The Pioneer 3 spacecraft was a spin-stabilized spacecraft launched on December 6, 1958, by...

The Pioneer 3 spacecraft was a spin-stabilized spacecraft launched on December 6, 1958, by the U.S. Army Ballistic Missile agency in conjunction with NASA. It was designed with a despin mechanism consisting of two equal masses A and B, each of mass m, that could be spooled out to the end of two wires of variable length ℓ(t) when triggered by a hydraulic timer. As the masses unwound, they would slow the spacecraft’s spin from an initial angular velocity ωs(0) to the final angular velocity (ωs)final, and then the weights and wires would be released. Assume that masses A and B are initially at positions A0 and B0, respectively, before the wire begins to unwind, that the mass moment of inertia of the spacecraft is IO (this does not include the two masses A and B), and that gravity and the mass of each wire are negligible. Hint: Refer to Prob. 6.117 if you need help with the kinematics.

Figure P7.65

Derive the equation(s) of motion of the system in terms of the dependent variables ℓ(t) and ωs(t). After doing so:

(a) Use a computer to solve the equations of motion for 0 ≤ t ≤ 4 s, using R = 12.5 cm, m = 7 g, IO = 0.0277 kg · m2, and the initial conditions ωs(0) = 400 rpm, ℓ(0) = 0.01 m, and .

(b) Determine the time at which the angular velocity of the spacecraft becomes zero (this can be done by plotting the solution for ωs and then estimating the time or by using numerical root finding to determine when ωs = 0).

(c) Determine the length ℓ of each of the wires at the instant that the angular velocity of the spacecraft becomes zero.