Consider a right triangular lamina of areal density ρ, with one edge of length a along the x-axis and another edge of length b along the y-axis, as shown in the diagram.

(a) Find the center of mass (X,Y,Z) in this coordinate system.

(b) Find the components of the inertia tensor in this coordinate system.

(c)
For a = b, transform the inertia tensor to principal axes, giving the angle between the
principal axes and those shown in the diagram, and find the three moments of inertia in
the principal axes system.

(d) Use the parallel axis theorem to find the
principal moments of inertia about the center of mass.

Assume a perfectly spherical Earth of radius R with a
frictionless surface. On the surface of this Earth an object with mass m is
moving with constant speed v towards the north pole. When the object is at
latitude λ, find the external force required to keep it moving on that
trajectory.

For m = 1 kg, v = 500 m/s, λ = 45^{o} give a numerical answer.

M_{Earth} = 5.97*10^{24} kg, R_{Earth} = 6378 km.

A rigid, symmetrical spaceship is shaped in the form of a
cone with a uniform density. The height of the cone is h, the radius of the base is
r, and the total mass is m. Being suspended in outer space
without any external forces acting on it, the space ship has a center of mass velocity
**v**
and angular momentum **L** not quite parallel to the symmetry axis. Thus it
experiences precession.

(a) Calculate the principal moments of inertia of the spaceship about its
CM in terms of h,
r, and m.

(b) Show that the symmetry axis rotates in space about the fixed direction of the
angular momentum **L**.

Two identical billiard balls of radius R and mass M, rolling with CM velocities
±v**i**, collide elastically, head-on.
Assume that after the collision they have both reversed motion and are
still rolling.

(a) Find the
impulse which the surface of the table must exert on each ball during its
reversal of motion.

(b) What
impulse is exerted by one ball on the other?

A stationary space station can be approximated as a hollow spherical shell of
mass 6 tons (6000 kg) and inner and outer radii of 5 m and 6 m. To change its
orientation, a uniform fly wheel of radius 10 cm and mass 10 kg located at the
center of the station is spun quickly from rest to 1000 rpm.

(a) How long (in minutes) will it take the station to rotate by 10^{o}?

(b) What energy (in Joules) is needed for the whole operation?

(Moment of Inertia of hollow spherical shell: [2M(R_{o}^{5 }-
R_{i}^{5 })]/[5(R_{o}^{3 }- R_{i}^{3})]
where R_{o} is outer radius and R_{i} is inner radius)