Exercises

- Derive Equations (5.2) and (5.7).
- Prove that in the case of a central force varying inversely as the cube of the
distance,
- The orbit of a particle moving in a central field is a circle that passes
through the origin; that is,
, where . Show that the force law
is inverse fifth power. (Modified from Fowles and Cassiday 2005.)
- The orbit of a particle moving in a central field is the cardoid
, where . Show that the force law
is inverse fourth power.
- A particle moving in a central field describes a spiral orbit
, where , .
Show that the force law is inverse cube, and that varies logarithmically with .
Demonstrate that there are two other possible types of orbit in this force field, and give their
equations. (Modified from Fowles and Cassiday 2005.)
- A particle moves in the spiral orbit
, where . Suppose that increases linearly
with . Is the force acting on the particle central in nature? If not, determine how would have to
vary with in order to make the force central. Assuming that the force is central,
demonstrate that the particle's potential energy per unit mass is
- A particle moves under the influence of a central force per unit mass of the form
- A particle moves in a circular orbit of radius in an attractive
central force field of the form
, where and .
Demonstrate that the orbit is only stable provided that .
- A particle moves in a circular orbit in an attractive
central force field of the form
, where . Show
that the orbit is unstable to small perturbations.
- A particle moves in a nearly circular orbit of radius under the action of the radial
force per unit mass
*apse line*, joining successive apse points, rotates in the same direction as the orbital motion through an angle each revolution. (From Lamb 1923.) - A particle moves in a nearly circular orbit of radius under the action of the central potential
per unit mass
- Suppose that the solar system were embedded in a tenuous uniform dust cloud. Demonstrate that the apsidal
angle of a planet in a nearly circular orbit around the Sun would be
- Consider a satellite orbiting around an idealized planet that takes the form of a uniform spheroidal mass
distribution of mean radius and ellipticity (where
). Suppose that the orbit
is nearly circular, with a major radius , and lies in the equatorial plane of the planet. The potential
energy per unit mass of the satellite is thus (see Chapter 3)