Angular Displacement, Velocity & Acceleration (HL) (HL IB Physics)

• A rigid rotating body can be described using the following properties:
  • Angular displacement
  • Angular velocity
  • Angular acceleration

• These properties can be inferred from the properties of objects moving in a straight line combined with the geometry of circles and arcs

Angular Displacement

• Angular displacement is defined as:

The change in angle through which a rigid body has rotated relative to a fixed point

• Angular displacement is measured in radians

Angular displacement to linear displacement

• The linear displacement s at any point along a segment that is in rotation can be calculated using:

• Where:
  • θ = angular displacement, or change in angle (radians)
  • s = length of the arc, or the linear distance travelled along a circular path (m)
  • r = radius of a circular path, or distance from the axis of rotation (m)

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

Angular Velocity

• The angular velocity ω of a rigid rotating body is defined as:

The rate of change in angular displacement with respect to time

• Angular velocity is measured in rad s^–1
• This can be expressed as an equation:

• Where:
  • ω = angular velocity (rad s^–1)
  • Δθ = angular displacement (rad)
  • Δt = change in time (s)

Angular velocity to linear velocity

• The linear speed v is related to the angular speed ω by the equation:

• Where:
  • v = linear speed (m s^–1)
  • r = distance from the axis of rotation (m)

• Taking the angular displacement of a complete cycle as 2π, angular velocity ω can also be expressed as:

• Rearranging gives the expression for linear speed:

• Where:
  • f = frequency of the rotation (Hz)
  • T = time period of the rotation (s)

Angular Acceleration

• Angular acceleration α is defined as

The rate of change of angular velocity with time

• Angular acceleration is measured in rad s⁻²
• This can be expressed as an equation:

• Where:
  • α = angular acceleration (rad s⁻²)
  • = change in angular velocity, or (rad s⁻¹)
  • = change in time (s)

Angular acceleration to linear acceleration

• Using the definition of angular velocity ω with the equation for angular acceleration α gives:

• Rearranging gives the expression for linear acceleration:

• Where:
  • a = linear acceleration (m s⁻²) 
  • r = distance from the axis of rotation (m)
  • = change in linear velocity, or  (m s⁻¹)

Graphs of Rotational Motion

• Graphs of rotational motion can be interpreted in the same way as linear motion graphs

Graphs of angular displacement, angular velocity and angular acceleration

• Angular displacement is equal to...
  • The area under the angular velocity-time graph

• Angular velocity is equal to...
  • The gradient of the angular displacement-time graph
  • The area under the angular acceleration-time graph

• Angular acceleration is equal to...
  • The gradient of the angular velocity-time graph

Summary of linear and angular variables

Variable	Linear	Angular
displacement
velocity
acceleration		$\alpha = \frac{a}{r}$