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)

Consider the matrix of the operator **A**,

**A** =

0 | α | 0 | ||

α | 0 | 0 | ||

0 | 0 | β |

.

(a) Find the matrix sin(**A**).

(b) Find the matrix cos(**A** - (π/2)**I**), where **I** is the identity matrix.

(c) Find the matrix exp(**A**^{n}), for arbitrary n = 0, 1, 2, ... . (Hint: even and odd
n are different.)

A system consists of four point masses are located in the xy plane at the
corners of a rectangle as follows:

mass m_{1} = m at (0,0); mass m_{2} = 2m at (a,0); mass m_{3}
= 3m at (a,b); and mass m_{4} = 4m at (0,b).

(a) Find the center of mass of the system.

(b) Find the moments I_{ii} and products I_{ij} (i ≠ j) of
inertia about the origin for the given choice of coordinate axes.

Consider a 2 by 2 matrix M with matrix elements m_{11},
m_{12}, m_{21}, and m_{22}.

(a) What relationships between these matrix elements must exist for the matrix
M to be a Hermitian matrix, M = H?

(b) Find the eigenvalues of M = H and show that they are real.

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?