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 represent by C (for a sphere, C = 2/5, a disk would have C = 0.5 a hollow cylinder would have C = 1, and so forth) multiplied by its mass and radius squared as usual. Set an object rolling without slipping starting from rest down an inclined plane of angle θ. Find the time for this object to roll distance D along the plane. Which object rolls down in the least time?

 

3. An object of mass m is on a frictionless table rotating with a given tangential speed Vo with a radius of a string ri. The string does down through a hole (no friction) in the table, an applied force, Fa pulls the string down distance d. As the constant force Fa pulls, the mass keeps rotating, but in a smaller and smaller circle. Find the speed of the mass after the string has been pulled by distance d.

 

4. A large disk of mass Md and radius R is spinning in a horizontal plane about a vertical axis through its center with angular velocity ωi . A person of mass Mp (initially at the center of the disk and for simplicity sake imagine the person is not spinning) walks from the center of the disk to the edge of the disk. Treat the person as a point mass. Find the final angular velocity of the disk.

 

5. A stick of length L and mass M is in free space (no gravity) and not rotating. A point mass m has initial velocity v heading in a trajectory perpendicular to the stick. The mass as a perfectly inelastically collision a distance b from the center of the stick. Find the velocity of the center of mass and the final angular velocity.

 

6. A sphere of radius R and mass M is spinning about its center with an initial angular velocity ωi. The axis of rotation is horizontal, parallel to the table. It is then placed on the level table with its center of mass velocity zero. Kinetic friction is present between the sphere and the table as the sphere is slipping on the table surface. How far does the sphere travel before it stops slipping? How long does that take? What is the speed of the 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 that it is free to rotate. Find the distance from the top of the stick where an impulse, FΔt, is applied such that there is no horizontal component to the force of the hinge on the stick. This point is called the center of percussion.

 

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

 

9. A disk of radius R and mass M has initial angular velocity ωi. It is flat on a horizontal surface and friction is present. The center of mass speed is zero (always) in this problem. Find the time it takes for the disk to stop spinning.

 

10. An Atwood machine has two hanging masses, m1 and m2, attached with a massless string over a pulley. If the pulley spins, rather than allowing the string to change direction without spinning, has mass M3, radius R, and moment of inertia equal to that of a disk, what is the tension force down on each side of the pulley?