Angular displacement

Beginner

Angle swept, measured in radians.

Key relationθ = s/r

θ = s/r in radians measures the turn. A full circle is 2π. The right-hand rule gives it a direction along the axis for small rotations.

Worked example

An arc of 6 m on a radius of 3 m gives 2 rad.

Common mistake

Finite rotations in 3D don't add like vectors.

In more detail

Angular displacement is the angle through which a body rotates, measured in radians. It is the rotational counterpart of linear displacement.

A clock hand moves through an angular displacement every hour.
Car wheel rotation is measured by counting angular displacement with sensors.
A GPS receiver or a compass reading uses angles swept out by a rotating body, measured in radians rather than degrees.

How many radians is a full turn?

2π radians, about 6.28.

Is angular displacement a vector?

For rotation about a fixed axis it is treated as a signed scalar, though in general it is a vector.

Why are radians used in physics?

They connect angles directly to arc length, so formulas such as s = rθ stay simple.