This document defines key terms related to angular displacement, velocity, and motion. It explains that angles can be measured in degrees or radians, with 360° equal to 2π radians. Angular displacement is the angle an object rotates through, measured in radians. Angular velocity is the rate of change of the angular displacement with respect to time. Angular velocity and period are related to frequency, as angular velocity equals 2π divided by the period. Linear speed can also be calculated from angular velocity and radius of rotation. Examples are provided to illustrate these relationships and calculations.

2. Book Reference : Pages 22-23
1.
To summarise the relationship between
degrees and radians
2.
To understand the term angular
displacement
3.
To define angular velocity
4.
To connect angular velocity to the period
and frequency of rotation
5.
To connect angular velocity to linear speed

3. Angles can be measured in both degrees & radians :
Arc
length
θ
r
The angle θ in radians is defined as
the arc length / the radius
For a whole circle, (360°) the arc
length is the circumference, (2π r)
∴ 360° is 2π radians
Common values :
45° = π /4 radians
90° = π /2 radians
180° = π radians
Note. In S.I. Units we use “rad”
How many degrees is 1 radian?

4. Angular velocity, for circular motion, has
counterparts which can be compared with linear
speed s=d/t.
Time (t) remains unchanged, but linear distance
(d) is replaced with angular displacement θ
measured in radians.
Angular displacement θ
r
θ
r
Angular displacement is the number of
radians moved

5. For a watch calculate the angular displacement in
radians of the tip of the minute hand in
1. One second
2. One minute
3. One hour
Each full rotation of the London eye takes 30
minutes. What is the angular displacement per
second?

6. Consider an object moving along the arc of a circle
from A to P at a constant speed for time t:
Arc length
Definition : The rate of change of
angular displacement with time
A
“The angle, (in radians) an object
rotates through per second”
P
r
θ
r
ω =θ /t
Where θ is the angle turned through in radians, (rad),
yields units for ω of rads-1
This is all very comparable with normal linear speed, (or velocity)
where we talk about distance/time

7. The period T of the rotational motion is the time
taken for one complete revolution (2π radians).
Substituting into : ω = θ / t
ω = 2π / T
∴ T = 2π / ω
From our earlier work on waves we know that the
period (T) & frequency (f) are related T = 1/f
∴ f = ω / 2π

8. Considering the diagram below, we can see that
the linear distance travelled is the arc length
P
Arc length
r
θ
r
A
∴ Linear speed (v) = arc length (AP) / t
v = rθ / t
Substituting... (ω = θ / t)
v = rω

9. A cyclist travels at a speed of 12ms-1 on a bike
with wheels which have a radius of 40cm.
Calculate:
a. The frequency of rotation for the wheels
b. The angular velocity for the wheels
c. The angle the wheel turns through in 0.1s in
i radians ii degrees

10. The frequency of rotation for the wheels
Circumference of the wheel is 2π r
= 2π x 0.4m = 2.5m
Time for one rotation, (the period) is found using
s =d / t rearranged for t
t = d / s = T = circumference / linear speed
T = 2.5 / 12 = 0.21s
f = 1 / T = 1 / 0.21 = 4.8Hz

11. The angular velocity for the wheels
Using T = 2π / ω , rearranged for ω
ω = 2π / T
ω = 2π / 0.21
ω = 30 rads-1

12. The angle the wheel turns through in 0.1s in
i radians ii degrees
Using ω = θ / t
re-arranged for θ
θ = ωt
θ = 30 x 0.1
θ = 3 rad
= 3 x (360°/ 2π ) = 172°