Newton’s law of universal gravitation

The law

📖 Definition (learn the wording)

Newton’s law of universal gravitation: every particle with mass in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

F = G m₁m₂ ÷ r² G = 6,67 × 10⁻¹¹ N·m²·kg⁻²

The force is always attractive, and the two masses exert equal and opposite forces on each other (Newton’s third law).
r is measured between the centres of the masses — for a planet, from the centre of the planet.

Proportional reasoning

Many multiple-choice questions ask what happens to F when a quantity changes. Use the formula as a ratio:

Change	Effect on F
One mass doubled	F doubles
Both masses doubled	F × 4
Distance doubled	F ÷ 4
Distance tripled	F ÷ 9
Distance halved	F × 4

Gravitational acceleration and weight

On a planet of mass M and radius R, the weight of an object is the gravitational force on it, so mg = GMm ÷ R² and

📖 Gravitational acceleration

g = GM ÷ r² (r = R at the surface; r = R + h at a height h above the surface)

✏️ Worked example 1: g on Earth

1M = 5,98 × 10²⁴ kg, R = 6,38 × 10⁶ m.

2g = (6,67 × 10⁻¹¹)(5,98 × 10²⁴) ÷ (6,38 × 10⁶)² = 9,80 m·s⁻².

✏️ Worked example 2: at a height of 2R

1Distance from the centre = R + 2R = 3R.

2g′ = GM ÷ (3R)² = (1/9) × GM/R² = g ÷ 9.

Mass and weight

Mass is the amount of matter in an object (kg). It is the same everywhere.
Weight is the gravitational force the Earth (or another planet) exerts on an object (N): w = mg. It depends on where the object is.

✏️ Worked example 3: force between two objects

1Two 1 000 kg cars are 2 m apart (between centres).

2F = (6,67 × 10⁻¹¹)(1 000)(1 000) ÷ 2² = 1,67 × 10⁻⁵ N — far too small to notice, which is why gravity between everyday objects is usually ignored.

Answer in your exercise book, then open each memo to mark your work.

Question 1 · (2 marks)

State Newton’s law of universal gravitation in words.

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Every particle with mass in the universe attracts every other particle with a force which is directly proportional to the product of their masses ✓ and inversely proportional to the square of the distance between their centres ✓.

Question 2 · (4 marks)

Calculate the gravitational force between the Earth (5,98 × 10²⁴ kg) and a 70 kg person standing on the surface (R = 6,38 × 10⁶ m).

Show memo

F = Gm₁m₂/r² ✓ = (6,67 × 10⁻¹¹)(5,98 × 10²⁴)(70) ÷ (6,38 × 10⁶)² ✓✓ = 686 N (accept 685–687 N) ✓

Question 3 · (3 marks)

The gravitational force between two objects is F. What is the new force if the mass of one object is doubled and the distance between them is also doubled? Show your reasoning.

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F ∝ m₁m₂/r² ✓. New force = (2)/(2²) × F ✓ = F/2 ✓.

Question 4 · (4 marks)

A planet has the same mass as the Earth but twice its radius. (a) Calculate the gravitational acceleration on its surface in terms of g on Earth. (b) What would a 60 kg astronaut weigh there? (Use g = 9,8 m·s⁻² for Earth.)

Show memo

(a) g′ = GM/(2R)² = g/4 ✓✓ = 2,45 m·s⁻²
(b) w = mg′ = 60 × 2,45 ✓ = 147 N ✓