A satellite in geosynchronous orbit has cylindrical symmetry with principal mome

Question

A satellite in geosynchronous orbit has cylindrical symmetry with principal moments of inertia I1 = I2 = I3. A camera pointing along its symmetry axis is trained on one spot of the earth. Due to a computer malfunction, the positioning thrusters all fire for an instant of time, applying impulsive torques to the principal axes of the spacecraft. Like an impulsive force, which instantaneously changes the momentum by p, the net impulsive torque instantaneously changes the angular momentum by L = (L1, L2, L3) along the spacecraft’s principal axes. If the spacecraft has L=0 initially, what is the angle between the angular momentum vector immediately after the thrusters have fired, and the symmetry axis of the spacecraft? How long does it take before the spacecraft has tumbled back into its original position immediately after the thrusters fire? Will the camera point at the same location on which it was originally trained?

Explanation / Answer

Solution :-

Given

A satellite in geosynchronous orbit has cylindrical symmetry with principal moments of inertia I1 =I2 = I3. A camera pointing along its symmetry axis is trained on one spot of the earth.

Due to a computer malfunction, the positioning thrusters all fire for an instant of time, applying impulsive torques to the principal axes of the spacecraft.

Like an impulsive force, which instantaneously changes the momentum by p, the net impulsive torque instantaneously changes the angular momentum by L = (L1, L2, L3) along the spacecraft’s principal axes.

a)  If the spacecraft has L=0 initially, then the angle between the angular momentum vector immediately after the thrusters have fired, and the symmetry axis of the spacecraft= tan = L

Hence

tan = L = 0

Therefore the angle between the angular momentum vector immediately after the thrusters have fired, and the symmetry axis of the spacecraft= 0 degrees. (because tan 0 = 0)

b) It takes 2t units of time before the spacecraft has tumbled back into its original position immediately after the thrusters fire

We know t = 2*pi()/L

Therefore 2t = 4* pi()/L

Hence 4* pi()/L units of time it takes before the spacecraft has tumbled back into its original position immediately after the thrusters fire

c) Yes ofcourse the camera point at the same location on which it was originally trained because after 2t units it will come back to its original position.

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