#### Problem 1:

A train has a whistle, which emits a 400 Hz sound. You are stationary
and you hear the whistle, but the pitch is 440 Hz. How fast is train
moving towards or away from you?

#### Problem 2:

Two masses, 1 kg and 2 kg, are fixed horizontally to fixed side supports
with springs as shown below. The masses are constrained to move along the horizontal line. From their equilibrium position m_{1} is given a displacement L to
the right, while m_{2} is held fixed. At t = 0 they are
released from rest. Give the equation for the positions of m_{1} and m_{2}
as a function of time.

#### Problem 3:

Consider a "molecule" made up of three equal masses
m connected by three
equal springs with spring constant k. The equilibrium position is an equilateral
triangle. Consider only motion in the plane of this triangle. Find all the normal modes
for motion in this plane.

#### Problem 4:

A space scientist proposes to measure the Gravitational constant G by
locating a solid gold sphere of mass 8 x 10^{4} kg and radius 1 m in a
spaceship. A hole is to be drilled through the diameter of the sphere and
a small gold ball of mass 80 gram is released from rest at its surface so that it
oscillates back and forth within the tube passing through the large sphere.

(a) Calculate the period of the oscillatory motion of the particle,
assuming that there is no friction present.

(b) If air friction is
present so that the particle returns to a radial distance of 99 cm rather than
to 100 cm at the end of one full oscillation, obtain an estimate of the
frictional force (assumed to be of form **F**_{f} = -b**v**), and
the change in period caused by the frictional force.

#### Problem 5:

A set of coupled masses is constrained to move on a circular path of fixed
radius. The chain consists of four light masses m alternating with four
heavy masses M, joined by identical springs with force constant k.

(a) Find the (coupled) equations of motion for the ith light mass m and the ith
heavy mass M.

(b) Solve these equations of motion for the normal modes of the system.

(c) Sketch the motion associated with each of the allowed frequencies