Cambridge A Levels A2 Physics Chapter 12 Motion in a Circle
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Course
Cambridge A Levels A2 Physics
Institution
Chapter 12 Motion in a Circle: 21 pages
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Chapter 12 Motion in a Circle
Definition of a Radian
One radian is the angle subtended at the centre
of a circle by an arc length equal to the radius.
Angular Displacement, 𝜽
The angle through which the object has moved.
𝒔
𝜽=
𝒓
θ = angular displacement Unit of θ: radian
s = arc length
r = radius
In one revolution,
𝑠
s = 2πr 𝜃=𝑟
2𝜋𝑟
𝜃=
𝑟
𝜃 rad = 2π rad
2π rad = 360o
1 rad = 57.3o
Angular Velocity, ω
The rate of change of angular displacement.
𝚫𝜽
𝝎= Unit of 𝜔 : rad s-1
𝚫𝒕
𝜔 = angular velocity
𝜃 = angular displacement
𝑡 = time
, v = linear/tangential velocity
𝜔 = angular velocity
Angle of a full circle = 2π
Circumference of
a full circle = 2πr
Period, T
The time to complete and revolution.
If t = T, 𝜃 = 2𝜋 rad
𝟐𝝅 𝟐𝝅𝒓
𝝎= 𝑽=
𝑻 𝑻
Frequency, f
Number of revolutions per unit time.
𝝎 = 𝟐𝝅𝒇 𝒗 = 𝟐𝝅𝒓𝒇
Unit of f : hertz, Hz
1 Hz = 1 revolution s-1
Relationship between Speed & Angular Velocity
Derivation:
𝒗 = 𝒓𝝎
𝑠
𝑣=𝑡 𝑠 = 𝑟𝜃
𝜃
𝒗 𝑣 = 𝑟( )
𝑡
𝝎=
𝒓 = 𝑟𝜔
, Period Frequency
TA = TB fA = fB
Angular speed Linear speed
𝜔𝐴 = 𝜔𝐵 𝑣𝐴 < 𝑣𝐵
ϴ is the same. rA < rB.
Examples of Relationship between Speed and Angular Velocity
1) Pulley
∴ 𝑉 = 𝑟𝜔
2) Vehicles If a small tire is replaced with a bigger tire:
▪ Power from engine is constant, 𝜔 is
constant.
∴ 𝑉 = 𝑟𝜔 constant
▪ 𝑉 = 𝑟𝜔
▪ r increases, v (linear speed) increases for
the same force produced from the engine.
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