# 5.4.3 Angular Velocity

## Angular Velocity

#### Angular Displacement

• In circular motion, it is more convenient to measure angular displacement in units of radians rather than units of degrees
• Angular displacement is defined as:

The change in angle, in radians, of a body as it rotates around a circle

• This can be summarised in equation form:

• Where:
• Δθ = angular displacement, or angle of rotation (radians)
• S = length of the arc, or the distance travelled around the circle (m)
• r = radius of the circle (m)

• Note: both distances must be measured in the same units e.g. metres

#### Angular Speed

• Any object travelling in a uniform circular motion at the same speed travels with a constantly changing velocity
• This is because it is constantly changing direction, and is therefore accelerating
• The angular speed (⍵) of a body in circular motion is defined as:

The rate of change in angular displacement with respect to time

• Angular speed is a scalar quantity and is measured in rad s-1
• It can be calculated using:

• Where:
• Δθ = change in angular displacement (radians)
• Δt = time interval (s)

When an object is in uniform circular motion, velocity constantly changes direction, but the speed stays the same

• Taking the angular displacement of a complete cycle as 2π, the angular speed ⍵ can be calculated using the equation:

• Where:
• v = linear speed (m s-1)
• r = radius of orbit (m)
• T = the time period (s)
• f = frequency (Hz)

• Angular velocity is the same as angular speed, but it is a vector quantity
• This equation shows that:
• The greater the rotation angle θ in a given amount of time, the greater the angular velocity ⍵
• An object rotating further from the centre of the circle (larger r) moves with a smaller angular velocity (smaller ⍵)

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