Radians & Angular Displacement (Cambridge International A-Level Physics)

Radians & angular displacement

Angles in radians

• 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

Visual definition of radian

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

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

$\frac{\mathrm{circumference} \mathrm{of} \mathrm{circle}}{\mathrm{radius}}=\frac{2πr}{r}=2π$

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

θ degree×(straight π / 180)=θ rad

Table of common degrees to radians conversions

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:

$\Deltaθ=\frac{\mathrm{distance} \mathrm{travelled} \mathrm{around} \mathrm{the} \mathrm{circle}}{\mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{circle}}$

• 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)

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

Visual representation of angular displacement equation

An angle in radians, subtended at the centre of a circle, is the arc length divided by the radius of the circle

Worked Example Convert the following angular displacement into degrees:

Answer:

Step 1: Rearrange the degrees to radians conversion equation

$\mathrm{degrees}\rightarrow\mathrm{radians}\Rightarrowθ^{\circ}\times\frac{π}{180}=θ\mathrm{rad}$

$\mathrm{radians}\rightarrow\mathrm{degrees}\Rightarrowθ\mathrm{rad}\times\frac{180}{π}=θ^{\circ}$

Step 2: Substitute the values to calculate

$\frac{π}{3}\mathrm{rad}\times\frac{180}{π}=\frac{180^{\circ}}{3}=60^{\circ}$

Exam Hint

• 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!