Question 1. Consider the motion of an axially symmetric body in the absence of torques. Find the analytical solution for the Euler angles as a function of time.

Question 2. Show that the ellipsoid of inertia of a cube of uniform density having an edge of length a, is a sphere for a set of axes whose origin is at the cube's center. What is the magnitude of the radius?

Question 3 A thin uniform circular disk of mass M and radius r, is mounted on a shaft which is pivoted clockwise at 0 about the vertical axis z. The shaft is perpendicular to the face of the disk and it defines the x₂-axis; the mass of the shaft can be ignored. If the disk rolls without slipping and makes one complete revolution in a circle of radius R in a time τ, write the expression for the angular momentum of the disk with respect to the axes x₂ and x₃ through 0.

Question 4 A football is thrown such that on release its axis of symmetry is 10^o above the horizon and its angular velocity is 2.5 rev/sec at an angle of 15° above the axis of symmetry. The moment of inertia of the football about the axis of symmetry is 4/9 times the moment of inertia about a perpendicular axis through its center of mass. Assuming torque free motion determine:

a) The angle between the symmetry axis of the football and the axis of precession.

b) The rate of precession of the football's symmetry axis

c) The rate of the football's spin.