Question 1:  (Momentum & Energy)

A homogeneous disk of mass m and radius R spins at a rate col about its skew axis, which rotates about the horizontal at rate 0)2. Determine

(a) the angular momentum of the disk relative to the center of mass C.

(b) the kinetic energy of the disk.

Question 2:  (Gyroscopic Effect)

An airplane has just cleared the runway with a takeoff speed v. Each of its freely spinning wheels has a mass m, with a radius of gyration k about its axle. As seen from the front of the airplane, the wheel processes at the angular rate S2 as the landing strut is folded into the wing about the pivot 0. As a result of the gyroscopic action, the supporting member A exerts a torsional moment M on B to prevent the tubular member from rotating in the sleeve at B. Determine M and identify whether it is in the sense of M1 or M2.

Question 3: (Football)

The figure shows a football in three common in-flight configurations.  Case (a) is a perfectly thrown spiral pass with a spin rate of 120 rev/min.  Case (b) is a wobbly spiral pass again with a spin rate of 120 ev/min about its own axis, but with the axis wobbling through a total angle of 20°. Case (c) is an end-over-end place kick with a rotational rate of 120 rev/min. For each case, specify the values of s, θ, ß, and Ψ . The moment of inertia about the long axis of the ball is 0.3 of that about the transverse axis of symmetry.

Question 4: The turbine rotor in a ship's power plant has a mass of 1000 kg, with center of mass at G and a radius of gyration of 200 mm. The rotor shaft is mounted in bearings A and B with its axis in the horizontal fore-and-aft direction and turns counterclockwise at a speed of 5000 rpm when viewed from the stern. Determine the vertical components of the bearing reactions at A and B if the ship is making a turn to port (left) of 400-m radius at a speed of 25 knots (1 knot = 0.514 m/s). Does the bow of the ship tend to rise of fall because of the gyroscopic action?