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 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 10 cm radius disk is rotating about an axis through its center with a small ant at the rim. From t = 0s to t = 5s, it rotates at a constant angular speed of 5 rad/s. From 5s to 10s, it has an angular acceleration is 15 rad/s2. From 10s to 17s, the disk slows to a stop under constant angular acceleration.
(a) At 7s, what is the linear, tangential acceleration of the ant?
(b) At 7s, what is the linear, centripetal acceleration of the ant?
(c) At 7s, what is the linear acceleration of the ant?

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 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.

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?

At t=0 a grinding wheel has an angular velocity of 26.0 rad/s . It has a constant angular acceleration of 31.0 rad/s2 until a circuit breaker trips at time t exttip{t}{t}= 2.10 s. From then on, it turns through an angle 436 rad as it coasts to a stop at constant angular acceleration.a) Through what total angle did the wheel turn between t=0 and the time it stopped?b) At what time did it stop?c) What was its acceleration as it slowed down?

An airplane propeller is rotating at 1910 rev/min.a) Compute the propellers angular velocity in rad/s.b) How long in seconds does it take for the propeller to turn through 36°?

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?