Concept #1: Equations of Rotational Motion

Example #1: Rotational velocity of disc

Practice: A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s^{2} . Calculate its angular speed 8 s after starting to accelerate.

BONUS: Calculate its linear displacement in the 8 s.

Practice: The turntable of a DJ set is spinning at a constant rate just before it is turned off. If the turntable decelerates at 3 rad/s^{2} and goes through an additional 30 rotations before stopping, how fast (in RPM) was the turntable initially spinning?

BONUS: How long (in seconds) does the turntable take to stop?

A solid sphere rolls along a horizontal, smooth surface at a constant linear speed without slipping.
What is the ratio between the rotational kinetic energy about the center of the sphere and the sphere’s total kinetic energy?
A. 3/7
B. None of these
C. 7/2
D. 2/5
E. 2/7
F. 3/5
G. 5/3

A wheel with radius 0.20 m starts from rest and turns through 8.0 revolutions in 5.0 s. At t = 5.0 s, what is the radial acceleration of a point on the rim of the wheel?

A disk is rotating at 150 rpm. Answer the following questions:
(a) What angle, in radians, does the disk rotate through in 25 s if the angular speed is constant?
(b) If a brake is applied to the disk, and it takes 5 revolutions to stop, what was the angular acceleration applied by the break, in rad/s2?

A car’s wheel rotates along the floor without slipping. If the car is moving at 30 m/s, and the wheels are 45 cm in diameter, what is the rotational speed of the wheels?

A car is initially driving at 20 m/s when the driver sees a dog in the road 100 m in front of him. In order to break in time to not hit the dog, what angular acceleration do the breaks have to apply on the wheels? Assume the wheels have a diameter of 45 cm.

An ant is walking on a disk of radius 15 cm rotating at an angular speed of 150 rad/s. At the center of the disk, what is the ant’s linear speed? What about at the rim of the disk?

A car moves forward at a speed of 15 m/s. In order for the car to maintain traction, the wheels have to rotate without slipping. What, then, is the linear speed at the bottom of the wheel, where it meets the ground? What is the linear speed at the top of the wheel?

A wheel with radius 0.20 m starts from rest and turns through 8.0 revolutions in 5.0 s. At t = 5.0 s, what is the tangential acceleration of a point on the rim of the wheel?

A CD-ROM is accelerated from rest with constant angular acceleration αo.Find the time for the disk to complete the first full loop.
[a] t = 1/2 (αo2 )
[b] t = 2π/αo
[c] t = 4π/αo
[d] t = (2π/αo)1/2
[e] t = (4π/αo)1/2

A fan is turned off, and its angular speed decreases from 10.0 rad/s to 6.3 rad/s in 5.0 s. What is the magnitude of the angular acceleration of the fan?A) 0.37 rad/s2B) 11.6 rad/s2C) 0.74 rad/s2D) 0.86 rad/s2E) 1.16 rad/s2

A wheel accelerates from rest to 59 rad/s at a rate of 74 rad/s2. Through what angle (in radians) did the wheel turn while accelerating?
A) 24 rad
B) 30 rad
C) 19 rad
D) 47 rad

A wheel with radius 0.200 m starts from rest at t = 0 and then starts to rotate with constant angular acceleration about an axis at its center. At t = 5.0 s the wheel has turned through 4.00 rev. What is the angular acceleration of the wheel, in rad/s2?

A wheel with radius 0.200 m starts from rest at t = 0 and then starts to rotate with constant angular acceleration about an axis at its center. At t = 5.0 s the wheel has turned through 4.00 rev. At t = 5.0 s, what is the magnitude of the linear velocity of a point on the rim of the wheel?

A large wheel of radius 0.300 m is initially at rest and then starts to rotate with a constant angular acceleration of 0.400 rad/s2 about an axle at its center. What is the tangential velocity of a point on the rim of the wheel after the wheel has turned through 1.60 rad?

A wheel of radius 0.300 m is mounted with frictionless bearings about an axle through its center. A light rope is wrapped around the wheel around the wheel and a block is suspended from the free end of the rope. When the system is released from rest, the block descends with a constant linear acceleration of 4.00 m/s2.What is the angular velocity (in rad/s) of the wheel after it has turned through 5.00 rev?