A form of energy associated with the presence of conservative interactions such as gravity or a spring.
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Motivation for Concept
Conservative interactions like [gravity] have the ability to "store" kinetic energy. Consider an object thrown up to a high roof. If the object is thrown perfectly, the force of gravity will slow it to a stop just as it reaches the roof. The object will then remain at rest on the roof until disturbed. But, as it falls, gravity will restore all the speed that was removed on the way up. Because we can perfectly "recover" the kinetic energy "removed" by gravity, we can consider the total energy to be constant in such a situation if we can associate some energy with the object's height. That energy is the [gravitational potential energy].
Definition
Finding Potential Energy From Force
The work done by a general force is given by:
\begin
[ W = \int_
\vec
\cdot d\vec
]\end
The [work energy theorem] tells us that when work is done on a system, the system's kinetic energy will change:
\begin
[ K_
+ W = K_
]\end
Suppose that we consider a system acted upon by a single, conservative force. If we want to define a potential energy U to represent this interaction in such a way that the mechanical energy of the system is conserved, we must take:
\begin
[ U_
- U_
= W^
]\end
With this definition, the work-energy theorem takes the form:
\begin
[ K_
+ (U_
-U_
) = K_
]\end
which is equivalent to the conservation of mechanical energy:
\begin
[ K_
+ U_
= K_
+ U_
]\end