Find the normal, longitudinal modes of vibration for three
masses connected by identical springs of spring constant k. The masses are
collinear. The end masses have mass m, while the inner mass has mass 2m.

(a) Calculate the normal modes of the system.

(b) Describe the
relative motion of the particles for each normal mode.

A triple pendulum consists of masses αm, m and m attached to a single light
string at distances a, 2a, and 3a respectively from its points of suspension.
Consider only motion in a plane.

(a) Determine the value of α such that
one of the normal frequencies of the system will equal the frequency of a simple
pendulum of length a/2 and mass m. You may assume the displacements of the
masses from equilibrium are small.

(b) Find the mode corresponding to
this frequency and sketch it.

Consider an ideal gas of N particles in a cylinder with a piston so that the
volume and pressure may change. Suppose that each particle has f = 5 quadratic degrees
of freedom in its energy. Consider the three paths in a PV diagram shown. Suppose ways 1
and 3 are straight lines on the PV diagram and way 2 is an adiabatic process.

(a) How much work W is done on the gas for ways 1, 2, and 3?

(b) How much heat Q is transferred to the gas for ways 1, 2, and 3?

(c) What is the change in internal energy ΔU for ways 1, 2, 3?

(d) What is the change in entropy ΔS for way 3?

Consider a ring of radius, r, with 4 identical point particles of mass m
interconnected by identical springs with spring constant k to their nearest
neighbors. The particles move without friction. The interconnected springs can
be viewed as causing harmonic oscillations.

(a) Determine the number of normal modes of oscillations, and establish the
Lagrangian.

(b) Find the frequencies for small oscillations and describe the corresponding
eigen-vibrations.

A metal block of mass m and specific heat c with temperature T_{b} is
placed into the ocean which has temperature T. Assume T_{b} > T. What
is the total change in the entropy, ΔS_{total}, for the system? Express
your answer in terms of m, c, T_{b}, and T.