1. A mechanical system with one degree of freedom oscillatesabout a stable equilibrium state. Its displacement from theequilibrium, x(t), satisfies the simple harmonicsoscillator equation:

d2x + ω2 x = 0.dt2

(a) What is the characteristic period of the oscillation?
(b) Write a solution to the above equation for which x(0)= x0 and ẋ(0) = v0. (c) Demonstrate that E =ẋ 2 + ω 2 x 2 does not vary in time. What is thephysical significance of E?

2. An damped mechanical system with one degree of freedomoscillates about a stable equilibrium state. Its displacement fromthe equilibrium, x(t), satisfies the dampedharmonic oscillator equation:

ẍ+νẋ+ω2 x=0, where ν > 0.

(a) Demonstrate that

x = A sin(ω1 t)eγt

is a particular solution of the damped harmonic oscillatorequation, and determine the values of ω1 and γ (assuming that ν< 2 ω).

(b) Demonstrate that

dE ≤ 0,dt

where E is defined in Q1(c). What is the physical significanceof this equation?

3. Consider a mass-spring system consisting of two identicalmasses, of mass m, which slid over a frictionlesshorizontal surface. In order, from the left to the right, thesystem consists of a spring of spring constant k′ whoseleft end is attached to an immovable wall and whose right end isattached to the first mass, a spring of spring constant kwhose left end is attached to the first mass and whose right end isattached to the second mass, and a spring of spring constantk′ whose left end is attached to the second mass and whoseright end is attached to an immovable wall. Let ω0 =√k/m and α = k′/k.

(a) Demonstrate that the equations of motion of the system canbe written:

ẍ1 = −(1+α)ω02 x1 +ω02 x2,

ẍ2 = ω02 x1 −(1+α)ω02 x2,

(b) Demonstrate that the normal frequencies of the system are ω= α1/2 ω0 and ω = (2 +α)1/2 ω0.

(c) Demonstrate that the low frequency normal mode is such thatthe two masses oscillate in phase with the same amplitude, and thatthe high frequency mode is such that the two masses oscillate inanti-phase with the same amplitude.