1.
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
2.
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 (or “rad”)
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?
3.
Angular velocity, for circular motion, has
counterparts which can be compared with linear
speed s=Δx/Δt.
Period of time (Δt) remains unchanged, but linear
distance (Δx) is replaced with angular
displacement Δθ measured in radians.
Angular displacement Δθ
r
Δθ
r
Angular displacement is the number of
radians moved
4.
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?
5.
Consider an object moving along the arc of a circle
from A to P at a constant linear speed for time Δt:
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 = Δθ / Δt
Where Δθ is the angle turned through in radians, (rad),
yields units for ω of rads-1
This is all very comparable with linear speed, (or velocity) where
we talk about distance/time
6.
The period T of the rotational motion is the time
taken for one complete revolution (2π radians).
Substituting into : ω = Δθ / Δt
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
7.
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) / Δt
v = r Δθ / Δt
Substituting... (ω = Δθ / Δt)
v = ωr
8.
A cyclist travels at a linear speed of 12 ms-1 on a
bike with wheels which have a radius of 40 cm.
The wheels rotate clockwise. Calculate:
a. The frequency of rotation for the wheels
b. The angular velocity for the wheels
c. The angle the wheel turns through in 0.10 s
in
i. radians
ii. degrees
9.
The frequency of rotation for the wheels
Circumference of the wheel is 2π r
= 2π x 0.40m = 2.5m
Time for one rotation, (the period) is found using
s = Δd / Δt rearranged for Δt
Δt = Δd / s = T = circumference / linear speed
T = 2.5 / 12 = 0.21s
f = 1 / T = 1 / 0.21 = 4.8Hz
10.
The angular velocity for the wheels
Using T = 2π / ω , rearranged for ω
ω = 2π / T
ω = 2π / 0.21
ω = 30 rads-1 Clockwise
11.
The angle the wheel turns through in 0.10s in
i radians ii degrees
Using ω = Δθ / Δt
re-arranged for Δθ
Δθ = ω t
Δθ = 30 x 0.10
Δθ = 3.0 rad
= 3.0 x (360°/ 2π ) = 172° ≈ 1.7 x 102 °
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