Motion in a potential

Although you first learn about
Newton's second law
and the
dynamics that results from it, much of the discussion in
the more advanced physics texts is in terms of ``potentials''
.
A particle undergoes motion ``in a potential''.
Note that *V* is a *scalar*, while is a vector.
It is often easier to work with the potential unless you
are forced to work with the force. Actually even motion
in a potential is carried out using Newton's second law.
However visualizing the potential is very
helpful in developing physical insight into the trajectories.
It is also useful in
understanding thermodynamic processes, which are statistical in
nature. Anyway for our purposes, we just need to know how to
relate the the force to the potential, and that is via the equation:

(1) |

Often it is easier to work in polar co-ordinates . If we work with

(2) |

Almost all that you do in undergrad. physics (and most of postgrad. physics courses) is with central potentials.

This week we study motion in two different central
potentials: The gravitational potential near of a mass *m*, near a mass *M*:

(3) |

The ``Lennard-Jones'' potential between two inert gas atoms:

(4) |

The constants

**Assignment 8. - Hand in by Thursday Mar. 24**

**Problem 1**.

(i) Make a plot of the
Lennard-Jones potential.

(ii) Find the value, *r*_{0}, at which
the Lennard-Jones Potential is a minimum. Evaluate
*V*_{LJ}(*r*_{0}).
What is the physical meaning of
*V*_{LJ}(*r*_{0}).

(iii) By expanding around the minimum of the Lennard-Jones
potential(use the ``Series'' function),
show that, at low kinetic energies, two inert
gas atoms undergo simple harmonic motion with respect to
each other. For what kinetic energies would you
expect this to be true (compare the kinetic energy
with the ``depth'' of the potential well).

**Problem 2**.

(i) Make a plot of the gravitational potential energy.

(ii) Write a piece of Mathematica code to study the motion of
a comet as it approaches the sun(ignore the planets in this
calculation). Sun mass= 1.991x 10^{30} kg, Sun radius = 6.96x 10^{8} m,
Assume that the ratio
(mass of comet/mass of sun)
zero. For a few
initial conditions, plot out the trajectory
of the comet as it passes by the sun (e.g. try to find
initial conditions that lead to trapping of the comet).
Find a set of initial
conditions which makes the comet's orbit a circle.