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# Gravitational potential and gravitational potential energy

## Change in gravitational potential energy

The change in gravitational potential energy is usually more relevant than the absolute level of potential energy.

It indicates the energy needed to bring an object from one point to another (assuming that no other forces, such as friction, are acting on it).

In a uniform field, the change in gravitational potential energy is given by: $$\Delta U=\Delta E_{\text{p}}=mg\Delta h$$ $\Delta h=$change in height relative to the gravitational field, $m=$mass of the object being moved, $g=$gravitational field strength.

## Gravitational potential

The gravitational potential ($\Phi$ or $V$) is the gravitational potential energy per unit mass.

$\Phi$ at a point is defined as the work done by a gravitational field on a unit mass in bringing the mass from infinity to the point in question. The potential is measured in $\text{ J}\text{ kg}^{-1}$ or $\text{ kg m}^2\text{ s}^{-2}$.

The equation for the gravitational potential is: $$\Phi=V = \frac{U}{m} = -\frac{GM}{r}$$

$r=$distance from mass $M$; $G=$universal gravitational constant.

## Gravitational potential and gravitational force

The gravitational force is the negative of the gradient (or the derivative) of the gravitational potential energy.

The gradient tells you by how much the gravitational potential energy changes per unit of distance that you move towards or away from the gravitational force.

The proof for the equation for the potential around a unit mass $M$ is as follows:

The potential is the work done by a gravitational field on a unit mass in bringing the mass from infinity to its current position $R$.

If the unit mass is at distance $S$ from the planet (the source of the gravitational field), the work done to move it infinitesimally from $S$ to $S+dS$ is $F(S) dS = -GM/S^{2}$ (the force is opposite to the movement).

Integrating between the current position $R$ to infinity and using the fact that the integral of $S^{-2}$ is $-S^{-1}$, we get:$$V =\int_R^\infty F(S) dS= -\int_R^\infty \frac{GM}{S^2}\, dS = \left[\frac{GM}{S}\right]_R^\infty = -\frac{GM}{R}$$

## Representing gravitational fields

A gravitational field can be represented graphically in two different ways:

• Gravitational field lines are lines indicating the direction of the gravitational force. For a point mass, the gravitational field lines point radially towards the mass. The density of field lines in an area indicates the gravitational field strength.

• An equipotential line connects points of equal gravitational potential in a gravitational field. In two-dimensional diagrams, the equipotential lines of a point mass are represented as concentric circles around the mass.

The potential difference between neighbouring equipotential lines is always the same. A greater density of equipotential lines (a smaller distance between the lines) reflects a sharper change in the potential in the corresponding region of space.

An equipotential line is always perpendicular to the gravitational field lines and the arrows representing the force vectors.

Gravitational field lines in a uniform field (left) and around a point mass (centre) and equipotential lines around a point mass (right).