Problem

Following up on parts (b) and (c) of the Pioneer 3 despin in Prob. 7.65, it turns out that...

Following up on parts (b) and (c) of the Pioneer 3 despin in Prob. 7.65, it turns out that it is possible to analytically determine the length of the unwound wire needed to achieve any value of ωs by making use of conservation of energy and conservation of angular momentum. In doing so, let the masses of A and B each be m, and the mass moment of inertia of the spacecraft body be IO. Let the initial conditions of the system be ωs (0) = ω0, ℓ(0) = 0, and , and neglect gravity and the mass of each wire.

(a) Find the velocity of each of the masses A and B as a function of the wire length ℓ(t) and the angular velocity of the spacecraft body ωs(t) (and the radius of the spacecraft R). Hint: This part of the problem involves just kinematics—refer to Prob. 6.117 if you need help with the kinematics.

(b) Apply the work-energy principle to the spacecraft system between the time just before the masses start to unwind and any arbitrary later time. You should obtain an expression relating ℓ, , ωs, and constants. Hint: No external work is done on the system.

(c) Since no external forces act on the system, its total angular momentum must be conserved about point O. Relate the angular momentum for this system between the time just before the masses start unwinding and any arbitrary later time. As with part (b), you should obtain an expression relating ℓ, , ωs, and constants.

(d) Solve the energy and angular momentum equations obtained in parts (b) and (c), respectively, for and ωs. Now, letting ωs = 0, show that the length of the unwound wire when the angular velocity of the spacecraft body is zero is given by

(e) From your solutions for and ωs in part (d), find the equations for ℓ(t) and ωs(t). These are the general solutions to the nonlinear equations of motion found in Prob. 7.64.