Velocity and acceleration

Velocity in circular motion

We have seen that velocity in circular motion is always tangential (perpendicular to the radius drawn to the current position). It changes its direction from point to point.

In uniform circular motion, the magnitude of the velocity (the speed) remains constant; in non-uniform circular motion, the magnitude also changes.

The relationship between velocity and speed:
\[\mathbf{v} = \lim\limits_{\Delta t \to 0}\frac{\Delta \mathbf{r}}{\Delta t} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t},\]

\[v = |\mathbf{v}| = \frac{\mathrm{d}s}{\mathrm{d}t},\]

where \(s\) is the arc swept by the object.

Speed can be expressed in terms of angular frequency:
\[v = \frac{\mathrm{d}s}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\left(r \phi\right) = r \frac{\mathrm{d}\phi}{\mathrm{d}t} = r \omega.\]

Centripetal acceleration

The acceleration vector in uniform motion is always directed towards the centre of the circle. We call this type of acceleration centripetal acceleration (from Latin, meaning ‘centre-seeking’). It is also called radial acceleration, because it is in the direction of the radius.

Notations: \(\mathbf{a}_r\), \(\mathbf{a}_{\mathrm{cp}}.\)

The magnitude of the centripetal acceleration depends on the speed (or the angular speed) and on the radius:

\[a_r = \frac{v^2}{r} = \frac{\omega^2 r^2}{r} = \omega^2 r.\]

In non-uniform circular motion, the centripetal component is still present, but there is a tangential acceleration component as well.

Tangential acceleration

If the circular motion is not uniform, there is a tangential acceleration component present in addition to the centripetal acceleration.

The tangential acceleration is always perpendicular to the radius and its magnitude is the rate of change of the speed:
\[a_{\mathrm{t}} = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}|\mathbf{v}|}{\mathrm{d}t} \neq \left|\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}\right|.\]

Tangential acceleration can be related to angular acceleration:

\[a_{\mathrm{t}} = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\left(r \omega\right) = r \frac{\mathrm{d}\omega}{\mathrm{d}t} = r \alpha.\]