5.2 Connecting Linear and Rotational Motion

In rotational motion, any point on a rotating body moves in a circular path around the axis. Although the motion can be described angularly, each point also experiences linear motion. The distance from the axis to the point is denoted by ${r}$.

By definition of a radian: ${s = r \theta}$

${s}$ = linear (arc) distance traveled

${r}$ = radius from axis of rotation

${\theta}$ = angular displacement (radians)

From this relationship, we can directly connect angular and linear quantities:

Linear position: ${s = r \theta}$

Linear velocity: ${v = r \omega}$

Linear acceleration: ${a = r \alpha}$

These relationships show that the farther a point is from the axis of rotation, the greater its linear velocity and linear acceleration, even if ${\omega}$ and ${\alpha}$ are the same for the entire rigid body.