# 14.1.1 Uniform Circular Motion

• In circular motion, it is more convenient to measure angular displacement in units of radians rather than units of degrees
• The angular displacement (θ) of a body in circular motion is defined as:

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

• The angular displacement is the ratio of:

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

The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle

• Angular displacement can be calculated using the equation:

When the angle is equal to one radian, the length of the arc (Δs) is equal to the radius (r) of the circle

• 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)

• Radians are commonly written in terms of π
• The angle in radians for a complete circle (360o) is equal to:

• If an angle of 360o = 2π radians, then 1 radian in degrees is equal to:

• Use the following equation to convert from degrees to radians:

Table of common degrees to radians conversions

#### Exam Tip

• This is shown by the “D” or “R” highlighted at the top of the screen
• Remember to make sure it’s in the right mode when using trigonometric functions (sin, cos, tan) depending on whether the answer is required in degrees or radians
• It is extremely common for students to get the wrong answer (and lose marks) because their calculator is in the wrong mode – make sure this doesn’t happen to you!

### 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

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

### Calculating Angular Speed

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

• Where:
• Δθ = change in angular displacement (radians )
• Δt = time interval (s)
• T = the time period (s)
• f = frequency (Hz)
• Angular velocity is the same as angular speed, but it is a vector quantity
• When an object travels at constant linear speed v in a circle of radius r, the angular velocity is equal to:

• Where:
• v is the linear speed (m s-1)
• r is the radius of orbit (m)
• This equation tells us:
• 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 faster angular velocity (larger ⍵)

### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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