1.1. To summarise the relationship between To summarise the relationship between degrees and radiansdegrees and radians

2.2. To understand the term angular To understand the term angular displacementdisplacement

3.3. To define angular velocityTo define angular velocity

4.4. To connect angular velocity to the period To connect angular velocity to the period and frequency of rotationand frequency of rotation

5.5. To connect angular velocity to linear speedTo connect angular velocity to linear speed

Book Reference : Pages 22-23Book Reference : Pages 22-23

Angles can be measured in both degrees & radians :Angles can be measured in both degrees & radians :

The angle The angle in radians is defined as in radians is defined as the arc length / the radiusthe arc length / the radius

For a whole circle, (360°) the arc For a whole circle, (360°) the arc length is the circumference, (2length is the circumference, (2r)r)

360° is 2360° is 2 radians radians

Arclength

r

Common values :Common values :

45° = 45° = /4 radians/4 radians90° = 90° = /2 radians/2 radians180° = 180° = radians radians

Note. In S.I. Units we use “rad”Note. In S.I. Units we use “rad”

How many degrees is 1 radian?How many degrees is 1 radian?

Angular velocity, for circular motion, has Angular velocity, for circular motion, has counterparts which can be compared with linear counterparts which can be compared with linear speed speed s=d/ts=d/t..

Time (t) remains unchanged, but linear distance Time (t) remains unchanged, but linear distance (d) is replaced with (d) is replaced with angular displacement angular displacement measured in radians.measured in radians.

Angular displacement Angular displacement

r

r Angular displacement is the number of Angular displacement is the number of radians movedradians moved

For a watch calculate the angular displacement in For a watch calculate the angular displacement in radians of the tip of the minute hand inradians of the tip of the minute hand in

1.1. One secondOne second

2.2. One minuteOne minute

3.3. One hourOne hour

Each full rotation of the London eye takes 30 Each full rotation of the London eye takes 30 minutes. What is the angular displacement per minutes. What is the angular displacement per second?second?

Consider an object moving along the arc of a circle Consider an object moving along the arc of a circle from A to P at a constant from A to P at a constant speedspeed for time t: for time t:

Definition : The rate of change of Definition : The rate of change of angular displacement with timeangular displacement with time

““The angle, (in radians) an object The angle, (in radians) an object rotates through per second”rotates through per second”

= = / t / t

Arc length

r

r

P

A

This is all very comparable with normal linear speed, (or velocity) This is all very comparable with normal linear speed, (or velocity) where we talk about distance/timewhere we talk about distance/time

Where Where is the angle turned through in radians, (rad), is the angle turned through in radians, (rad), yields units for yields units for of rads of rads-1-1

The period T of the rotational motion is the time The period T of the rotational motion is the time taken for one complete revolution (2taken for one complete revolution (2 radians). radians).

Substituting into : Substituting into : = = / t / t

= 2= 2 / T / T

T = 2T = 2 / /

From our earlier work on waves we know that the From our earlier work on waves we know that the period (T) & frequency (f) are related T = 1/fperiod (T) & frequency (f) are related T = 1/f

f = f = / 2 / 2

Considering the diagram below, we can see that Considering the diagram below, we can see that the linear distance travelled is the arc lengththe linear distance travelled is the arc length

Linear speed (v) = arc length (AP) / tLinear speed (v) = arc length (AP) / t

v = rv = r / t / t

Substituting... (Substituting... ( = = / t) / t)

v = rv = r

Arc length

r

r

P

A

A cyclist travels at a speed of 12msA cyclist travels at a speed of 12ms-1-1 on a bike on a bike with wheels which have a radius of 40cm. with wheels which have a radius of 40cm. Calculate:Calculate:

a.a. The frequency of rotation for the wheelsThe frequency of rotation for the wheels

b.b. The angular velocity for the wheelsThe angular velocity for the wheels

c.c. The angle the wheel turns through in 0.1s inThe angle the wheel turns through in 0.1s in

i radians ii degrees i radians ii degrees

The frequency of rotation for the wheelsThe frequency of rotation for the wheels

Circumference of the wheel is 2Circumference of the wheel is 2r r

= 2= 2 x 0.4m = 2.5m x 0.4m = 2.5m

Time for one rotation, (the period) is found usingTime for one rotation, (the period) is found using

s =d / t rearranged for ts =d / t rearranged for t

t = d / s = T = circumference / linear speedt = d / s = T = circumference / linear speed

T = 2.5 / 12 = 0.21sT = 2.5 / 12 = 0.21s

f = 1 / T = 1 / 0.21 = f = 1 / T = 1 / 0.21 = 4.8Hz 4.8Hz

The angular velocity for the wheelsThe angular velocity for the wheels

Using T = 2Using T = 2 / / , rearranged for , rearranged for

= 2= 2 / T / T

= 2= 2 / 0.21 / 0.21

= 30 rads= 30 rads-1-1

The angle the wheel turns through in 0.1s inThe angle the wheel turns through in 0.1s in

i radians ii degreesi radians ii degrees

Using Using = = / t / t re-arranged for re-arranged for

= = tt

= 30 x 0.1= 30 x 0.1

= 3 rad = 3 rad

= 3 x (360°/ 2= 3 x (360°/ 2) ) = 172°= 172°