Relate the rotational inertia, the angular and linear velocities, and the rotational kinetic energy for a variety of rotational systems.
At right is a solid disk with a mass of 2.00 kg, and radius of 10.5 cm. A force of 5.75 Newtons is applied for 18 revolutions of the disk about the axis through Point O. Assume the bearing in the rotating disk is frictionless.
How much work is done on the disk by the torque produced by the tension force?
Work
Joules
What is the angular velocity of the disk after 18 revolutions if it starts from rest?
Angular Velocity
rad/sec
Below right is a solid disk of mass 1.16 kg, and radius of 15.50 cm. The hanging mass of 300 grams is released from rest. Assume the bearing in the rotating disk is frictionless.
What is the velocity of the hanging mass after it has fallen 75 cm.
Velocity
m/s
What if 10% of its initial energy is lost to friction in the disk's bearing as it falls. What would be its velocity after 75 cm then?
Two masses hang over a solid disk pulley with a frictionless bearing as shown at right. Mass 1 is 9.0 kg, mass 2 is 4.0 kg, pulley mass is 10.2, and the pulley radius is 12.6 cm.
If the masses are released from rest, how fast will the hanging masses be moving when they have switched places vertically? The initial separation h is 42.0 cm.
Speed
Compare wheels of different shapes (ring, solid disk, solid sphere, thin spherical shell) rotating around an axis through their center of mass as shown at right being brought to a stop by applying a constant braking force of 48 Newtons at the outside edge. Consider all of them to have the same mass of 4.30 kg, radius of 14.0 cm, and initial angular speed of 72.0 rad/s.
What is the magnitude of the work done by the torque to bring the ring to rest?
What is the angular distance covered while stopping for the ring?
Angular distance
Revolutions
What is the magnitude of the work done by the torque to bring the solid disk to rest?
What is the angular distance covered while stopping for the solid disk?
Angular Distance
What is the magnitude of the work done by the torque to bring the solid sphere to rest?
What is the angular distance covered while stopping for the solid sphere?
What is the magnitude of the work done by the torque to bring the thin, spherical shell to rest?
What is the angular distance covered while stopping for the thin, spherical shell?
A merry-go-round is being ridden by 2 children; one is 22.0 kg, and the other one is 24.5 kg as shown below. Both are sitting at the outside edge of a ring that has a mass of 54.5 kg and radius of 1.3 meters. The father gets them started by pushing at the outside edge with a tangential force of 258 Newtons through an angular distance of 96˚.
In an ideal situation without friction from the bearing in the middle, what would be the angular speed of the ride after the force is removed?
Angular Speed
If the final angular speed actually reached is 2.54 rad/sec, what is the frictional torque that exists in the ride?
Torque from Friction
N•m
Given this frictional torque, how many revolutions will the ride make before it comes to a stop after the father's force is removed?
A solid sphere, hollow sphere, ring, and solid disk (left to right in the image at right) are all released from rest at the same height, 87.5 cm from the floor. All roll without slipping down a ramp that is 1.80 meters long. All have the same mass and radius. What linear speed does each object's center of mass have as it reaches the bottom?
Solid Disk?
Center of Mass Speed
Ring?
Hollow Sphere?
Solid Sphere?
Put the objects in order of reaching the bottom using the associated values below:
Solid Disk - 1
Ring - 2
Hollow Sphere - 3
Solid Sphere - 4
A possible answer should have the digits 1 through 4 in the proper order without spaces. If all reach at the same time put the value "5" in the answer field.
Order of finish
A 1.85-kg pole 2.5 meters long is initially situated in a straight up position on end connected by a fricitonless pin joint as shown at right. It begins to rotate from rest and falls through the final postion where it is perfectly horizontal. Consider the pole to be a long, thin rod.
What is the angular speed of the pole as it passes through the final position?
What is the tangential speed of Point A on the pole at the final position?
Tangential Speed
What is the magnitude of the torque on the pole about Point O at the final poisition?
Torque
What is the magnitude of the angular acceleration of the pole at the final position?
Angular Acceleration
rad/s/s
What is the magnitude of the tangential acceleration of Point A on the pole at the final position?
Tangential Acceleration
m/s/s
A 1.00-kg pole 3.0 meters long is initially situated in a perfectly horizontal position connected to a frictionless pin joint as shown at right. It is released from rest and falls through the final postion where it is perfectly vertical. Consider the pole to be a long, thin rod.
What is the magnitude of the torque on the pole about Point O at the initial position just after being released?
What is the magnitude of the angular acceleration of the pole at the initial position just after being released?
What is the magnitude of the tangential acceleration of Point A on the pole at the initial position just after being released?
What is the tangential speed of Point A on the pole it passes through the final position?
What is the magnitude of the tangential acceleration of Point A at the final position?
What is the magnitude of the centripetal acceleration of Point A at the final position?
Centripetal Acceleration
A yoyo is released from rest and allowed to fall from a person's finger as shown at right. Consider the yoyo as a uniform solid disk of mass 72 grams and radius of 8.0 cm and that the string wraps around at the outer radius.
What is the speed of the yoyo's center of mass after it has fallen 30.5 cm.