4.17 Consider the cylindrical spacecraft illustrated in Fig. P4.17. The space- craft has an inertia I, about the ez axis. Two masses m, and m2 are attached symmetrically to this craft through two tethers. Initially the body is spinning at a constant rate o(0) = woēz. At time to=0, the bodies are released from the main spacecraft body. The tethers now begin to unspool from the cylindrical spacecraft body. Assume that the cables will remain tangential to the body surface as shown in Fig. P4.17. Use the coordinate frames S:{$p, ŜR, $3}, E: {êı, êz, êz}, and the inertial frame N:{n1, n2, n3}. Assuming m=2mı = 2m2, (a) Express the inertial position and velocity vectors r and r2 of both point masses my and m2. (b) Derive the inertial angular momentum vector H of the entire system. (c) Derive the total kinetic energy expression T for the entire system.
SR e2 Rom R SO ei m2 Fig. P4.17 Illustration of cylindrical spacecraft with two tethered masses. (d) Using conservation laws, find expressions for $(t) and m(t) in terms of the initial conditions. Use the parameter y=1/(mR?) + 1. (e) Assume we would like the craft to be at rest at time t=T with 0(T)=0. How much cable length l=Ro will you need? (f) How long I will it take for the spacecraft hub to be at rest?
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