Evans SPH 4U1
Physics Grade 12
Unit 3: Circular Motion
Note 5: Angular Velocity, Angular Acceleration
Reference: None in text
We will now discuss rotation of a rigid body about a fixed axis.
Recall that for a circle:
radian (no units) =
Angular Velocity:
where is the angle through which the body has rotated in time.
Angular Acceleration:
Tangential Velocity: (note that angular velocity will be the same for every point in a rotating body BUT tangential velocity is greater for points farther away from the axis).
Tangential Acceleration:
Centripetal Acceleration:
Frequency (The number of revolutions per second):
The Equations that we derived for Uniform Linear Acceleration can similarly be derived for Constant Angular Acceleration:
where
Example 1:
A disc of radius 10 cm rotates about its axis from rest with constant angular acceleration of 10 rad/s2. At t=5.0s what is
a) the angular velocity of the disk. [50 rad/s]
b) the tangential acceleration of a point on the edge of the disc? [1.0 m/s2]
c) the centripetal acceleration of a point on the edge of the disc? [250 m/s2]
Note: Be careful to get the units correct!
Example 2:
A carousel has a radius of 5.0 m. It starts from rest and reaches an angular velocity of 0.5 rad/s in 20 s. Find:
a) The angular acceleration [0.025 rad/s2]
b) The number of revolutions completed in 20 s [0.8]
c) The tangential acceleration [0.125 m/s2]
d) The centripetal acceleration at 20 s. [2.5 m/s2]
Example 3:
A turntable rotating at 33 1/3 rev/min is shut off. It brakes with constant angular acceleration and comes to rest in 2 min.
a) Find the angular acceleration. [2.91x10-2 rad/s2]
b) What is the average angular velocity of the turntable? [1.75 rad/s]
c) How many revolutions does it make before stopping? [33.3 rev]
Example 4:
A 20. cm diameter grinding wheel rotates at 2000. rpm. Calculate its angular velocity in rad/s [2.1 x 10-2 rad/s]
Example 5:
A 70. cm diameter wheel rotating 1200. rpm is brought to rest in 15s. Calculate its angular acceleration [-8.4 rad/s2]
Example 6:
A car engine slows down from 4000. rpm to 100. rpm in 4.5 s. Calculate
a) its angular acceleration (assumed uniform) [-56 rad/s]
b) the total number of revolutions the engine makes in this time.[3.0 x 10-2]
Example 7:
A turntable reaches its rated speed of 33 rpm after making 1.5 revolutions. What was its angular acceleration? [0.63 rad/s]
Example 8::
A 40. cm diameter wheel accelerates uniformly from 80. rpm to 300. rpm in3.6 s. How far will a point of the edge of the wheel have travelled in this time? [14.3 m]