Study Guide

Angular displacement and speed

Physics· 25 min read

1. Angular Displacement in Radians★★☆☆☆⏱ 8 min

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📘 Definition

Angular Displacement

θ\theta

The angle (in radians) that a rotating body has turned from its initial position around a fixed axis, with positive values for counter-clockwise rotation by convention

Example:

A full rotation gives \theta = 2\pi radians

Radians are the standard unit for circular motion because they create a simple proportional relationship between arc length (the distance a point travels along the circular path), radius, and angular displacement:

s=rθs = r\theta
📐 Worked Example

A point on a bicycle wheel of radius 30 cm rotates by 2.5 radians. Calculate the arc length the point travels.

  1. 1

    Identify known values, convert radius to SI units:

  2. 2
    r=30 cm=0.30 m,θ=2.5 radr = 30 \text{ cm} = 0.30 \text{ m}, \quad \theta = 2.5 \text{ rad}
  3. 3

    Substitute into the arc length formula:

  4. 4
    s=rθ=0.30×2.5=0.75 ms = r\theta = 0.30 \times 2.5 = 0.75 \text{ m}
  5. 5

    The point travels 0.75 m (75 cm) along its path.

2. Angular Speed, Period and Frequency★★☆☆☆⏱ 10 min

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📘 Definition

Angular Speed

ω\omega

The rate of change of angular displacement over time, measured in radians per second (rad s⁻¹)

Example:

A ceiling fan rotating 4π radians per second has \omega = 4\pi rad s⁻¹

🔬 Derivation
Goal:

Derive the relationship between angular speed, period and frequency

Starting from:

Angular speed is defined as \omega = \frac{\Delta \theta}{\Delta t}

  1. 1

    For one full rotation, the total angular displacement is (2\pi) radians, and the time taken is the period (T). Substitute into the definition:

  2. 2
    ω=2πT\omega = \frac{2\pi}{T}
  3. 3

    Frequency (f) is defined as the number of rotations per second, so (f = \frac{1}{T}). Substitute to get:

  4. 4
    ω=2πf\omega = 2\pi f
Result:

Angular speed is directly proportional to rotation frequency and inversely proportional to the period of rotation.

📐 Worked Example

A car engine crankshaft rotates at 3000 revolutions per minute (rpm). Calculate its angular speed in rad s⁻¹.

  1. 1

    Convert rotation rate from rpm to frequency in Hz (revolutions per second):

  2. 2
    f=300060=50 Hzf = \frac{3000}{60} = 50 \text{ Hz}
  3. 3

    Substitute into the angular speed formula:

  4. 4
    ω=2πf=2π(50)=100π≈314 rad s−1\omega = 2\pi f = 2\pi (50) = 100\pi \approx 314 \text{ rad s}^{-1}

3. Relationship Between Angular and Linear Speed★★★☆☆⏱ 12 min

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For a rigid rotating body, all points rotate with the same angular speed, but the linear (tangential) speed of a point depends on its distance from the axis of rotation. Starting from the arc length relationship (s = r\theta), differentiate both sides with respect to time to get:

v=rωv = r\omega
📐 Worked Example

A merry-go-round rotates with constant angular speed of 0.8 rad s⁻¹. Calculate the tangential speed of a rider sitting 3.0 m from the center of rotation.

  1. 1

    Identify known values: (r = 3.0) m, (\omega = 0.8) rad s⁻¹

  2. 2

    Substitute into the relationship between angular and linear speed:

  3. 3
    v=rω=3.0×0.8=2.4 m s−1v = r\omega = 3.0 \times 0.8 = 2.4 \text{ m s}^{-1}
✓ Quick check

Test your understanding:

  1. A wheel of diameter 0.8 m rotates with angular speed 10 rad s⁻¹. What is the tangential speed of a point on its edge?

    • 4 m s⁻¹

    • 8 m s⁻¹

    • 16 m s⁻¹

    • 10π m s⁻¹

4. Common Pitfalls

Wrong move:

Using degrees instead of radians in the formulas (s = r\theta) or (v = r\omega)

Why:

These proportional relationships are only valid when angular displacement is measured in radians, not degrees

Correct move:

Always convert angles from degrees to radians by multiplying by (\pi/180) before substituting into formulas

Wrong move:

Assuming all points on a rigid rotating body have the same linear speed

Why:

Only angular speed is constant across all points on a rigid rotating body; linear speed increases with distance from the axis

Correct move:

Remember that for any point on a rigid rotating body, (v = r\omega), so linear speed scales with radius

Wrong move:

Forgetting to convert frequency from rpm to Hz before calculating angular speed

Why:

SI units for frequency are revolutions per second (Hz); using rpm directly gives an incorrect result 60 times too small

Correct move:

Always divide rpm by 60 to get frequency in Hz before substituting into (\omega = 2\pi f)

Wrong move:

Using diameter instead of radius in (v = r\omega)

Why:

Exam questions commonly give the diameter of wheels/circles to test attention to detail

Correct move:

Always check if the given length is radius or diameter, and divide diameter by 2 to get radius before substitution

5. Quick Reference Cheatsheet

Quantity

Symbol

Formula

SI Unit

Angular Displacement

radians (rad)

Angular Speed

rad s⁻¹

Tangential Speed

m s⁻¹

Period of Rotation

seconds (s)

Frequency of Rotation

hertz (Hz)

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

  • 2022 · 1

    Convert angular to linear speed

  • 2023 · 2

    Calculate angular speed from rpm

  • 2021 · 1

    Compare speed of two points on wheel

What's Next

The concepts of angular displacement and speed covered here form the foundation for all further work on circular motion, the next core topic being centripetal acceleration and force, which explain why circular motion requires a net force even when speed is constant. Mastery of these angular-linear relationships is essential for all subsequent circular motion problems, and they extend to rotational dynamics, torque, and moment of inertia topics that appear in later sections of the CIE 9702 syllabus. Exam questions regularly combine these basic kinematic relationships with force calculations, so consistent recall is key.