Although you first learn about
Newton's second law
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:
This week we study motion in two different central
potentials: The gravitational potential near of a mass m, near a mass M:
Assignment 8. - Hand in by Thursday Mar. 24
(i) Make a plot of the Lennard-Jones potential.
(ii) Find the value, r0, at which the Lennard-Jones Potential is a minimum. Evaluate VLJ(r0). What is the physical meaning of VLJ(r0).
(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).
(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 1030 kg, Sun radius = 6.96x 108 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.