A mass m moves in a central force field. The force is  , where f(r) = -kr and k > 0.
For part (a) assume the mass moves at a constant speed in a circular path of radius R.

    (a) Calculate the angular velocity of the mass and show that its energy E is E = kR².

For parts (b) and (c) consider the more general case in which the motion is not circular.

    (b) Write an expression for the energy of the mass.  Solve the resulting differential equation for r(t).

    (c) Calculate the maximum and minimum values of r(t) and show that E  =(1/2)k(r²_max + r²_min).

A particle moves in two dimensions under the influence of a central force determined by the potential V(r) = ar^p + br^q. Find the powers p and q which make it possible to achieve a spiral orbit r = cq², with c a constant.

A particle of mass m is acted on by an attractive force whose potential is given by V µ r^-4.  It is incident from infinity with an initial velocity v_¥.  Sketch the effective potential of the particle U_eff(r).  Find the total cross section for capture of the particle.