State Newton's law of gravitation. Derive the expression for the acceleration due to gravity and find the mass of the Earth.

1. 1 · State the law in one line. Every particle attracts every other with a force ∝ product of masses and ∝ 1/(distance)²: F = Gm₁m₂/r².
2. 2 · Explain G. G = 6.674 × 10⁻¹¹ N m² kg⁻²: numerically equal to the force between two 1 kg masses 1 m apart. It is the same everywhere.
3. 3 · Derive g.
   • Force of gravity on a body of mass m: F = GMm/R²
   • By the second law: F = mg
   • So g = GM/R²; g does not depend on m
4. 4 · Find the mass of the Earth. M = gR²/G = 9.8 × (6.378 × 10⁶)² ÷ (6.674 × 10⁻¹¹) ≈ 5.97 × 10²⁴ kg.
5. 5 · Say how g varies. g decreases with height and depth, is zero at the centre, and is greatest at the poles.
6. 6 · Real-life link. Applications: masses of planets and stars, paths of astronomical bodies, roots growing downward.
7. 7 · Close in one line. The same law explains a falling coconut and the motion of the planets.

Scoring tip: Keep the derivation to four clean lines and box g = GM/R². Show the substitution for the mass of the Earth with units.