Gravitational fields

This module now moves from circles to orbits, and the force behind every orbit is gravity. This page sets up the two equations everything else is built on: the force between two masses, and the gravitational field around one.

Newton's law of universal gravitation

Every mass attracts every other mass. For two masses $M$ and $m$ whose centres are a distance $r$ apart:

$F = \dfrac{GMm}{r^2}$

$G = 6.67 \times 10^{-11}\ \text{N m}^2\,\text{kg}^{-2}$ is the universal gravitational constant. It's tiny, which is why you don't feel the pull of the person next to you (about $10^{-7}$ N), and why gravity only matters when at least one of the masses is planet-sized.

Three features to know:

• It is an inverse square law: double the distance and the force drops to a quarter; triple it and the force is one-ninth.
• Distance is measured between centres. For a satellite, $r$ = planet's radius + altitude.
• The forces form a Newton's third law pair: Earth pulls the Moon exactly as hard as the Moon pulls Earth. The Moon simply responds more, because its mass is smaller.

Gravitational field strength

A gravitational field is the region around a mass where another mass feels a force. The field strength $g$ is the force per kilogram on a small test mass placed there: $g = F/m$. Dividing Newton's law by $m$:

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

The units of $g$ are N kg⁻¹, which are the same as m s⁻². It is both the force on each kilogram and the acceleration of anything in free fall there. It depends only on the mass $M$ creating the field and the distance from its centre, not on the object placed in it.

Field lines show the direction a test mass would accelerate. Around a planet they point radially inward. Closer lines mean a stronger field. Near Earth's surface, over a region much smaller than Earth, the lines are almost parallel and evenly spaced: the field is close to uniform, which is the assumption we used for projectiles.

So gravity at the ISS is far from zero. Astronauts float because they and the station are falling together, which is the subject of 5H.

The data-sheet mass gives 9.86 rather than the familiar 9.8 because it is rounded to two significant figures (the precise value is $5.97 \times 10^{24}$ kg). Use the data sheet values in exams; the markers expect them.

What affects g

From $g = GM/r^2$, a planet with more mass has a stronger surface field, and a planet with a larger radius has a weaker one, for the same mass. Mars has about one-ninth of Earth's mass but just over half its radius, so its surface $g$ is $3.7\ \text{N kg}^{-1}$, about 38% of Earth's. Jupiter is 318 times as massive as Earth but 11 times as wide, so its "surface" (cloud-top) $g$ is only about 2.5 times Earth's.

On Earth itself, $g$ isn't exactly the same everywhere:

• Latitude: about 9.78 N kg⁻¹ at the equator and 9.83 N kg⁻¹ at the poles. Earth bulges at the equator (you are further from the centre) and the spin means part of gravity's pull is used as centripetal force.
• Altitude: $g$ falls by about 0.3% for every 10 km up.
• Local geology: dense ore bodies give a slightly stronger field; low-density sediments or salt domes a weaker one. Mineral explorers in Australia map these tiny differences with gravimeters, from the ground and from aircraft.