Concept #1: Rotational Velocity & Acceleration

Example #1: Rotational velocity of Earth

Practice: Calculate the rotational velocity (in rad/s) of a clock’s minute hand.

EXTRA: Calculate the rotational velocity (in rad/s) of a clock’s hour hand.

Practice: A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. Calculate its angular acceleration.

As a particle with a velocity v in the negative x direction passes through the point (0, 0, 1), it has an angular velocity relative to the origin that is best represented by vector
A. 1
B. 2
C. 3
D. 4
E. Zero

A wheel of radius 0.5 m rolls without slipping on a horizontal surface. Starting from rest, the wheel moves with a constant angular acceleration of 6 rad/s2 . The distance traveled by the center of the wheel from t = 0 to t = 3 s is about:
A. none of these
B. 2.1 m
C. 13.5 m
D. 18 m
E. 27 m

When you look up into the sky, you always see the same part of the moon, no matter what time of the month or year it is. In order to achieve this, the rotational period of the moon must be equal to its orbital period (how long it takes to orbit the Earth). Given this fact, what is the angular velocity of the moon due to its spinning about its own axis?

When you ride your bicycle, in what direction is the angular velocity of the wheels?
A) to your left
B) to your right
C) forward
D) backward
E) up

Two solid discs are rotating about a perpendicular shaft through their centers, as shown in the figure. Disc A, has a radius that is twice as large as disc B,? Which of the following statements is NOT true?
A) A point on the rim of disc A, has twice the linear speed as a point on the rim of disc B.
B) The direction of the angular velocity is to the right.
C) Every point on the body has the same angular acceleration.
D) The linear acceleration of a point on the rim of disc B is the same as the linear acceleration of a point halfway from the center to the rim on disc A.
E) The angular velocity at a point on the rim of disc A is twice the angular velocity of a point on the rim of disc B.

What is the acceleration experienced by the tip of the 1.6 cm -long sweep second hand on your wrist watch?

A fan blade rotates with angular velocity given by ωz(t) = γexttip{gamma }{gamma} - βexttip{eta }{beta}t2.a) Calculate the angular acceleration as a function of time.b) If γexttip{gamma}{gamma_1} = 5.05 rad/s and βexttip{eta}{beta_1} = 0.805 rad/s3 , calculate the instantaneous angular acceleration αz at t exttip{t}{t_0}= 3.10 s .c) If γ exttip{gamma}{gamma_1} = 5.05 rad/s and βexttip{eta}{beta_1} = 0.805 rad/s3 , calculate the average angular acceleration αav - z for the time interval t = 0 to t exttip{t}{t_0}= 3.10 s .

A roller in a printing press turns through an angle θ(t) given by θ(t) = γt2 - βt3 , where γ = 3.20 rad/s2 and β eta= 0.500 rad/s3{
m{ rad/s}}^3.a) Calculate the angular velocity of the roller as a function of time.b) Calculate the angular acceleration of the roller as a function of time.c) What is the maximum positive angular velocity?d) At what value of t does it occur?

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to θ(t)= γt+ βt3, where γ = 0.406 rad/s and β = 1.30×10−2 rad/s3.a) Calculate the angular velocity of the merry-go-round as a function of time.b) What is the initial value of the angular velocity?c) Calculate the instantaneous value of the angular velocity ωz at t = 5.05 s .d) Calculate the average angular velocity ωav - z for the time interval t = 0 to t = 5.05 s .

Two children are riding on a merry-go-round. Child A is at a greater distance from the axis of rotation than child B. Which child has the larger tangential speed?A) Child BB) They have the same zero tangential speedC) Child AD) They have the same non-zero tangential speedE) There is not enough information given to answer the question.

a) Calculate the angular velocity of the second hand of a clock. State in rad/s.b) Calculate the angular velocity of the minute hand of a clock. State in rad/s.c) Calculate the angular velocity of the hour hand of a clock. State in rad/s.d) What is the angular acceleration in each case?

The angular acceleration of a wheel, as a function of time, is α = 5.0 t2 - 8.5 t, where α is in rad/s2 {
m{rad/s^2}} and t is exttip{t}{t}in seconds. If the wheel starts from rest (θheta = 0, ω omega= 0, at t exttip{t}{t}= 0), determine a formula fora) the angular velocity ωomega as a function of timeb) the angular position θheta as a function of timec) evaluate ω at t = 4.0 sd) evaluate θ at t = 4.0 s