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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.

Real world connections: energy of a bowling ball

How much energy does a bowling ball have? (Just think about it for a minute.)

If you are thinking that you need more information, you’re right. If we can measure the ball’s velocity, then determining its kinetic energy is simple. Note that this does require defining a reference frame in which to measure the velocity. Determining the ball’s potential energy also requires more information. You need to know its height above the ground, which requires a reference frame of the ground. Without the ground—in other words, Earth—the ball does not classically have potential energy. Potential energy comes from the interaction between the ball and the ground. Another way of thinking about this is to compare the ball’s potential energy on Earth and on the Moon. A bowling ball a certain height above Earth is going to have more potential energy than the same bowling ball the same height above the surface of the Moon, because Earth has greater mass than the Moon and therefore exerts more gravity on the ball. Thus, potential energy requires a system of at least two objects, or an object with an internal structure of at least two parts.

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

PE s = 1 2 kx 2 , size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {}

where k size 12{k} {} is the spring’s force constant and x size 12{x} {} is the displacement from its undeformed position. The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance x size 12{x} {} . The potential energy of the spring PE s size 12{"PE" rSub { size 8{s} } } {} does not depend on the path taken; it depends only on the stretch or squeeze x size 12{x} {} in the final configuration.

Questions & Answers

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
why god made humenity
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
when we face electric shock these elements work as a conductor that's why we got this shock
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
why electromagnetic induction is not used in room heater ?
Gopi Reply
What is position?
Amoah Reply
What is law of gravition
sushil Reply
Practice Key Terms 5

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