Velocities and Accelerations of Points in Fixed-Axis Rotation

During rotation of a rigid body about a fixed axis, all points share the same angular velocity $\omega$ and angular acceleration $\varepsilon$, but their linear velocities and accelerations depend on perpendicular distance from the axis.

Linear velocity

For a point of a rigid body at perpendicular distance $r$ from the fixed rotation axis, the linear speed is:

$$v=\omega r.$$

The velocity vector is tangent to the circle traced by the point and perpendicular to the radius $r$.

The farther a point is from the axis, the greater its speed for the same $\omega$. Points located directly on the axis have $r=0$ and remain fixed.

Vector velocity relation

For a point whose position from the axis is represented by $\vec r$, velocity can be written as $\vec v=\vec\omega\times\vec r$. The cross product automatically gives the tangential direction according to the right-hand rule.

Tangential acceleration

A change in speed produces the tangential component $a_\tau=\varepsilon r$. It is tangent to the circular path, with direction determined by the sign of angular acceleration.

Normal acceleration

The change in velocity direction produces the normal component $a_n=\omega^2r=v^2/r$, directed from the point toward the rotation axis.

Total acceleration

The tangential and normal components are perpendicular, so $a=\sqrt{a_\tau^2+a_n^2}=r\sqrt{\varepsilon^2+\omega^4}$. In vector form, $\vec a=\vec\varepsilon\times\vec r+\vec\omega\times(\vec\omega\times\vec r)$.

Example

A point on a disk is $r=0.20$ m from the axis. With $\omega=10$ rad/s and $\varepsilon=4$ rad/s², $v=2$ m/s, $a_\tau=0.8$ m/s², $a_n=20$ m/s², and $a\approx20.02$ m/s².

Common mistakes

• using distance from the body center instead of perpendicular distance from the axis;
• confusing $a_\tau=\varepsilon r$ with $a_n=\omega^2r$;
• assuming point acceleration is zero when $\omega$ is constant;
• adding tangential and normal accelerations algebraically instead of vectorially.