Question

A space station is approximately a ring of radius,R, and mass m, which rotates about its symmetry axis with angular velocity,~ω=ω0ˆe3. A meteor is traveling with momentum,~p, that is parallel to the original ˆe3, and strikes the space station at a point on the rim,transferring the entire momentum to the space station (an inelastic collision where the meteor sticks to the space station). Further, though the meteor has significant momentum,it is of very small mass so that the moment of inertia tensor elements are approximately the same before and after the collision

a) What is the vector angular momentum of the space station with respect to a coordinate system with origin at the center of the ring and one axis along ˆe3 just before the collision?

b) What is the vector angular momentum of the space station in the same coordinate system (and defining the ˆe2 axis as in the direction from the origin to the point of impact on the edge of the ring) just after the collision?

c) After the collision, there are no further torques acting on the space station. Assume that the angular momentum of the space station after the collision differs by only a small (vector) amount from the initial angular momentum. Write down equations of motion that describe how the components of~ωfor the space station evolve with time.

d) Use these equations to describe how the rotational velocity vector of the space station evolves with time. If you predict simple rotation about a new direction, say so and describe the new direction. If you predict precessional motion, say so and predict the precession frequency. If you think something else happens, say so and describe the motion. In all cases, Explain: Back up your prediction with reasoning and (possibly approximate) solutions of the equations from part (c).

Answer 1

(a) angular momentum of space station when axis passes through center (just before collision)

(b) Conservation of angular momentum for space station and meteor system

for the combined system, no external forces act on it. thus

(let dm be mass of meteor which is very small - given assumption that moment of inertial tensor elements remain same after collision, implies we can ignore small mass dm)

(we define axis e1 perpedicular to both e2 and e3, and pR is along

thus equation becomes

(c) we obtained the equation  in section (b)

    Let

thus

  

(d) Thus there is rotation along a new axis e1 with freqeuncy dw. This is a change in the orientation of rotation and thus we can say motion precesses. i.e. the space station will show precession.