Conservative Forces and Potential Energy Quiz

PDF page 315: Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative. That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

PDF page 315: Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable. A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy for any conservative force. The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added.

PDF page 315: Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative. That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

PDF page 315: Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable. A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy for any conservative force. The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added.

PDF page 315: First, let us obtain an expression for the potential energy stored in a spring ( ). We calculate the work done to stretch or compress a spring that obeys Hooke’s law. (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force on the spring and the resulting deformation are proportional, .) (See Figure 7.10.) For our spring, we will replace (the amount of deformation produced by a force ) by the distance that the spring is stretched or compressed along its length. So the force needed to stretch the spring has magnitude , where is the spring’s force constant. The force increases linearly from 0 at the start to in the fully stretched position. The average force is . Thus the work done in stretching or compressing the spring is . Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of vs. is the work done by the force. In Figure 7.10(c) we see that this area is also . We therefore define the potential energy of a spring, , to be

PDF page 315: First, let us obtain an expression for the potential energy stored in a spring ( ). We calculate the work done to stretch or compress a spring that obeys Hooke’s law. (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force on the spring and the resulting deformation are proportional, .) (See Figure 7.10.) For our spring, we will replace (the amount of deformation produced by a force ) by the distance that the spring is stretched or compressed along its length. So the force needed to stretch the spring has magnitude , where is the spring’s force constant. The force increases linearly from 0 at the start to in the fully stretched position. The average force is . Thus the work done in stretching or compressing the spring is . Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of vs. is the work done by the force. In Figure 7.10(c) we see that this area is also . We therefore define the potential energy of a spring, , to be

PDF page 317: where is the total work done by all conservative forces. Thus, 7.45 Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, . Therefore, 7.46 or 7.47 This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,

PDF page 317: where is the total work done by all conservative forces. Thus, 7.45 Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, . Therefore, 7.46 or 7.47 This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,

PDF page 317: where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy, . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between and the various types of , with the total energy remaining constant.

PDF page 317: where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy, . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between and the various types of , with the total energy remaining constant.

PDF page 317: where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy, . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between and the various types of , with the total energy remaining constant.

PDF page 316: FIGURE 7.11 Work is done to deform the guitar string, giving it potential energy. When released, the potential energy is converted to kinetic

PDF page 316: energy and back to potential as the string oscillates back and forth. A very small fraction is dissipated as sound energy, slowly removing energy from the string.