Physics › Gravitational fields › Newton's law of gravitation
Newton's law of gravitation
One equation covers the apple and the Moon: every pair of masses attracts with a force set by their product and the inverse square of their separation. Divide by the test mass and the same law hands you the field strength anywhere.
Builds on The field concept.
IN THIS TOPIC
- Use Newton's law of gravitation for point masses.
- Define g as force per unit mass and use g = F/m.
- Use g = GM/r² in a radial field, with r measured from the centre.
WHAT YOU PROBABLY THINK
There's no gravity in space.
The universal law
Gravity is a universal attractive force acting between all matter: every mass pulls every other. For point masses, and for spheres treated from outside as points at their centres, the magnitude is
where G, the gravitational constant, is 6.67 × 10−11 N m2 kg−2, printed in the data booklet. Its smallness is why gravity between everyday objects goes unnoticed: two people a metre apart attract with well under a millionth of a newton.
The force is mutual: the Earth pulls you and you pull the Earth with exactly equal force, the accelerations differing only because the masses do. Exam questions probe this precisely because intuition resists it.
Inverse square
The r2 downstairs is the law's character. Double the distance, quarter the force; treble it, a ninth. Any question comparing forces at two separations is really asking you to square a ratio, and setting the ratio up before touching numbers is the fastest route through.
From force to field strength
The gravitational field strength at a point is the force per unit mass a body placed there would feel:
measured in N kg−1, which is the same unit as m s−2: field strength and free-fall acceleration are one quantity in two costumes. Substitute Newton's law and the test mass cancels, leaving the field of a mass M in its radial region:
One habit prevents most lost marks in this unit: r is measured from the centre, never from the surface. An orbit “400 km up” sits at r = 6.37 × 106 + 4.0 × 105 m, and forgetting the planet's own radius is the classic error.
THE EXAM BIT
- The law applies to point masses, with spherical bodies treated as points at their centres. Stating that assumption is often a mark in itself.
- Ratio questions: (r₁/r₂)2 first, numbers second. Doubling r quarters both F and g.
- The pull is mutual and equal on both bodies, however unequal the masses. “The Earth pulls harder on you than you pull on it” is the planted lie.
- r from the centre. Add the planet's radius to any altitude before squaring anything.
- g at the International Space Station's altitude is about 89% of the surface value; astronauts float because they are in free fall, not because gravity has stopped. The calculation and the explanation pair up in exam questions.
CHECK YOURSELF
The ISS orbits about 400 km above the Earth's surface. Taking M = 5.97 × 1024 kg and RE = 6.37 × 106 m, calculate g at the station, and explain why astronauts float despite your answer.
Show a hint
Build r from the centre first, then ask what free fall feels like from inside.
Show the answer
r = 6.37 × 106 + 4.0 × 105 = 6.77 × 106 m.
g = GM/r2 = (6.67 × 10−11 × 5.97 × 1024) / (6.77 × 106)2 = 8.7 N kg−1, about 89% of the surface value.
Astronauts float because station and crew are both in free fall, accelerating identically under that g while perpetually missing the ground. Gravity up there is nearly full strength; support forces are what vanished.
Every mass pulls every other, as the inverse square.
Field strength is force per unit mass, from the centre.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.