4A Homework Set 8
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.
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
(a sphere, a disk, or a hollow cylinder) will roll down in the least
3. An object of mass m is on a frictionless table rotating with a given
tangential speed vo with a radius of string ri. The string goes down
through a hole (no friction) in the table where a given applied force, FA
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 Md 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 Mp (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
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.