3.7.2.1
Newton's law of gravitation: $F = GMm/r^{2}$
Gravitational Fields & Orbits — AQA A-Level Physics
Key Definition
Newton's law of gravitation — The gravitational force between two point masses is proportional to the product of their masses and inversely proportional to the square of their separation.
- This is an inverse square law. Doubling the separation reduces the force to (1/2)² = 1/4 of its original value.
- r is measured from centre to centre, not surface to surface. For a satellite above a planet: $r = planet radius + altitude$.
- Planets are not point masses, but their separation is much larger than their radius, so the law applies.
Worked Example
A satellite of mass 6500 kg orbits at 2000 km above the Earth's surface. The gravitational force is 37 kN. Calculate the mass of the Earth. Radius of Earth = 6400 km.
Show Solution
1
Write Newton's law
$$F = \frac{Gm_1 m_2}{r^2}$$
2
Rearrange for M (Earth's mass)
$$M = \frac{r^2 F}{G m}$$
3
Calculate the distance r
r is from the centre of the Earth to the satellite:
$$r = 2000 + 6400 = 8400 \text{ km} = 8.4 \times 10^6 \text{ m}$$4
Substitute values
$$M = \frac{(8.4 \times 10^6)^2 \times 37 \times 10^3}{(6.67 \times 10^{-11}) \times 6500}$$
$$= 6.0 \times 10^{24} \text{ kg}$$Answer
$M = 6.0 \times 10^{24}$ kg
Common Mistake
MEDIUM
Students often: Using the altitude above the surface as r.
Instead: r is always measured from the centre of the planet. Add the planet's radius to the altitude: $r = R + h$.
Instead: r is always measured from the centre of the planet. Add the planet's radius to the altitude: $r = R + h$.