- Joe has a barrel full of sand. The barrel is a cylinder
L = 2 m long, with radius R = 0.5 m.
The sand has density rho = 2000 kg/m^3.
Joe suspends the barrel horizontally high above the ground
so it is free to spin around its central horizontal axis.
Joe wraps a rope several time around the barrel, and attaches the end of the rope to a bucket of mass m = 5 kg. The bucket hangs motionless H = 2.5 m above the ground because Joe is holding onto the barrel.

At time t = 0, Joe releases the barrel. The bucket starts to fall downwards, pulling on the rope and causing the barrel to spin. As the bucket falls, the barrel spins faster and faster.

How long does it take the bucket to reach the ground?

- the answer
- detailed solution: part I part II part III

- A fancy flagpole consists of a uniform thin rod of
length
**L = 5 m**and mass**m = 7 kg**, and a compact gold ball of mass**M = 2 kg**. It is connected to the wall by a hinge at point P. A wire of length**d = 4 m**runs from the wall at an angle**alpha = 70 degrees**to support the pole.- When the flagpole is in equilibrium, what is the tension in the wire?
- When the pole has swung so that there is an
angle
**theta = 20 degrees**remaining between it and the wall, what is the pole's angular acceleration around the hinge? - Is the motion of the pole an example of simple harmonic motion?
- Just before the pole slams into the wall, what is the linear speed of the golden ball?

The wire breaks! The pole starts to swing downwards around the hinge without friction of any kind.

- the answers
- detailed solution: part I part II part III part IV

- Figure skaters Fred (mass mF = 60 kg) and Jane (mJ = 45 kg)
prepare to spin together. They skate towards each
other so that their outstretched arms (lengths AJ = 0.8 m,
AF = 0.6 m)
will just meet at the point marked "X". Their plan is
to join hands so that they revolve around the point "X"
without any translation.
- Jane has initial speed 4 m/s. What speed should Fred have?
- What is their initial angular momentum around point "X"?
- What is their moment of inertia around point "X"?
- What is their angular speed?
- What are their linear speeds?
- What is their rotational KE?
- What is their angular speed now?
- What is their rotational KE now?

It works! They are now holding hands and spinning around point "X", Jane AJ from the center and Fred AF from the center.

To end their program, they pull each other close, until their centers of mass are only bJ = 0.28 meters and bF = 0.21 meters from point "X".

- the answers
- detailed solution: part I part II part III

- Fred makes a big flat disk of radius R = 0.9 m and
mass M = 12 kg.
He pins it to the wall with a pin in its very center, so that the disk
is free to spin around the pin.
Fred then glues to the outer edge of the disk a tiny blob
of lead, with mass m = 0.5 kg.
The lead hangs straight down below the pin.
Fred now turns the disk a small distance

**theta**from its resting position and releases it. The disk starts to oscillate back and forth.- Show that the motion of the disk is SHM.
- What is the period of the motion?

- the answers
- detailed solution: part I part II