Problem 1:

Consider a damped harmonic oscillator.  Let us define T1 as the time between adjacent zero crossings, 2T1 as its “period”, and ω1 = 2π/(2T1) 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 - (8π2n2)-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) = U0[1 - cos(x/R0)] for  πR0/2 < x < πR0/2,
where x denotes distance, and U0 and R0 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) = -U0/[1 + (x/R0)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 qi (i = 0 to N - 1) denote the displacement of the ith mass from its equilibrium position.  Assume periodic boundary conditions, qi = qi+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 qj(t)  = |A|exp(iφj) exp(-iωt)  = |A|exp(i(φj - ωt)) exist, where ω is a normal mode frequency.
Assume the phase of the amplitude depends on the position of the mass and write φj = p*ja.
What are the restrictions on p due to the boundary conditions?
(d)  Find the N normal mode frequencies ωn.  Make a sketch of ωn 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 L1 = 0.25 m, and the next resonance is heard when the length of the air column is L2 = 0.8 m.  The speed of sound in air (at 20 oC, 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.