angular displacement

Angular displacement answers a simple question about anything that turns: how far around has it gone? When the minute hand of a clock sweeps from the 12 to the 3, or a merry-go-round swings you a quarter of the way around, the amount it has rotated is its angular displacement. It is the spinning-world cousin of ordinary displacement — but instead of measuring a distance in metres, we measure a turn in an angle.

Precisely, angular displacement is the change in the rotation angle, written as a change in theta: delta-theta = theta_final - theta_initial. We usually measure it in radians (the natural unit of angle), where one full turn is 2 pi radians, the same as 360 degrees, and a half turn is pi radians. There is a neat link to the distance a point on the object actually travels: a point sitting a distance r from the axis moves along an arc of length s = r theta, provided theta is in radians.

One honest subtlety: for large turns, angular displacements do not add up like ordinary vectors — turning a book 90 degrees one way then another gives a different result depending on the order, so finite rotations do not commute. Very small (infinitesimal) rotations do behave like vectors, which is why angular velocity and angular acceleration can be treated as vectors pointing along the axis.