Problem

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 neglect gravity and the mass of each wire. With this in mind, you are given the task of despinning the spacecraft body from an angular velocity of 400 rpm to any angular velocity in the range −400 rpm < ωs < 400 rpm. To do this, a transducer is used that senses the tension in one of the wires at every instant. Design the total length of the wires to achieve the desired range of angular velocities, and determine the tension in the wires as a function of the angular velocity of the spacecraft so that the sensor can know when the spacecraft has achieved the desired velocity and release the wires and masses.

Use R = 12.5 cm, m = 7 g, IO = 0.0277 kg·m2, and the initial conditions ℓ(0) = 0.01 m and , with the despin mechanism shown. Hint: Refer to Prob. 6.117 on p. 532 if you need help with the kinematics.