CIE A Level Physics (9702) 2019-2021

Revision Notes

14.1.1 Uniform Circular Motion

Radians & Angular Displacement

  • 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:

Radians & Angular Displacement equation 1

  • Note: both distances must be measured in the same units e.g. metres
  • A radian (rad) is defined as:

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:

Radians & Angular Displacement equation 2a

 

Radians definition, downloadable AS & A Level Physics revision notes

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:

Radians & Angular Displacement equation 3

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

Radians & Angular Displacement equation 4

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

 

1. Radians & Angular Displacement equation 2

Table of common degrees to radians conversions

Table of common degrees to radians conversions, downloadable AS & A Level Physics revision notes

 

Worked example: Radians conversion

Worked example - radians conversion, downloadable AS & A Level Physics revision notes

Exam Tip

  • You will notice your calculator has a degree (Deg) and radians (Rad) mode
  • 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!

 

Radians on calculator, downloadable AS & A Level Physics revision notes

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

Angular speed diagram, downloadable AS & A Level Physics revision notes

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:

Calculating Angular Speed equation 1

  • 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:

Calculating Angular Speed equation 2

  • 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 ⍵)

 

Worked example: Angular speed

Worked example - angular speed, downloadable AS & A Level Physics revision notes

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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