Relation Between Linear and Angular Quantities

The connection between rotation and linear motion

A point at distance $r$ from the axis of rotation moves in a circle. Its linear motion is connected to the rotation of the body by three key relations.

Linear velocity

Arc length: $s = r\theta$

Differentiate with respect to time:

$\frac{ds}{dt} = r\frac{d\theta}{dt}$

$\boxed{v = r\omega}$

The linear speed of a point is proportional to its distance from the axis. Points farther from the axis move faster — this is why the rim of a wheel moves faster than points closer to the hub.

Tangential acceleration

Differentiate $v = r\omega$ with respect to time:

$\frac{dv}{dt} = r\frac{d\omega}{dt}$

$\boxed{a_t = r\alpha}$

Tangential acceleration is the rate of change of speed — it is directed tangentially (along the direction of motion). It arises from angular acceleration.

Centripetal acceleration

A point moving in a circle always has centripetal acceleration directed toward the axis:

$\boxed{a_c = r\omega^2 = \frac{v^2}{r}}$

This acceleration changes the direction of velocity (keeps the point moving in a circle), not its magnitude.

Total acceleration

The total acceleration of a point on a rotating body:

$a = \sqrt{a_t^2 + a_c^2} = \sqrt{(r\alpha)^2 + (r\omega^2)^2} = r\sqrt{\alpha^2 + \omega^4}$

Direction: at angle $\phi = \tan^{-1}\left(\frac{a_t}{a_c}\right) = \tan^{-1}\left(\frac{\alpha}{\omega^2}\right)$ from the radial direction.

Different points on the same body

All points on a rigid body have the same $\omega$ and $\alpha$ at any instant — this is what makes it a rigid body.

But their linear speeds and accelerations differ:

Vector form

In vector notation:

$\vec{v} = \vec{\omega} \times \vec{r}$

where $\vec{\omega}$ is along the rotation axis and $\vec{r}$ is from the axis to the point.