For several of these problems you'll need to look up some standard moments of inertia.

Calculate the total moment of inertia before and after. Part A relies on conservation of angular momentum. In Part B and C you find you out whether rotational kinetic energy is conserved. If it isn't, where does the work come from?

The model used for this problem is slightly different to the one we used in class, here when the arms are at the side of the person we treat them as vertical rods on the side of the cylinder, in class we just added them in to the mass of the cylinder. Other than that (and the lack of dumbbells) the problem is very similar.

Use the cross-product $\vec{r}\times\vec{F}$ to find the torque.

You can use conservation of angular momentum to solve this problem. **Just** before the collision the angular momentum is all in the bullet, **just** afterward some of it is transferred to the rod. You can treat the bullet as if it were moving in a circle around the axis at the moment of the collision as it is moving tangentially to the direction of rotation. This is only true at this time, at any significant time before and after the collision this would be inaccurate. This is the same idea as we used to justify neglecting the force due to gravity when we look at collisions of freely moving objects.

This problem is essentially similar to that of precession of a gyroscope. Here the force is not gravity but a force given to you in the question. You will also need to calculate the angular momentum of the asteroid by finding first it's moment of inertia and angular velocity. Once you have the velocity of precession you can find the time taken for the rotation axis to move by 15$^{o}$ .

This gyroscope precesses under it's own weight. Find the torque due that weight, and the angular momentum of the gyroscope, then you can find the precession velocity, and from this the period of precession.

This problem is similar to the crane boom example we did in class. Here however we do include the weight of the arm as force acting at it's center of mass.

You need to use geometry to find the force the two wires exert in the horizontal direction. Then use the balance of the torques around the base of the net to find a simple relation between the tension in the net and this force.

Redraw the problem with the weight of the door expressed as a force going through it's center of gravity. A smart choice ox axis to calculate the torques around can make this easier (try to choose an axis where all the points at which forces are exerted lie on a coordinate axis).

In a ladder problem like this one the wall exerts a normal force, which is horizontal, and the weight of the ladder which acts through it's center of mass is vertically down. The force at the base, has a normal component, which balances the weight, and a frictional component which balances the normal force from the wall. As the frictional force cannot exceed $\mu N$ use this as the condition for when the ladder begins to slip.