Summary

General Principles

Newton’s Theory of Gravity

  1. Two objects with masses M and m a distance r apart exert attractive gravitational forces on each other of magnitude

    FM on m=Fm on M=GMmr2

    where the gravitational constant is G=6.67×10−11 N m2/kg2.

    Gravity is an inverse-square force.

  2. Gravitational mass and inertial mass are equivalent.

  3. Newton’s three laws of motion apply to all objects in the universe.

The figure illustrates the characteristics of the force vectors acting on two masses labeled lowercase m and uppercase M.

Important Concepts

Orbital motion of a planet (or satellite) is described by Kepler’s laws:

  1. Orbits are ellipses with the sun (or planet) at one focus.

  2. A line between the sun and the planet sweeps out equal areas during equal intervals of time.

  3. The square of the planet’s period T is proportional to the cube of the orbit’s semimajor axis.

The figure illustrates the elliptical orbit of a planet of mass lowercase m depicted as a small sphere around the Sun of mass uppercase M.

Circular orbits are a special case of an ellipse. For a circular orbit around a mass M,

v=GMr   and  T2=(4π2GM)r3

Conservation of Angular Momentum

The angular momentum L=mrv sin β remains constant throughout the orbit. Kepler’s second law is a consequence of this law.

Orbital Energetics

A satellite’s mechanical energy Emech=K+UG is conserved, where the gravitational potential energy is

UG=−GMmr

For circular orbits, K=−12UG and Emech=12UG. Negative total energy is characteristic of a bound system.

Applications

The figure illustrates the orbital motion of a satellite due to the force of gravity.

To move a satellite to a larger orbit:

Terms and Notation