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Radians and angular velocity

Key idea: Radians link arc length to angle through s = rθ. Points on a rotating wheel share angular velocity, but their linear speeds increase with radius: v = rω.

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  • GCE A-Level H1 Physics 2027
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Every point on a spinning wheel turns through the same angle in the same time, but a point on the rim travels much further than a point near the hub. To describe rotation you need both: an angle that all the points share, and the distance that each point travels. Measuring the angle in radians links the two directly, which is why radians are used for all circular motion.

Radians

For an arc of length s on a circle of radius r, the angle at the centre in radians is

θ = s/r

One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. A full circle has a circumference of 2π r, so a full turn is 2π rad, which is 360°. To convert from degrees, multiply by π/180.

Because θ = s/r, the arc length is simply s = rθ, but only when θ is in radians.

Angular displacement measured by arc lengthThe highlighted arc has length equal to the circle's radius r. Its two bounding radii make an angle of one radian at the centre. The general radian relation is theta equals arc length divided by radius.rarc s = r1 raddivide by rθ = s/rif s = rθ = 1 rad2π rad per turn
Angular displacement measured by arc lengthThe highlighted arc has length equal to the circle's radius r. Its two bounding radii make an angle of one radian at the centre. The general radian relation is theta equals arc length divided by radius.rarc s = r1 raddivide by rθ = s/rif s = rθ = 1 rad2π rad per turn
Radians connect rotation to distance directly: θ = s/r, so an arc equal to the radius subtends 1 rad.

Check your understanding 1 An arc in degrees

A point on a circle of radius 0.300 m turns through 60°. How far does it travel along the circle?

Unit: rad
Show the steps Optional working
Unit: rad
Unit: m
Show answer

60° = π/3 rad, so s = rθ = 0.300 × π/3 = 0.314 m. Putting 60 straight into s = rθ would treat it as 60 radians.

Angular velocity

The angular velocity ω is the rate of change of angular displacement, the angle turned through per unit time:

ω = Δθ/(Δ t)

measured in rad s⁻¹. For steady rotation, one full turn of 2π rad takes one period T, and the frequency f = 1/T is the number of turns per second. So

ω = 2π/T = 2π f

Linear speed and angular velocity

In a time Δ t a point at radius r turns through Δθ and travels Δ s = rΔθ. Dividing by Δ t gives its speed:

v = rω

All the points of a rigid wheel have the same angular velocity, so a point twice as far from the axis moves twice as fast. The velocity of each point is along the tangent to its circle.

Worked example 1

A steadily rotating wheel

Problem

A wheel makes 3.00 revolutions in 1.50 s. Find its angular velocity and period, and the speed of a point 0.120 m from the axis.

Show full solution

  1. Angle turned

    Method

    Convert revolutions to radians.

    Reason

    Angular velocity must be in rad s⁻¹ to use v = rω.

    Working

    Answer

    Δθ = 3.00 × 2π = 6π rad

  2. Angular velocity and period

    Method

    Divide the angle by the time; the period is the time for one turn.

    Reason

    The rotation is steady.

    Working

    Answer

    ω = 6π/1.50 = 12.6 rad s⁻¹ and T = 1.50/3.00 = 0.500 s

  3. Speed of the point

    Working

    Answer

    v = rω = 0.120 × 12.6 = 1.51 m s⁻¹

Guided practice 2

A turntable

About 4 min

Problem

A turntable rotates steadily at 45.0 revolutions per minute. Find its frequency, its angular velocity and the speed of a point 0.100 m from its centre.

Convert to revolutions per second first

Unit: Hz
Show the steps Optional working
Unit: Hz
Unit: rad s⁻¹
Show the steps Optional working
Unit: Hz
Unit: rad s⁻¹
Unit: m s⁻¹

Hints

Hint 1

frequency

Divide the revolutions per minute by 60 to get revolutions per second.

Hint 2

angular velocity

ω = 2π f, then v = rω.

Show solution step by step

  1. Frequency and angular velocity

    Method

    Convert to hertz, then multiply by 2π.

    Reason

    Each revolution is 2π radians.

    Working

    Answer

    f = 45.0/60 = 0.750 Hz and ω = 2π × 0.750 = 4.71 rad s⁻¹

  2. Speed

    Working

    Answer

    v = 0.100 × 4.71 = 0.471 m s⁻¹

Check your understanding 2 Two wheels joined by a belt

A belt moves at 1.20 m s⁻¹ round two wheels, of radii 0.0600 m and 0.150 m, without slipping. Find the angular velocity of each wheel. Why are they different?

Show answer

The rim of each wheel moves at the belt’s speed. ω = v/r: 1.20/0.0600 = 20.0 rad s⁻¹ and 1.20/0.150 = 8.00 rad s⁻¹. The larger wheel has a longer rim, so it needs fewer turns per second to pass the same length of belt.

Common mistakes

  • Using an angle in degrees in s = rθ or v = rω. Both need radians.
  • Using revolutions per minute as if they were rad s⁻¹.
  • Thinking all points of a wheel have the same speed. They share the same angular velocity; the speed grows with radius.
  • Forgetting the 2π in ω = 2π/T.

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