How is potential energy calculated in gravitational fields?

Potential energy in gravitational fields is calculated using the formula PE = mgh, where m is mass, g is gravity, and h is height.

In more detail, potential energy is the energy that an object possesses due to its position relative to other objects. In the context of gravitational fields, this energy is often associated with the object's height above the ground or another reference point. The formula PE = mgh is a simplified version of the more general formula for gravitational potential energy, which is PE = -GMm/r, where G is the gravitational constant, M is the mass of the Earth (or other large body), m is the mass of the object, and r is the distance from the centre of the Earth (or other large body) to the object.

The simplified formula PE = mgh is often used in situations where the height h is much less than the radius of the Earth, so the gravitational field strength g can be considered constant. In this formula, m is the mass of the object, g is the acceleration due to gravity (approximately 9.81 m/s² on the surface of the Earth), and h is the height above the reference point.

The more general formula PE = -GMm/r takes into account the fact that the gravitational field strength decreases with distance from the centre of the Earth (or other large body). The negative sign indicates that the potential energy decreases as the distance r increases, which means that the potential energy is less than zero when the object is above the reference point (the centre of the Earth or other large body). This is because work would have to be done against the gravitational field to move the object to a higher position, which would increase its potential energy.

In both formulas, the potential energy is directly proportional to the mass of the object: the greater the mass, the greater the potential energy. This is because a larger mass would require more work to lift to a given height, or to move to a given distance from the centre of the Earth (or other large body).