The universal law
Isaac Newton proposed that any two point masses attract each other along the line joining them. The gravitational force between them is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. This single idea explains why an apple falls, why the Moon orbits the Earth, and why planets orbit the Sun.
Key formula / 关键公式
F = GMm / r²
F = gravitational force (N); M, m = the two masses (kg); r = distance between centres (m); G = 6.67 × 10⁻¹¹ N m² kg⁻² (universal gravitational constant).
What the formula tells us
- If either mass doubles, F doubles (F ∝ M and F ∝ m).
- If the separation r doubles, F becomes one quarter, because F ∝ 1/r² (the inverse-square relationship).
- The forces on the two bodies are equal in size and opposite in direction (Newton's third law), even when one mass is huge and the other tiny.
Because G is so small, gravitational forces between everyday objects are far too weak to feel. The force only becomes large when at least one mass is astronomical, such as a planet or star.
Link to weight
Near a planet's surface the pull of the planet on an object of mass m is its weight. Setting F = mg gives g = GM / r², so the gravitational field strength (and hence g) depends only on the planet's mass and radius, not on the object placed in the field.
Worked example / 例题
A 500 kg satellite orbits at a distance r = 8.0 × 10⁶ m from the centre of the Earth (mass M = 6.0 × 10²⁴ kg). Find the gravitational force on it.
F = GMm / r² = (6.67 × 10⁻¹¹ × 6.0 × 10²⁴ × 500) / (8.0 × 10⁶)²
= (2.0 × 10¹⁷) / (6.4 × 10¹³) ≈ 3.1 × 10³ N, directed toward the Earth's centre.
Remember / 记住
- r is measured between centres, not surfaces.
- Inverse-square: triple the distance divides the force by 9.
- G is a universal constant; g is not (it varies from planet to planet).