Angular Displacement
Simple Explanation
Angular displacement is the angle through which an object rotates about an axis, measured in radians. It plays the same role in rotational motion that ordinary displacement plays in straight-line motion.
Why Do We Need It?
Every other rotational quantity — angular velocity, angular acceleration — is built from angular displacement, so it is the natural starting point for describing any spinning or turning motion.
See It
A circle centered at O, with two radii drawn to a starting point and an ending point, the angle θ between them marked at the center, and the arc between the two points labelled s
Formula
Angular Displacement
θ = s / r
The angle swept out by a rotating object, measured in radians, found from the arc length traveled divided by the radius.
- θ
- — angular displacement, in radians (rad)
- s
- — arc length traveled along the circular path, in metres (m)
- r
- — radius of the circular path, in metres (m)
When to use it: Whenever the arc length and radius of a circular path are known, and the angle swept (in radians) is needed.
Worked Example
Find an angular displacement from an arc length
A point on the rim of a merry-go-round of radius 4 m travels an arc length of 6 m. Find its angular displacement.
Why Does This Work?
A full circle has circumference 2πr and sweeps through a full angle of 2π radians — so the ratio of arc length to radius, s/r, always gives exactly the fraction of a full circle traveled, expressed directly in radians.
Real-Life Example
How far a Ferris wheel car has turned
A rider tracks how far their Ferris wheel car has traveled around the wheel by comparing the arc length it has moved through to the wheel's radius.
The angular displacement θ=s/r tells the rider exactly what fraction of a full rotation they have completed, regardless of how big the wheel is.
Practice
A point on a wheel of radius 3 m travels an arc length of 9 m. Find its angular displacement in radians.
EasyCommon mistake
Assuming θ = s/r gives an answer in degrees — this formula always gives radians directly; converting to degrees requires multiplying by 180/π afterward.
Quick Review
- θ = s/r, always in radians.
- A full circle is 2π radians (360°).
- The rotational counterpart of ordinary (straight-line) displacement.