Use −GM/r for Potential and Add Contributions Scalar by Scalar
Gravitational potential is a scalar, so contributions from different masses add like numbers, and the zero is taken at infinity. That makes it simpler than the field, which has to be added as vectors.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Potential, field and energy
Outside a spherical mass M the potential is V = -GM/r and the potential energy of a mass m is U = -GMm/r, both zero at infinity, with the field g = GM/r² = -dV/dr.
Several bodies and shells
Add potentials as scalars: a mass m at distance R from the Earth and r from the Moon has U = -Gm (M_earth / R + M_moon / r), and with M_earth = 81 M_moon this is -G m M_moon (81/R + 1/r). Inside a hollow spherical shell of mass M and radius a, the potential is constant and equal to -GM/a (the field is zero).
For a solid shell the graph of V against r is flat inside and follows -GM/r outside.
Falling in, and bound systems
Energy conservation gives the speed at the surface for a body falling from a distance r0: v² = 2GM(1/R - 1/r0). Falling from infinity gives v² = 2GM/R, while falling from a height of 10 R (r0 = 11 R) gives v² = (10/11) x 2GM/R, so the speeds are in the ratio sqrt(11) : sqrt(10). A system is bound when its total energy, kinetic plus potential, is negative; read the points on an energy graph where the sum KE + PE is below zero.
OlympIQ exam lens
- ✓V = -GM/r, U = -GMm/r, both zero at infinity; r = -V/g links the two
- ✓Add potentials as scalars; add fields as vectors
- ✓Inside a thin shell the potential is constant (-GM/a) and the field is zero
- ✓Falling from r0: v² = 2GM(1/R - 1/r0)
- ✓Total energy KE + PE < 0 means a bound system
Do not forget the potential of the particle at the centre of a shell when finding the potential inside it. The shell contributes a constant -GM/a, but the central mass contributes -GM/r at the actual distance r from it, and the two must be added.
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