Newton's law of gravitation, gravitational field strength for radial and uniform fields, gravitational potential, and orbital motion of satellites and planets.

What this dot point is asking

Edexcel wants you to apply Newton's law of gravitation, define and calculate gravitational field strength in radial and uniform fields, work with gravitational potential and potential energy, and analyse the circular orbits of satellites and planets, deriving Kepler's third law.

The answer

Newton's law of gravitation

Like Coulomb's law this is an inverse-square law, but gravity has only one sign (always attractive) and is extremely weak: $G$ is tiny, so gravitational forces matter only when at least one mass is astronomical. Treat spherical bodies as point masses at their centres.

Gravitational field strength

Around a point or spherical mass the field is radial, $g = \frac{GM}{r^2}$, pointing inwards. Close to a planet's surface, over distances small compared with the radius, the field is effectively uniform, which is why we treat $g$ as constant at about $9.8$ N per kg near the ground. The radial $g$ against $r$ graph falls as an inverse square outside the body.

Gravitational potential

Potential is a scalar that falls off as $\frac{1}{r}$. Equipotential surfaces are concentric spheres around a mass, and no work is done moving along one. The (positive) energy needed to escape to infinity from a surface defines escape velocity, $v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$.

Orbital motion

For a circular orbit, gravity provides the centripetal force: $\frac{GMm}{r^2} = \frac{mv^2}{r}$. Cancelling $m$ gives the orbital speed $v = \sqrt{\frac{GM}{r}}$, so satellites in lower orbits move faster. Substituting $v = \frac{2\pi r}{T}$ yields Kepler's third law, $T^2 = \frac{4\pi^2 r^3}{GM}$, so the square of the period is proportional to the cube of the orbital radius. A geostationary satellite has a period of exactly one sidereal day and orbits over the equator at about $3.6 \times 10^{7}$ m altitude.

Examples in context

Geostationary communications and weather satellites sit at the orbital radius where the period equals one day, so they appear fixed over a point on the equator. GPS satellites in medium orbit must correct for both special and general relativistic time effects to keep nanosecond accuracy. Kepler's third law, applied to a planet's moons, lets astronomers weigh the planet, and applied to stars orbiting a galactic centre it reveals the supermassive black hole there.

Try this

Q1. State Newton's law of gravitation in words. [1 mark]

• Cue. The attractive force between two point masses is proportional to the product of their masses and inversely proportional to the square of their separation.

Q2. A planet has mass $3.0 \times 10^{24}$ kg and radius $4.0 \times 10^{6}$ m. Find its surface gravitational field strength. [2 marks]

• Cue. $g = \frac{GM}{r^2} = \frac{6.67 \times 10^{-11} \times 3.0 \times 10^{24}}{(4.0 \times 10^{6})^2} = 13$ N per kg.

Q3. Explain why gravitational potential is always negative. [2 marks]

• Cue. Gravity is attractive and potential is zero at infinity, so work must be done against the field to remove a mass, making the potential negative everywhere else.