- A particle is projected vertically upward from the Earth's surface with a
velocity that would, were gravity uniform, carry it to a height
.
Show that if the variation of gravity with height is allowed for, but the
resistance of air is neglected, then the height reached will be greater by
, where
is the Earth's radius. (From Lamb 1923.)
- A particle is projected vertically upward from the Earth's surface with a velocity
just sufficient for it to reach infinity (neglecting air resistance). Prove that the time needed to
reach a height
is
- Assuming that the Earth is a sphere of radius
, and neglecting air resistance, show that
a particle that starts from rest a distance
from the Earth's surface will reach the surface with
speed
after a time
, where
is the surface gravitational acceleration.
(Modified from Smart 1951.)
- Demonstrate that if a narrow shaft were
drilled though the center of a uniform self-gravitating sphere then a test mass moving in this
shaft executes simple
harmonic motion about the center of the sphere with period
- Consider an isolated system consisting of
point objects interacting via
gravity.
The equation of motion of the
th object is
- Consider a function of many variables
. Such a function that satisfies
*homogeneous function of degree*. Prove the following theorem regarding homogeneous functions: - Consider an isolated system consisting of
point particles interacting via
attractive central forces. Let the mass and position vector of the
th particle be
and
, respectively. Suppose that magnitude of the force exerted on particle
by
particle
is
. Here,
measures
some constant physical
property of the
th particle (e.g., its electric charge).
Show that the total potential energy
of the system is written
*virial theorem*. Demonstrate that when the system possesses no virial equilibria (i.e., states for which does not evolve in time) that are bounded. - Demonstrate that the gravitational potential energy of a spherically symmetric mass
distribution of mass density
that extends out to
can be
written
- A globular star cluster can be approximated as an isolated self-gravitating virial equilibrium consisting of a great
number of equal mass stars. Demonstrate, from the virial theorem, that
- A star can be through of as a spherical system that consists of a very large number of particles, of mass
and
position vector
, interacting
via gravity. Show that, for such a system, the virial theorem implies that