Evans SPH 4U1

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:

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]