Problem 1:

A mass m is connected by a string of negligible mass to a pulley of mass M = 6m and radius R as shown.  The pulley has uniform mass density.  Assume that at t = 0 the mass is at z = 0 and θ = 0.  For the time the mass is falling, find the equations of motion for z and θ and the forces of constraint using the method of Lagrange multipliers.

• he torque acting on the disk.  It is the generalized force of constraint associated with the coordinate θ.  The torque is pointing into the page.

Problem 2:

A uniform hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R as shown in the figure.  The only external force is that of gravity.  If the hoop starts rolling from rest on top of the big cylinder, find, by the method of Lagrange multipliers, the point at which the hoop falls off the cylinder.

Problem 3:

A particle of mass m is acted on by an attractive force whose potential is given by U(r) = -Ar^-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.

Problem 4:

A beam of particles of mass m and non-relativistic kinetic energy E is scattered from a central potential.  The potential energy of a particle is U = -U₀,  for r < a, U = 0 for r > a.
(a)  Show that the scattering produced by such a potential is identical to the refraction of light rays by a sphere of radius a and relative index of refraction n = ((E + U₀)/E)^½, i.e. show that sinα/sinβ = n, where α and β are defined in the diagram.

(b)  Show that the classical differential scattering cross section for this potential is
σ(θ) = dσ/dΩ = [n²a²/(4 cos(θ/2))](n cos(θ/2) - 1)(n  - cos(θ/2))/(1 + n² - 2n cos(θ/2))².

Problem 5:

In a scattering experiment, a beam of particles of mass m and energy E is sent towards a target.  The number of particles scattered per unit area per unit time into a specific direction is measured.  The potential energy function U(r) is that of a central attractive potential with a repulsive core.

(a)  Give the definition of the differential scattering cross section dσ/dΩ in terms of the impact parameter b and the scattering angle θ.
(b)  Relate the impact parameter b to the angular momentum, and thus find the dependence of θ(b) on U(r) and b in the form of an integral.
(c)  What is the value of the scattering angle θ for the special cases b = 0 and b = ∞?
(d)  Show that θ should become negative for suitable E and b.
(e)  Given the result of (d)  does the differential cross section ever diverge for any value of b?  Explain!

This problem just asks for definitions and reasoning, not for calculations.