Central(3)

Motion in a Central Potential

Problem:
A particle of mass m moves in a plane under the influence of a central force of potential V(r) and also of a linear viscous drag -mk(dr/dt). Set up Lagrange's equations of motion in plane polar coordinates and show that the angular momentum decays exponentially.

Solution:

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Problem:

A particle of mass m moves in a circle of radius R under the influence of a central attractive force

(a) Determine the conditions on the constant a, such that the circular motion will be stable.
(b) Compute the frequency of small radial oscillations about this circular motion.

Solution:

Problem:

A particle of mass m is moving in a central potential U(r) = -(a/r), a > 0. The motion is in a plane. Each point of the orbit is described by two coordinates, r and f.
(a) Give an expression for r as a function of f.
(b) Describe the orbits for different values of the total energy E.
(c) Find the period for a circular orbit of radius R. Is the orbit stable?
(d) What is the period of oscillations about this orbit?

Solution:

Problem:

A mass m, which is attached to the end of a string, moves on a frictionless, horizontal table. The string passes through a hole in the table under which it is pulled to make it taut. Initially, the mass moves in a circle of radius R and has kinetic energy E₀. The string is then slowly pulled until the radius of the circle is halved. Calculate the work done and compare it to E₀.

Student solution: