Gravitational fields

13.1 Gravitational field

A gravitational field is a field of force: a region where a mass feels a force. Gravitational field strength at a point is the force per unit mass:

$g = \frac{F}{m}$

The unit is N/kg, which is the same as m/s².

Field lines

Field lines show the direction of the force on a small mass. Around a planet they point radially inwards, and they get further apart with distance, so the field weakens. Near the surface over a small region, the lines are close to parallel and equally spaced, so the field is nearly uniform.

13.2 Gravitational force between point masses

Newton's law of gravitation

For a point outside a uniform sphere, the mass of the sphere may be treated as a point mass at its centre. Two point masses attract with:

$F = \frac{Gm_1m_2}{r^2}$

where $G = 6.67 \times 10^{-11}$ N m² kg⁻² and $r$ is the distance between their centres. It is an inverse square law, so doubling $r$ makes the force four times smaller.

Circular orbits

For a satellite in a circular orbit, gravity provides the centripetal force:

$\frac{GMm}{r^2} = \frac{mv^2}{r}$

which gives $v^2 = GM/r$. The satellite's mass cancels, and closer orbits are faster. Using $v = 2\pi r/T$ gives:

$T^2 = \frac{4\pi^2 r^3}{GM}$

So $T^2 \propto r^3$. You should be able to derive this, not just quote it.

Geostationary orbits

A geostationary satellite stays above the same point on the Earth's surface. For this, it must:

• have an orbital period of 24 hours,
• orbit from west to east, the same direction as the Earth's rotation,
• be directly above the Equator.

There is only one possible radius, about $4.2 \times 10^{7}$ m from the Earth's centre.

13.3 Gravitational field of a point mass

Put a small mass $m$ in the field of a point mass $M$. Since $F = GMm/r^2$ and $g = F/m$:

$g = \frac{GM}{r^2}$

Why g is nearly constant near the surface

The Earth's radius is about 6400 km. Rising 1 km changes $r$ by a tiny fraction, so $r^2$ and therefore $g$ barely change. This is why we use $g = 9.81$ N/kg and $\Delta E_P = mg\Delta h$ for small height changes.

13.4 Gravitational potential

Definition

The gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point.

$\phi = -\frac{GM}{r}$

The unit is J/kg.

Potential is a scalar, so potentials from several masses simply add.

Gravitational potential energy

The potential energy of a mass $m$ at a point is $m\phi$:

$E_P = -\frac{GMm}{r}$

The change in potential energy between two radii is:

$\Delta E_P = GMm\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

where $r_2 > r_1$ for a mass moving outwards, and the potential energy rises.

Field and potential

The field strength is the negative of the potential gradient, $g = -\Delta\phi/\Delta r$. On a graph of $\phi$ against $r$, the gradient gives the size of $g$.