Motion in central potentials

Potential energy U = crn

Problem:

A mass m moves in a central force field.  The force is  F = f(r)(r/r), where f(r) = -kr and k > 0.  Assume the mass moves at a constant speed in a circular path of radius R.  Calculate the angular velocity of the mass, and show that its energy is E = kR2.

Solution:

Problem:

A particle of mass m moves in a central force field such that its potential energy is given by V = krn, where r is the distance from the center of force and k and n are constants.
(a)  Write down the Lagrangian for this system and determine the equations of motion in polar coordinates. 
(b)  Show that angular momentum is conserved for the system.
(c)  Find an expression for the total energy of the system that depends only on the radial variable.
(d)  Find the conditions (sign and magnitude of n and k) for a stable circular orbit by investigating the particle at stable equilibrium.

Solution:

Problem:

Two particles with reduced mass μ orbiting each have a potential energy function U = ½kr2, where k > 0 and r is the distance between them.
(a)  Find the equilibrium distance r0 at which the particles can circle each other at a constant distance as a function of the angular momentum M.
(b)  Determine if this is a stable equilibrium distance.
(c)  Assume the particles orbiting at the equilibrium distance r0 are slightly disturbed.  Determine if the disturbed orbits are closed.

Solution:

Problem:

A particle of mass m moves under the action of a central force whose potential energy function is U(r) = kr4,  k > 0. 
(a)  For what energy and angular momentum will the orbit be a circle of radius a about the origin?  What is the period of this circular motion? 
(b)  If the particle is slightly disturbed from this circular motion, what will be the period of small radial oscillations about r = a?

Solution:


Potential energy U proportional to 1/rn

Problem:

A particle of mass m moves under an attractive central force with magnitude A/r3
(a)  Find the condition for which it moves with constant radial speed.
(b)  For this special case, find the orbital equation r(Φ), where r and Φ are polar coordinates.

Solution:

Problem:

Consider the potential energy function U(r) = kr-1exp(-αr), where k < 0 and α > 0. 
(a)  Find the corresponding force F
(b)  Assuming a particle of mass m, subject to this force, moves in a circle of radius a, find its angular momentum M and energy E.  What is the period of the circular motion?

Solution:

Problem:

A particle of mass m moves in a circular orbit of radius r  = a under the influence of the central attractive force
f(r) = -g exp(-br)/r2,
where g and b are positive constants.
(a)  What is the effective potential energy, Ueff(M,r), for radial motion in terms of r and the angular momentum M?  (Your answer may contain an integral.)
(b)  For what values of b will this orbit be stable?
(c)  What is the frequency of small radial oscillation about these stable circular orbits?

Solution:


Motion with a given orbit

Problem:

(a) Find the force law for a central force field which allows a particle to move in a spiral orbit r = kΦ2, where k is a positive constant.
(b) Compute the effective one-dimensional potential energy.
(c) Find the total energy, E, for which this motion is allowed.

Solution:

Problem:

Under the influence of a central force f(r), a particle follows the trajectory described by r = a/(Φ+1)2, where a is a constant.
(a)  Find the force of f(r).
(b)  At Φ = 0, the particle receives an impulse which reduce to zero its radial component of the velocity (vr) and which doubles its transverse component on the velocity (vf).  What is the path of the particle following this impulse?

Solution:

Problem:

A particle of mass m moves with angular momentum M in a central potential V(r).  It moves in a elliptical orbit with semi-major axis a and semi-minor axis b.  The center of the ellipse is at the origin.  Find the potential V(r), the magnitude of the central force f(r) and the energy of the particle

Solution:


Motion in a central potential with other forces present

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: