Coupled oscillations, pendula

Problem:

A pendulum is composed of two masses, 3m and m, and two strings of equal length L as shown.  

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At t = 0 the system is released from rest with the upper (heavier) mass not displaced from its equilibrium position and the lower mass displaced to the right a distance a.
x2(0) = 0, x1(0) = a << L.
Find an expression for the subsequent motion of the lower mass, x1(t).

Solution

Problem:

A uniform thin rod of length 3l/2 and mass m is supported at one end A by a weightless string of length l under the gravitational force near the earth surface.
(a)  Calculate the normal modes and normal frequencies of small oscillations for motion in a vertical plane.
(b)  Now the end point A is slowly displaces by a small amount δ, (without gaining any kinetic energy.)  The system is then released from rest and allowed to move freely.  What is the subsequent motion?

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Solution: