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

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 (or “rad”) radians (or “rad”)

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=Δx/s=Δx/ΔΔtt..

Period of time (Period of time (ΔΔt) remains unchanged, but t) remains unchanged, but linearlinear distance (distance (ΔΔx) is replaced with x) is replaced with angularangular displacement Δdisplacement Δ measured in radians.measured in radians.

ΔΔ

Angular displacement Angular displacement ΔΔ

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 linearlinear speed for time speed for time ΔΔtt::

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”

= = ΔΔ / / ΔΔtt

Arc length

r

r

P

A

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

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

Substituting into : Substituting into : == ΔΔ / / ΔΔtt

= 2= 2 / / TT

TT = 2 = 2 / /

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

ff = = / 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 (Linear speed (vv) = arc length (AP) / ) = arc length (AP) / ΔΔtt

vv = = rrΔΔ / / ΔΔtt

Substituting... (Substituting... ( = = ΔΔ / / ΔΔt)t)

vv = = rr

Arc length

r

r

P

A

A cyclist travels at a linear speed of 12 msA cyclist travels at a linear speed of 12 ms-1-1 on a on a bike with wheels which have a radius of 40 cm. bike with wheels which have a radius of 40 cm. The wheels rotate clockwise. Calculate:The wheels rotate clockwise. 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 0.10 s The angle the wheel turns through in 0.10 s inin

i. radians i. radians

ii. degrees 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 0.40m = 2.5m x 0.40m = 2.5m

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

ss = = ΔΔdd / / ΔΔtt rearranged for rearranged for ΔΔtt

ΔΔtt = = ΔΔdd / s = / s = TT = circumference / linear speed = circumference / linear speed

TT = 2.5 / 12 = 0.21s = 2.5 / 12 = 0.21s

ff = 1 / = 1 / TT = 1 / 0.21 = = 1 / 0.21 = 4.8Hz 4.8Hz

The angular velocity for the wheelsThe angular velocity for the wheels

Using Using TT = 2 = 2 / / , rearranged for , rearranged for

= 2= 2 / / TT

= 2= 2 / 0.21 / 0.21

= 30 rads= 30 rads-1-1 Clockwise Clockwise

The angle the wheel turns through in 0.10s inThe angle the wheel turns through in 0.10s in

i radians ii degreesi radians ii degrees

Using Using = = ΔΔ / / ΔΔtt re-arranged for re-arranged for ΔΔ

ΔΔ = = tt

ΔΔ = 30 x 0.10 = 30 x 0.10

ΔΔ = 3.0 rad = 3.0 rad

= 3.0 x (360°/ 2= 3.0 x (360°/ 2) = 172° ≈ ) = 172° ≈ 1.7 x 101.7 x 1022 ° °