Gravitation
Gravitation is the domain that tests universal gravitation, Kepler's laws, free fall, variation of g, mass versus weight, and escape velocity. Questions often combine a formula with a conceptual comparison, so keep the meaning of each symbol clear.
Core rules
- Newton's law: F = G m1 m2 / r^2, where G = 6.67 x 10^-11 N m^2 kg^-2. The force is mutual and along the line joining centres.
- Acceleration due to gravity: g = GM/R^2 on a spherical planet. It is independent of the falling body's mass.
- Kepler: orbits are ellipses with the Sun at one focus; equal areas in equal times; T^2 is proportional to a^3.
- Weight is W = mg, a force in newtons; mass is the amount of matter in kilograms and is unchanged by location.
- Escape velocity v_e = sqrt(2GM/R); for Earth it is about 11.2 km/s and does not depend on the body's mass.
- With height h, g_h = g/(1 + h/R)^2; at depth d, g_d = g(1 - d/R). At the centre g = 0.
Worked example
Find g at a height equal to Earth's radius R above the surface. The distance from Earth's centre is R + R = 2R. Since g is inversely proportional to distance squared, g_h = g(R/2R)^2 = g/4. With g = 9.8 m/s^2, g_h = 2.45 m/s^2. The reasoning is to use centre-to-centre distance, not height alone.
Traps
Do not call mass a force or weight a mass. Do not use G and g interchangeably. Escape velocity is not larger for a heavier rocket. At the centre, g is zero, not maximum. Read whether a question gives height above the surface or distance from the centre.