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  • Define conservative force, potential energy, and mechanical energy.
  • Explain the potential energy of a spring in terms of its compression when Hooke’s law applies.
  • Use the work-energy theorem to show how having only conservative forces implies conservation of mechanical energy.

Potential energy and conservative forces

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     ( PE ) size 12{ \( "PE" \) } {} 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.

Potential energy and conservative forces

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 ( PE ) size 12{ \( "PE" \) } {} 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.

Potential energy of a spring

First, let us obtain an expression for the potential energy stored in a spring ( PE s size 12{"PE" rSub { size 8{s} } } {} ). 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 F size 12{F} {} on the spring and the resulting deformation Δ L size 12{ΔL} {} are proportional, F = k Δ L size 12{F=kΔL} {} .) (See [link] .) For our spring, we will replace Δ L (the amount of deformation produced by a force F ) by the distance x that the spring is stretched or compressed along its length. So the force needed to stretch the spring has magnitude F = kx size 12{ ital "F = kx"} {} , where k size 12{k} {} is the spring’s force constant. The force increases linearly from 0 at the start to kx size 12{ ital "kx"} {} in the fully stretched position. The average force is kx / 2 . Thus the work done in stretching or compressing the spring is W s = Fd = kx 2 x = 1 2 kx 2 size 12{W rSub { size 8{s} } = ital "Fd"= left ( { { ital "kx"} over {2} } right )""x= { {1} over {2} } ital "kx" rSup { size 8{2} } } {} . Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of F size 12{F} {} vs. x size 12{x} {} is the work done by the force. In [link] (c) we see that this area is also 1 2 kx 2 size 12{ { {1} over {2} } ital "kx" rSup { size 8{2} } } {} . We therefore define the potential energy of a spring    , PE s size 12{"PE" rSub { size 8{s} } } {} , to be

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Practice Key Terms 5

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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