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A2 · Practice questions · Angular quantities

Turn the angle into radians.

Six original Cambridge-style questions on radians, angular displacement and angular velocity, the link v = rω, and period and frequency for uniform circular motion.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Keep these straight

One full turn is 2π radians.

01
Calculation
[2 marks]

Convert 150° to radians, and convert 0.80 rad to degrees. Give each answer to two significant figures.

  • 150° × (π / 180°) = 2.6 rad. ✓
  • 0.80 rad × (180° / π) = 46°. ✓
02
Calculation
[3 marks]

A point on the rim of a bicycle wheel of radius 0.33 m moves at a steady 8.4 m s⁻¹. Calculate the angular velocity of the wheel and the time for one revolution.

  • ω = v / r = 8.4 / 0.33 = 25 rad s⁻¹ (25.5). ✓
  • T = 2π / ω = 2π / 25.5 = 0.25 s. ✓
  • Check: one revolution is a circumference of 2π × 0.33 = 2.07 m at 8.4 m s⁻¹, which is 0.25 s. ✓
03
Analysis
[2 marks]

Define the radian, and explain why an angle in radians has no unit in a formula such as s = rθ.

  • One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. ✓
  • The angle is a ratio of two lengths, arc length divided by radius, so the metres cancel and s = rθ gives a length from a length. ✓
04
Calculation
[3 marks]

A satellite completes one circular orbit of radius 7.0 × 10⁶ m every 98 minutes. Calculate its angular velocity and its orbital speed.

  • T = 98 × 60 = 5880 s, so ω = 2π / T = 2π / 5880 = 1.07 × 10⁻³ rad s⁻¹. ✓
  • v = rω = 7.0 × 10⁶ × 1.07 × 10⁻³ = 7.5 × 10³ m s⁻¹. ✓
  • Both the angular velocity and the speed are constant, even though the velocity changes direction continuously. ✓
05
Analysis
[3 marks]

Two children sit on a roundabout turning at a steady rate, one at 1.0 m from the axis and the other at 2.0 m. Compare their angular velocities and their linear speeds, and state which child, if either, has the larger angular displacement after one minute.

  • They share the same angular velocity because every point of a rigid roundabout turns through the same angle in the same time. ✓
  • From v = rω the outer child moves at twice the linear speed of the inner child. ✓
  • Their angular displacements after one minute are equal, since θ = ωt and both ω and t are the same. ✓
06
Calculation
[4 marks]

A fairground ride turns at 12 revolutions per minute. A rider sits 5.5 m from the axis.

(a) Calculate the angular velocity of the ride in rad s⁻¹. (b) Calculate the rider's linear speed. (c) Calculate the angle, in radians, the rider turns through in 4.0 s.

  • (a) 12 rev min⁻¹ = 12 / 60 = 0.20 rev s⁻¹, so ω = 2π × 0.20 = 1.26 rad s⁻¹ (1.3). ✓
  • (b) v = rω = 5.5 × 1.26 = 6.9 m s⁻¹. ✓
  • (c) θ = ωt = 1.26 × 4.0 = 5.0 rad. ✓
  • That is less than one full turn (2π = 6.3 rad), which matches 4.0 s being less than the 5.0 s period. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.