If gravity moves an object it does work on that object. However, the amount of work done does not depend on the path over which gravity acted, but rather on the initial and final positions of the object. This means that gravity is a conservative force. We can sketch a proof of this. Imagine we have a fixed mass M and some other mass m that is moved from A to B by the gravitational force of M . It is clear that any two imaginable paths can be broken into infinitesimal steps perpendicular and parallel to the radius connecting M and m . Since gravity is a central force, the perpendicular steps make no contribution to the work, since no force is acting in this direction. Since both paths progress from A to B , the sum of their parallel-radial segments must be equal. Since the magnitude of the force is equal at equal radial distance, the work in each case must be equal.
This path independence allows us to assign a unique value to all points a distance r from a gravitating source. We call this value U(r) , the gravitational potential energy. As with any potential energy, we need to define some reference point as a zero. Therefore, we define U(∞) = 0 and then:
= - |
It remains to evaluate the integral. We can do this along any path we choose (since they are all equivalent). We will choose the simplest path: a straight radial path along the x -axis. In this case the force is given by = and d = dx . Thus:
U(r) = - dx = = - |
A useful concept when dealing with forces that act at a distance is the field. Gravitational field lines help us to imagine what sort of forces would act on a particle at a certain point near another gravitating object. The direction of the field lines indicates the direction of the force that a mass would experience if placed at a certain point, and the density of the field lines is proportional to the strength of the force. Since gravity is an attractive force, all field lines point towards masses.
Occasionally, another concept is defined with respect to gravitational potential energy. We define it here primarily to avoid possible confusion with the gravitational potential energy. Gravitational potential, Φ _{g} , is defined as the potential energy that a unit mass (usually 1 kilogram) would have at any point. Mathematically:
Φ _{g} = - |
We can see what happens to our expression for gravitational potential energy near the earth. In this case M = M _{e} . Consider a mass m at a distance r from the center of the earth. Its gravitational potential energy is:
U(r) = - |
U(r _{e}) = - |
ΔU = U(r)±U(r _{e}) - + = (GM _{e} m) |
ΔU = h = mgh |