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Here are some hints, suggestions, and comments on the problem set.

1. (K+K 6.39)
   • First draw a diagram adopting conventions for what variables you need to pin down in the final state.
   • Are any of the usual quantities (linear momentum, angular momentum, kinetic energy) conserved? If so, why? If not, why not?
   • For the case of angular momentum, what origin are you choosing? What would be different if you chose another origin?
   • When you have enough relations for the unknowns, solve!
   • For part (b), first imagine a point on the stick a distance say x from the collision point. Immediately after the collision it is moving to the right with a speed of the form v+ωr where v is the center of mass speed and r is the distance from the center of mass and the point x, i.e. r=R-x. Now we ask if there is any x such that v(x)=0.
2. (K+K 7.7)
   • What forces act on the circular rope?
   • One reasonable origin is the center of the circle traced out by the center of mass. Other choices might be the points of attachment.
   • Whatever the origin, compute the angular momentum including the RxP orbital term and the spin term. For the spin term, resolve the total spin in to two vectors: one perpendicular to the face of the hoop, and one through a diameter. In terms of those vectors, write L. Finally express L back in terms of r-hat and z-hat basis vectors.
   • With L in hand, take d/dt and equate with torque, where as usual we include all the rxF's
3. (K+K 7.8)
   • If the hoop is rolling w/o slipping, what is the spin rate?
   • If we pick an origin, say coincident with the center of the hoop at this moment, what is the initial angular momentum vector?
   • After the blow, what is the new angular momentum vector?
   • The gyroscopic approximation means that the precession rate is small compared to the spin rate. If this holds, then the angular momentum vector and the spin vector pretty much align, and you can find new direction the hoop is rolling.
   • For part (b), think of applying a constant small force F, and find the precession rate. Demand that it be smaller than the spin rate, and you get the condition they ask for.
4. (K+K 7.10) Look at the discussion of example 7.11, where the axle is mounted horizontally. Some questions to ask yourself:
   • About which axes can one apply a torque on the gyro?
   • What does this imply about the component (call it, say, L_axle) of angular momentum along the axle?
   • If the gyro is oriented at the latitude angle lambda, how does the angular momentum vector change as the earth turns?
   • What re the two contributions to L_axle?
   • What does torque=(d/dt)L tell you?
5. (K+K 8.1) a) Look at example 8.1! b) Is it a stable or unstable equilibrium point? More questions:
   • What are the forces on the rod in the frame of the car (including fictious forces)?
   • What are the torques about the pivot points? Where does the fictitious force act?
   • What is the condition for equilibrium?
   • What is the moment of inertia of the rod?
   • What is the effective direction and magnitude of gravity (i.e. real gravity plus fictitious forces)?
   • What does torque=(d/dt)L tell you?