**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/s^{2}.
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/s ^{2}]**

c) the centripetal acceleration
of a point on the edge of the disc? *[250 m/s ^{2}*

^{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/s ^{2}]**

b) The number of
revolutions completed in 20 s *[***0.8]**

c) The tangential
acceleration **[0.125 m/s ^{2}]**

d) The centripetal
acceleration at 20 s. *[***2.5 m/s ^{2}]**

**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/s^{2}]**

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 **

**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]**