Tipler Ch 9: 75, 76, 77, 78, 79, 80, 81, 89, 92, 100, 101, 102, 103
1980M3. A billiard ball has mass M, radius R, and moment of inertia about the center of mass I = 2 c
The ball is struck by a cue stick along a horizontal line through the ball's center of mass so that the ball initially slides with a velocity v as shown above. As the ball moves across the rough billiard table (coefficient of o
sliding friction ;), its motion gradually changes from pure translation through rolling with slipping to krolling without slipping.
a. Develop an expression for the linear velocity v of the center of the ball as a function of time while it is
rolling with slipping.
b. Develop an expression for the angular velocity ？ of the ball as a function of time while it is rolling
c. Determine the time at which the ball begins to roll without slipping.
d. When the ball is struck it acquires an angular momentum about the fixed point P on the surface of the
table. During the subsequent motion the angular momentum about point P remains constant despite the
frictional force. Explain why this is so.
1986M2. An inclined plane makes an angle of ； with the horizontal, as shown above. A solid sphere of
radius R and mass M is initially at rest in the position shown, such that the lowest point of the sphere is a vertical height h above the base of the plane. The sphere is released and rolls down the plane without 2slipping. The moment of inertia of the sphere about an axis through its center is 2MR/5. Express your
answers in terms of M, R. h, g, and B.
a. Determine the following for the sphere when it is at the bottom of the plane:
i. Its translational kinetic energy
ii. Its rotational kinetic energy
b. Determine the following for the sphere when it is on the plane.
i. Its linear acceleration
ii. The magnitude of the frictional force acting on it
The solid sphere is replaced by a hollow sphere of identical radius R and mass M. The hollow sphere, which is released from the same location as the solid sphere, rolls down the incline without slipping. c. What is the total kinetic energy of the hollow sphere at the bottom of the plane?
d. State whether the rotational kinetic energy of the hollow sphere is greater than, less than, or equal to
that of the solid sphere at the bottom of the plane. Justify your answer.
1990M2. A block of mass m slides up the incline shown above with an initial speed v in the position O
a. If the incline is frictionless, determine the maximum height H to which the block will rise, in terms of
the given quantities and appropriate constants.
b. If the incline is rough with coefficient of sliding friction ;, determine the maximum height to which
the block will rise in terms of H and the given quantities.
A thin hoop of mass m and radius R moves up the incline shown above with an initial speed v in the O
c. If the incline is rough and the hoop rolls up the incline without slipping, determine the maximum
height to which the hoop will rise in terms of H and the given quantities.
d. If the incline is frictionless, determine the maximum height to which the hoop will rise in terms of H
and the given quantities.