Gravitational Potential Energy
We can compute the amount of work done against the gravitational force of mass M to move the mass m from a distance r1 (position A) to a distance r2 (position B) as shown in figure.
This work is given by
which represents the increase in PE as m moves from A to B. we can apply this general expression to the special case of a mass near the surface of the earth.
Potential energy near the surface of the earth
Suppose a body of mass m is raised from the surface of the earth to a height h. In this case, setting r1=R and r2 = (R + h),
we have (R is the earth's radius)
For a body near the surface of the earth, h<<R and (R + h) in the denominator can be replaced by R. Thus we have
PE = mgh
Note: The work done W12 measures the change in potential energy when the mass m is moved from r1 to r2 which may also be written as
Change in PE =
- If r1 < r2, W12 will be positive. This means that an outside agent has to do positive work (force and displacement in the same direction) in separating the two masses against their mutual gravitational attraction.
- On the other hand, if r1>r2, W12 will be negative. This means that the work has to be done by the gravitational force in bringing the two masses closer. When the two masses are released from a given separation, they accelerate towards each other because of mutual attraction.
Gravitational Potential and Gravitational field
g = F/m.