Gravitational Potential Energy

Gravitational potential energy is a form of potential energy that an object possesses by virtue of its relative location within a gravitational field. When an object is lifted against the force of gravity, work is done on the object, and gravitational potential energy is stored in the object.

The relative height of the ball gives it gravitational potential energy. It’s position in a gravitational field imparts this form of energy.

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Formula for Gravitational Potential Energy

The formula for gravitational potential energy is:

$U_g = mgh$

where $U_g$ is the gravitational potential energy, $m$ is the mass of the object, $g$ is the acceleration due to gravity, and $h$ is the height of the object above a reference point. This formula states that the gravitational potential energy of an object is directly proportional to its mass, the acceleration due to gravity, and the height it is lifted (relative to some reference point at which $h=0$ . It is important to note that this feature of a “reference point” indicates a special quality of gravitaional potential energy: it can be negative! Kinetic energy, on the other hand, is a strictly positive quantity in classical mechanics.

General Formulation

The force due to gravity follows the inverse square principle–force decreases with $R^2$ (the square of the distance). Consequently, when we take the negative integral of the gravitational force $(F=- \dfrac{GMm}{R^2})$ with respect to the distance $R$ , we find that the general formulation of gravitational potential energy is given as:

$U_g = - \dfrac{GMm}{R}$

where:

$G=6.67430 \cdot 10^{-11} \dfrac{N \cdot m^2}{kg^2}$
$M$ is the mass of one body
$m$ is the mass of the other body
and $R$ is the distance between $M$ and $m$

The negative sign is meaningful because it indicates that gravity does positive work in pulling one object closer to another. It is a usesful (and simple) exercise to prove this, though this is left to the reader.

The moon has gravitational potential energy associated with it due to its position with respect to the Earth.

Units of Gravitational Potential Energy

The units of gravitational potential energy are Joules (J), which are the same units used for work and energy. One Joule is equal to the amount of energy required to move an object with a force of one Newton over a distance of one meter.

Note: In the English System, the unit of energy is the foot-pound.

Applications of Gravitational Potential Energy

The concept of gravitational potential energy is applied in many areas of science and engineering, including mechanics, geology, and astrophysics. For example, it is used in the design of roller coasters and other amusement park rides, where the height of the ride is used to create potential energy that is then converted into kinetic energy as the ride moves downward. Gravitational potential energy is also used in the study of tides as well as the motion of planets and stars in the universe.

Conservation of Energy

According to the Law of Conservation of Energy, energy cannot be created or destroyed, but it can be converted from one form to another. In the case of gravitational potential energy, when an object falls, its potential energy is converted into kinetic energy, which is the energy of motion. The total energy of the object remains constant, however, as energy is conserved. In reality, non-conservative forces are generally present which transfer energy out of the object and into the environment. Even in this case, the energy of the total system (object plus environment) remains constant.

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Gravitational Potential Energy