10 5 J (c) 65.7 m/s (d) 34.4 m/s
Example Ans: 4.3 m/s In the figure, a 5.00 kg block is moving at 5.00 m/s along a horizontal frictionless surface toward an ideal massless spring that is attached to a wall. After the block collides with the spring, the spring is compressed a maximum distance of 0.68 m. What is the speed of the block when it has moved so that the spring is compressed to only one half of the maximum distance?
Example Ans: 13.1 m/s A block slides down a frictionless inclined ramp. If the ramp angle is 17.0 and its length is 30.0 m, find the speed of the block as it reaches the bottom of the ramp, assuming it started sliding from rest at the top.
Example Ans: 13.7 m/s A small hockey puck slides without friction over the icy hill shown in the figure and lands 6.20 m from the base of the cliff with no air resistance. What was its speed, v o, at the bottom of the hill?