Assignment 8, solutions

Problem 1:

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.

Solution:

Problem 2:

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.

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Solution:

Problem 3:

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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?

Solution:

Problem 4:

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.
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(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.

Solution:

Problem 5:

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

Solution: