Gravitation

Newton's law of universal gravitation, gravitational potential, Kepler's laws, and orbits.

General idea of the topic

Gravitation explains the attraction between bodies with mass, from the fall of an apple to the motion of the planets. Newton's law of universal gravitation describes that force with a single expression, proportional to the masses and inversely proportional to the square of the distance.

The block starts from the law, introduces the gravitational field and potential, the gravitational potential energy that generalizes mgh, and applies everything to orbits and satellites. It ends with tides and perturbations, where the variation of the field with distance produces measurable effects. Between potential energy and orbits, motion under central forces is studied, which reduces the problem to a radial equation with an effective potential.

What you will truly understand

By the end you will be able to compute the force between two masses, the field strength at a point, the potential energy and the escape velocity, and derive Kepler's laws of orbital motion.

You will also know when mgh is a valid approximation and when −GMm/r must be used.

Recommended learning route

1. Universal Law of Gravitation: force between masses, the constant G and relation to weight.
2. Gravitational Field and Potential: field as force per unit mass and potential as energy per unit mass.
3. Gravitational Potential Energy: −GMm/r, reference level at infinity and relation to mgh.
4. Central Forces: motion under a central force: conservation of angular momentum, effective potential and the Binet equation.
5. Orbits and Satellites: orbital speed, orbital energy, escape velocity and Kepler's laws.
6. Tides and Perturbations: effect of field differences on extended bodies and perturbations.

Physical criterion you must acquire

The essential criterion is to choose the formulation according to the problem: force if the instantaneous acceleration matters, energy if two positions matter, and Kepler's laws if orbits are compared. With energy, the reference at infinity makes the energy of a bound system negative.

Connection with real problems

• communication and navigation satellites.
• space travel and escape velocity.
• tides and astronomy.
• geodesy and gravimetry.

Final goal of the topic

The final goal is to describe and compute motion under gravity coherently, from local fall to planetary orbits.