Kinematics of circular motion: (a) the radian as a measure of angle

The idea

The radian is the SI unit of angle, defined so that one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Because the circumference is 2πr, a full turn is 2π radians, so 360° = 2π rad. To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. For example, 90° = π/2 rad ≈ 1.57 rad. Using radians makes arc length s = rθ and angular speed ω = Δθ/Δt valid directly, without conversion factors, which is why circular-motion equations assume angles in radians.

What earns the marks

• Defines the radian as the angle subtended by an arc equal in length to the radius.
• States that a full circle is 2π radians, so 360° = 2π rad.
• Converts between degrees and radians using π/180 and 180/π respectively.
• Explains that arc length s = rθ requires θ in radians.
• Applies radian measure to angular quantities such as angular speed ω = Δθ/Δt.

Common mistakes and how to put them right

• Treating a full circle as π radians.

  Put it right: A full turn is 2π radians.

• Substituting degrees into s = rθ.

  Put it right: Convert the angle to radians first.

• Multiplying by 180/π when converting degrees to radians.

  Put it right: Multiply degrees by π/180.

• Assuming the radian is a dimensional unit like the metre.

  Put it right: It is a dimensionless ratio of two lengths.

Exam tips

• Write the conversion factor π/180 explicitly before substituting an angle in degrees.
• Check whether a calculator is in radian or degree mode before evaluating trigonometric functions in circular-motion work.
• Use s = rθ only after confirming θ is in radians.
• Quote angles in radians to an appropriate number of significant figures, often as multiples of π.