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Angular Speed, Period & Frequency

Key idea: Use angular velocity, period, frequency and v = rω to connect rotation with tangential speed.

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Relate angular velocity, period, frequency and tangential speed using v = rω.

1. Definitions (Must Know)

A. Angular velocity, ω

Angular velocity is the rate of change of angular displacement:

ω = Δθ/(Δ t).

For uniform circular motion, ω is constant. Its magnitude is the angular speed; a rotational sign convention distinguishes opposite senses of rotation.

B. Time period, T

The time period, T (s), is the time taken for one complete revolution.

C. Frequency, f

The frequency, f (Hz), is the number of revolutions per second.

Unit: s⁻¹ (hertz, Hz).

D. Relationship between T and f

f = 1/T and T = 1/f

2. Key Ideas (What Earns Marks)

  • One revolution is 2π radians, so: ω = 2π/T = 2π f
  • Linear speed at radius r: v = rω = (2π r)/T = 2π r f
  • Points on one rigid rotating body share the same ω, T and f, but a point at larger r has greater tangential speed v.
  • Convert units early:
    • rpm → Hz: f = rpm/60
    • km/h → m/s: divide by 3.6

3. Detailed Explanations

A. Why ω = 2π/T

Angular speed is angle per time.

In one full revolution, the angle is 2π rad. If that takes time T:

ω = Δθ/(Δ t) = 2π/T

Angular speed vs frequency (linear relationship)

Angular speed increases in direct proportion to frequency with gradient 2π.

Scroll across the graph to read all labels.

Angular speed increases in direct proportion to frequency with gradient 2π.Angular speed increases in direct proportion to frequency with gradient 2π.
Because ω = 2π f, doubling frequency doubles angular speed.
Open full-size graph
View figure data
Values for Angular speed vs frequency (linear relationship)
Frequency, f (Hz)ω = 2πf
00
16.28
212.57
318.85
425.13

B. Linking angular speed to linear speed

For an arc length s:

s = rθ

Divide by time:

s/t = rθ/t ⇒ v = rω

This is why points further from the centre have larger linear speed even if ω is the same.

4. Common Mistakes

  • Forgetting to convert rpm to Hz.
  • Mixing up f (Hz) and ω (rad s⁻¹).
  • Using v = ω r but using r in cm (convert to m).
  • Treating v as radial; instantaneous linear velocity is tangent to the circular path.

5. Exam Tips

  • Write the “2π per revolution” idea explicitly if you get stuck.
  • If you’re given “revolutions in a time”, compute f first, then use ω = 2π f.
  • Always include units: Hz, s, rad s⁻¹, m s⁻¹.

6. Worked Examples

Modelled example 1

Convert rpm to angular speed

Core

Problem

A motor spins at 1800 rpm. Find its frequency and angular speed.
Study the worked solution

  1. Convert minutes to seconds

    Method

    Divide revolutions per minute by 60.

    Reason

    Frequency counts revolutions each second.

    Working

    f = 1800/60 = 30 Hz

  2. Convert cycles to radians

    Method

    Use ω = 2π f.

    Reason

    Each revolution is 2π radians.

    Working

    ω = 2π(30) = 188 rad s⁻¹

Guided practice 2

Linear speed from period

About 5 min

Problem

A point is 0.25 m from a wheel’s centre and completes one revolution every 0.50 s. Find its linear speed.

Try this before viewing the solution

Unit: m s^-1

Hints

Hint 1: move through angular speed first
One revolution every T seconds gives ω = 2π/T.
View solution step by step

  1. Find angular speed

    Method

    Divide 2π by the period.

    Reason

    One cycle corresponds to 2π radians.

    Working

    ω = 2π/0.50 = 4π rad s⁻¹

  2. Find tangential speed

    Method

    Use v = rω.

    Reason

    The arc distance per time scales with radius.

    Working

    v = 0.25(4π) = 3.14 m s⁻¹

Common misconception 3

Linear speed from frequency

Find and correct the mistake

Learner working

For r = 0.80 m and f = 0.75 Hz, a learner writes v = rf = 0.60 m s⁻¹. Locate the first error and correct the speed.

Try this before viewing the solution

First error

View solution step by step

  1. Restore distance per cycle

    Method

    Use circumference 2π r.

    Reason

    One revolution travels once around the circle.

    Working

    v = (2π r)f.

  2. Calculate

    Method

    Substitute the radius and frequency.

    Reason

    Distance per cycle times cycles per second gives speed.

    Working

    v = 2π(0.80)(0.75) = 3.77 m s⁻¹

Examiner practice 4

Find rpm from linear speed and radius

4 marks

Examination question

A wheel has radius 0.30 m and rim speed 9.0 m s⁻¹. Find its rotation rate in rpm. [4 marks]

Try this before viewing the solution

View solution step by step

  1. Angular speed

    1 mark

    Method

    Use ω = v/r.

    Reason

    Tangential speed is rω.

    Working

    ω = 9.0/0.30 = 30 rad s⁻¹

  2. Frequency relation

    1 mark

    Method

    Use f = ω/(2π).

    Reason

    Each revolution is 2π radians.

    Working

    f = 30/(2π).

  3. Frequency value

    1 mark

    Method

    Evaluate f = 4.77 Hz.

    Reason

    This is revolutions per second.

    Working

    f = 4.77 s⁻¹.

  4. Convert to rpm

    1 mark

    Method

    Multiply by 60.

    Reason

    One minute contains 60 seconds.

    Working

    60(4.77) = 286 rpm

Challenge 5

Convert road speed to angular speed (unit conversions)

Minimal support

Rolling transfer

A car travels at 72 km h⁻¹ on tyres of radius 0.32 m without slipping. Find wheel angular speed and frequency.

Try this before viewing the solution

Hints

Hint 1: translate linear motion first
Convert 72 km h⁻¹ by dividing by 3.6, then use the no-slip relation v = rω.
View solution step by step

  1. Convert road speed

    Method

    Convert to 20 m s⁻¹.

    Reason

    Radius is given in metres and angular speed uses seconds.

    Working

    v = 72/3.6 = 20 m s⁻¹

  2. Find angular speed

    Method

    Use ω = v/r.

    Reason

    No slipping equates rim speed and car speed.

    Working

    ω = 20/0.32 = 62.5 rad s⁻¹

  3. Find frequency

    Method

    Divide angular speed by 2π.

    Reason

    Each revolution contains 2π radians.

    Working

    f = 62.5/2π = 9.95 Hz

7. Mind Stretchers

Mind stretcher 1: Same T, different vExtension

Two points on a disc have the same period T but different radii.

Which point has larger v and why?

Show Answer

They have the same ω = 2π/T.

But v = rω, so the point with the larger radius has larger linear speed.

Mind stretcher 2: What happens if frequency doubles?Extension

A point on a disc is at a fixed radius r. The disc’s frequency increases from f to 2f.

State what happens to:

  1. ω
  2. v
  3. a_c.
Show Answer

Using ω = 2π f, v = rω, and a_c = rω²:

  1. ω doubles.

  2. v doubles.

  3. a_c increases by a factor of 4 (because it depends on ω²).

Mind stretcher 3: Optional (Enrichment)Extension

A. Earth rotation (speed depends on latitude)

The Earth rotates once every 24 h, so ω ≈ 2π/86400.

At latitude λ, the radius of the circle traced is r cos λ, so:

v = ω r cos λ

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027