Newton's Law of Gravitation — Earth–Moon Force (Worked Example)

Given

• m₁ = 5.97 × 10²⁴ kg (Earth)

• m₂ = 7.35 × 10²² kg (Moon)

• r = 3.84 × 10⁸ m

• G = 6.674 × 10⁻¹¹ N·m²/kg²

Find

The gravitational force F.

Principle

Newton's law of universal gravitation: every mass attracts every other mass with F = Gm₁m₂/r², weakening with the square of the distance.

Formula

Solution

1. Write the law: $F = G\frac{m_1 m_2}{r^2}$

2. Substitute: $F = \frac{(6.674 \times 10^{-11})(5.97 \times 10^{24})(7.35 \times 10^{22})}{(3.84 \times 10^8)^2}$

3. Numerator: $2.929 \times 10^{37}$; denominator: $1.475 \times 10^{17}$

4. Divide: $F = 1.99 \times 10^{20}\ \mathrm{N}$

Unit check

N·m²/kg² × kg² / m² = N — a force, correct.

What this means physically

About 2 × 10²⁰ N — an almost unimaginable force, yet it only produces the Moon's gentle 0.0027 m/s² centripetal acceleration because the Moon is so massive. This same force raises Earth's ocean tides.