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# Gravitational Potential Energy

Formula for gravitational potential energy

Andrew Zimmerman Jones

### Gravity Index

When an object moves in a gravitational field, work must be done to get it from one place to another (starting point 1 to end point 2). Using calculus, we take the integral of the force from the starting position to the end position. Since the gravitational constants and the masses remain constant, the integral turns out to be just the integral of 1 / r2 multiplied by the constants.

We define the gravitational potential energy, U, such that W = U1 - U2. This yields the equation to the right, for the Earth (with mass mE. In some other gravitational field, mE would be replaced with the appropriate mass, of course.

### Gravitational Potential Energy on Earth

On the Earth, since we know the quantities involved, the gravitational potential energy U can be reduced to an equation in terms of the mass m of an object, the acceleration of gravity (g = 9.8 m/s), and the distance y above the coordinate origin (generally the ground in a gravity problem). This simplified equations yields a gravitational potential energy of:

U = mgy

There are some other details of applying gravity on the Earth, but this is the relevant fact with regards to gravitational potential energy.

Notice that if r gets bigger (an object goes higher), the gravitational potential energy increases (or becomes less negative). If the object moves lower, it gets closer to the Earth, so the gravitational potential energy decreases (becomes more negative). At an infinite difference, the gravitational potential energy goes to zero. In general, we really only care about the difference in the potential energy when an object moves in the gravitational field, so this negative value isn't a concern.

This formula is applied in energy calculations within a gravitational field. As a form of energy, gravitational potential energy is subject to the law of conservation of energy.