4A Homework Set 8
Rotation
1.  Find the rotational inertia of a sphere rotating about an axis through its center.
2. Let the rotational inertia of an object that rotates about its center of mass be given with some fractional coefficient represented as K (so for a sphere it's K= 2/5, disk K = 1/2, hollow cyinder K=1 and so forth) multiplied by its mass and radius squared in the usual fashion. Now, for rolling without slipping starting from rest down an inclined plane of a given angle theta, find the time it takes for the object to roll down a distance D along the plane (noting at the end that this is independent of the object's radius or mass!). Then (and here's the real question) find which of these three objects (a sphere, a disk, or a hollow cylinder) will roll down in the least time.

3. An object of mass m is on a frictionless table rotating with a given tangential speed v_o with a radius of string r_i. The string goes down through a hole (no friction) in the table where a given applied force, F_A pulls down on the string a given distance d. Then, the object still keeps rotating but in a smaller and smaller circle. Find the final speed of the object after the rope was pulled down the distance d.

4. A large disk of mass M_d and radius R is spinning in a horizontal plane about a vertical axis through its center with a given angular velocity of omega. A person of mass M_p (initially at the center of the disk and not spinning, let's say) then walks out to the edge of the disk (yes, the disk is that large!). Find the final angular velocity of the disk (with the person standing on its edge).Treat the person as a point mass.
5. A stick of length L and mass M is in free space and not rotating. A point mass m has an initial velocity v heading in a trajectory perpendicular to the stick. The mass collides and adheres to the stick a distance b from the center of the stick. Find the resulting motion of the two together in terms of their center of mass velocity and final angular velocity.

6. A sphere of radius R and mass M is initially spinning about its center with a given angular velocity of omega. The axis of its rotation is horizontal and therefore parallel to the table. It is then placed on the level table with its center of mass velocity zero and there is kinetic friction between the sphere and the table. Find how far the sphere travels before it stops slipping. Find how much time this takes and find the final speed of the sphere's center of mass at that time.

7. A stick of length L and mass M is hanging at rest from its top edge from a ceiling hinged at that point so it is free to rotate about that point. Find the distance vertically down from that point where an applied an impulse Fdeltat would strike the stick such that there was no horiztonal force of the hinge on the stick. This point is called the "center of percussion".

8. A rod of length L and mass M is balanced in a vertical position at rest. The rod tips over and rotates to the ground with its bottom attachment to the ground never slipping. Find the velocity of the center of mass of the stick just before it hits the ground and also find the velocity of the tip just before it hits.

9. A disk of radius R and mass M has an initial angular velocity of omega. There is sliding friction between the disk and the horizontal surface it is on. The center of mass speed of the disk is always zero in this problem. Find the time it takes for the disk to stop spinning.

10. A "real" Atwood's machine involves two hanging masses, m1 and m2, attached with a "massless" string over a pulley but the pulley has a significant mass M3 and a radius R. Consider the pulley a disk. Find the tension force on each side of the pulley.