Radians and angular velocity

Radians

An angle in radians is θ = s / r, where s is the arc length and r the radius. One radian is the angle at the centre when the arc equals the radius.

• A full circle: 2π rad = 360°
• To convert: degrees x π / 180 = radians; radians x 180 / π = degrees

Angular velocity

Angular velocity ω is the angle swept out per unit time: ω = Δθ / Δt, in rad/s. For one complete revolution in period T:

ω = 2π / T = 2πf

Linking to speed

A point at radius r moving with angular velocity ω has speed v = ωr. Every point on a rigid rotating object has the same ω, but points further out move faster.

Worked example

A wheel of radius 0.30 m turns at 2.0 revolutions per second. Find ω and the speed of a point on the rim.

1. ω = 2πf = 2π x 2.0 = 12.6 rad/s
2. v = ωr = 12.57 x 0.30

Answer: ω = 12.6 rad/s; v = 3.77 m/s

Check yourself

1. Convert 60° into radians, to 3 s.f.
   Show answer

   1.05 rad. 60 x π / 180 = π/3 = 1.05 rad.

2. Convert 2.5 rad into degrees, to 3 s.f.
   Show answer

   143 °. 2.5 x 180 / π = 143°.

3. An arc of length 0.45 m is part of a circle of radius 0.30 m. Find the angle it subtends at the centre, in rad.
   Show answer

   1.5 rad. θ = s / r = 0.45 / 0.30 = 1.5 rad.

4. A fan blade completes one revolution every 0.20 s. Find its angular velocity in rad/s, to 3 s.f.
   Show answer

   31.4 rad/s. ω = 2π / T = 2π / 0.20 = 31.4 rad/s.

5. The Earth rotates once every 24 hours. Find its angular velocity, in units of 10^-5 rad/s, to 3 s.f.
   Show answer

   7.27 x 10^-5 rad/s. ω = 2π / (24 x 3600) = 2π / 86400 = 7.27 x 10^-5 rad/s.

6. Using ω = 7.27 x 10^-5 rad/s and an Earth radius of 6.37 x 10⁶ m, find the speed of a point on the equator in m/s, to 3 s.f.
   Show answer

   463 m/s. v = ωr = 7.27 x 10^-5 x 6.37 x 10⁶ = 463 m/s.

7. A disc spins at a steady rate. How do a point near the rim and a point near the centre compare?
   • Same speed; the rim point has a greater angular velocity
   • Same angular velocity; the rim point has a greater speed
   • Same speed and same angular velocity
   • The centre point has a greater speed
   Show answer

   Same angular velocity; the rim point has a greater speed. Both sweep the same angle per second, but v = ωr is larger at larger r.

8. What is one radian?
   • The angle at the centre of a circle subtended by an arc equal in length to the radius
   • One full turn
   • The angle subtended by an arc equal to the diameter
   • Exactly 60°
   Show answer

   The angle at the centre of a circle subtended by an arc equal in length to the radius. θ = s / r = 1 when s = r. One radian is about 57.3°.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)
• Pearson Edexcel IAL Physics Unit 4 (WPH14) question paper, June 2024: list of data, formulae and relationships

Last updated 7 November 2026. Spotted a mistake? Tell us.