**UIUC
Physics 211 James Scholars Problem Set 5**

**Consider a hollow tube of mass M = 1.2 kg and length
L = 1.6 m that
rotates about an axle through its center and perpendicular to
its length.
Inside the tube are two masses, m_1 = 0.4 kg each. These
masses are initially
held a distance d = 0.8 m apart by a string and centered in
tube. The maximum
tension the string can sustain is 100 N. You may consider that
the radius of the
tube is negligible (i.e. its moment of inertia is that of a 'stick')
and that
the masses held by the string are point-like.**

1.1 Starting from rest, the cylinder starts to rotate as a
result of a constant driving torque applied to it. What is the work
done by this
torque up to the point at which the string breaks?

1.2 Immediately after the string breaks the driving torque
is
removed. For the period after the string breaks and the masses are
still inside
the tube, find an expression for omega(x), where x is the distance of
each of
the masses * m_1* from the center (axle) of the tube. Your
solution for omega(x)
should apply to the region * 0.4 m < x < 0.8 m*.

1.3 Now we want to think about the total angular momentum of
the
system as the masses * m_1* fly out of the tube. What is the
total angular momentum
of the cylinder and masses just before the masses exit? What is the
angular
momentum of each mass just after it has left the tube? What is the
angular
momentum of the tube just after the masses have exited? What do you
conclude
about the external torque acting on the system in this period?

**In this problem we will investigate a particular example of
damped
harmonic motion. A block of mass m rests on a horizontal table and is
attached
to a spring of force constant k. The coefficient of friction
between the block
and the table is mu. For this problem we will assume that the
coefficients of
kinetic and static friction are equal. Let the equilibrium position of
the mass
be x = 0. The mass is moved to the position x = +A,
stretching the spring, and
then released. **

2.1 Apply Newton's 2nd law to the block to obtain an
equation
for its acceleration for the first *half *cycle of its motion,
i.e. the
part of its motion where it moves from * x = +A to x < 0* and
(momentarily)
stops. Show that the resulting equation can be written *d ^{2}x'/dt^{2}
=
-omega^{2} * x'*, where

2.2 Repeat the above for the second half cycle, i.e. wherein
the
block moves from its maximum negative position to its (new) maximum
positive
position. First show that the differential equation for the block's
acceleration
can be written *d ^{2}x''/dt^{2} = -omega^{2}*x''*
where this time

2.3 Make a graph of the motion of the block for the first 5
half
cycles of the motion in the case where * A = 10.5*x0*. Plot the
position of the
block normalized to * x0* as a function of the fractional period,
* T = 2*Pi/omega*
(i.e. plot * x(t)/x0* vs *t/T*). Attach the graph to
your email.

2.4 Something interesting happens at the end of the 5th half
cycle - what changes in the physical situation and what is the motion
after this
half cycle?

**A heavy rope 3 m long is attached to the
ceiling and allowed to hang
freely. Let y = 0 denote the bottom end of the rope. To get
started on this
problem, imagine cutting the rope at an arbitrary value of y.
Draw a free body
diagram of each of the two pieces of rope to determine the tension at
the point
where the rope was cut. **

3.1 Determine the propagation speed of transverse waves on
the
rope and show that this speed is independent of the rope's mass and
overall
length.

3.2 How long would the rope have to be in order for the
maximum
propagation speed to be equal to the speed of sound in air (which we
will take
to be * 330 m/s*)?

3.3 Calculate the time it takes for a transverse wave to
travel
from the bottom of the * 3 m* long rope to the top and then back
to the bottom.

3.4 Compare this round-trip time to that for a horizontal
rope
with the same tension as the average tension of the vertical rope.