Assignment 3, solutions

Problem 1:

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 10o?
(b)  What energy (in Joules) is needed for the whole operation?

(Moment of Inertia of hollow spherical shell: [2M(Ro5 - Ri5 )]/[5(Ro3 - Ri3)] where Ro is outer radius and Ri is inner radius)


Problem 2:

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(An), for arbitrary n = 0, 1, 2, ... .  (Hint: even and odd n are different.)


Problem 3:

A system consists of four point masses are located in the xy plane at the corners of a rectangle as follows:
mass m1 = m at (0,0); mass m2 = 2m at (a,0); mass m3 = 3m at (a,b); and mass m4 = 4m at (0,b).
(a)  Find the center of mass of the system.
(b)  Find the moments Iii and products Iij (i ≠ j) of inertia about the origin for the given choice of coordinate axes.


Problem 4:

Consider a 2 by 2 matrix M with matrix elements m11, m12, m21, and m22.
(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.


Problem 5:

Two identical billiard balls of radius R and mass M, rolling with CM velocities ±vi, 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?