**Problem 1:**

Consider a damped harmonic oscillator. Let us define T_{1} as
the time between adjacent zero crossings, 2T_{1} as its “period”, and ω_{1}
= 2π/(2T_{1}) as its “angular frequency”. If the amplitude of the
damped oscillator decreases to 1/e of its initial value after n periods, show
that the frequency of the oscillator must be approximately [1 - (8π^{2}n^{2})^{-1}]
times the frequency of the corresponding undamped oscillator.

**Problem 2:**

Consider the motion of a point of mass m subjected to a
potential energy function of the form

U(x) = U_{0}[1 - cos(x/R_{0})]
for πR_{0}/2 < x < πR_{0}/2,

where x denotes distance, and U_{0} and R_{0} are positive
constants with dimensions of energy and length, respectively.

(a) ** **Find the position of stable equilibrium for the mass.

(b) **
**Show that the motion of the mass in proximity of the stable equilibrium
position is SHM.

(c) ** **Find the period of the small oscillations.

(d) ** **Find the period of the small oscillations for the same mass in
the potential

U(x) = -U_{0}/[1 + (x/R_{0})^{2}].

**Problem 3:**

A pendulum consisting of a mass m and a weightless
string of length l is mounted on a mass M, which in turn slides on
a support without friction and is attached to a horizontal spring with force
constant k, as seen in the diagram. There is a slot in the support in
order that the pendulum may swing freely.

(a) Set up Lagrange's equations.

(b) Find the normal mode frequencies for small oscillations.
What are those frequencies to
zeroth order in m/M, when m << M?

**Problem 4:**

A large number N (N = even) of point masses m are connected
by identical springs of equilibrium length a and spring constant k. Let q_{i}
(i = 0 to N - 1) denote the displacement of the ith mass from its equilibrium
position. Assume periodic boundary conditions, q_{i} = q_{i+N}.
(You can, for example imagine the masses arranged on a large circle of
circumference Na.)

(a) Write down the Lagrangian for the system of N point masses.

(b) Find the equation of motion for the jth point mass.

(c) Assume solutions of the form q_{j}(t) = |A|exp(iφ_{j})
exp(-iωt) = |A|exp(i(φ_{j} - ωt)) exist, where ω is a normal mode
frequency.

Assume the phase of the amplitude depends on the position of the mass and write
φ_{j }= p*ja.

What are the restrictions on p due to the boundary conditions?

(d) Find the N normal mode frequencies ω_{n}. Make a sketch
of ω_{n} as a function of mode number n.

**Problem 5:**

A vibrating tuning fork is held above a column of air as shown in the
diagram. The reservoir is raised and lowered to change the water level,
and therefore the length of the column of air. The shortest length of air
column that produces a resonance is L_{1} = 0.25 m, and the next
resonance is heard when the length of the air column is L_{2} = 0.8 m.
The speed of sound in air (at 20 ^{o}C, the temperature during the
experiment) is 343 m/s and the speed of sound in water is 1490 m/s.

(a) Calculate the wavelength of the standing sound wave produced by the
tuning fork.

(b) Calculate the frequency of the tuning fork that produces the standing
wave.

(c) Calculate the wavelength of the sound wave produced by this tuning
fork in water.