Gravitation

Complete revision notes

1. Introduction

Gravity is an attractive interaction. Near Earth's surface it is approximated by a constant downward acceleration g, but g varies with distance from Earth's centre.

2. Kepler's laws

Kepler's laws: planets orbit in ellipses with the Sun at one focus; the radius vector sweeps equal areas in equal times; and T² is proportional to the cube of semi-major axis for objects orbiting the same central mass.

3. Universal law of gravitation

Newton's law is F=Gm₁m₂/r² along the line joining the centres. Outside a spherically symmetric body, its field is equivalent to that of a point mass at the centre.

4. The gravitational constant

G is universal with SI unit N m² kg⁻²; g is local free-fall acceleration and depends on the attracting body's mass and distance.

5. Acceleration due to gravity of the earth

At a planet surface g=GM/R² and weight W=mg. Mass remains fixed, while weight changes with gravity. Free-fall apparent weight may be zero although gravity is present.

6. Acceleration due to gravity below and above the surface of earth

At altitude h, g_h=g[R/(R+h)]². In a uniform-density Earth model, g_d≈g(1−d/R) below the surface; actual density varies, so this is an approximation.

7. Gravitational potential energy

With zero potential energy at infinity, U=−GMm/r. For small height changes near Earth's surface, ΔU≈mgΔh. Gravitational potential is potential energy per unit mass.

8. Escape speed

Escape speed is v_e=√(2GM/R)=√(2gR), assuming no atmosphere or rotation. It does not depend on the launched body's mass.

9. Earth satellites

For a circular satellite orbit, gravity supplies centripetal force. Equating GMm/r²=mv²/r gives v=√(GM/r) and T=2π√(r³/GM). Measure r from the planet's centre.

10. Energy of an orbiting satellite

For a circular orbit, K=GMm/(2r), U=−GMm/r and total E=−GMm/(2r). Negative total energy indicates a gravitationally bound orbit.

Formula sheet

Use SI units, state assumptions and directions before substitution, show working and round only the final result.

Worked example

Common errors and exam checklist

Use orbital radius from the centre, not altitude alone; distinguish G from g; escape speed is not orbital speed.

• Write the relevant law before substituting values.
• Show steps, units and signs/directions.
• Check that the final answer is physically reasonable.