P01-230 – Instantaneous Angular Velocity

Instantaneous Angular Velocity

The average angular velocity vector gives some information regarding how fast the particle rotates along a circular path, on average, over a finite amount of time. However, it fails to adequately describe the angular velocity of the particle at any given instant as doing so requires letting $\mathrm{\Delta }t\to 0$. Indeed, this limit shrinks the time intervals between two consecutives measurements to an infinitely small amount thus providing, in a sense, a video of the particle’s motion rather than a sequence of snapshots.

With the help from calculus, we define the instantaneous angular velocity as the following limit

\boxed{\omega ={\mathop{\mathrm{lim}}_{\mathrm{\Delta }t\to 0} \frac{\mathrm{\Delta }\theta }{\mathrm{\Delta }t}\ }=\frac{d\theta }{dt}\ \ \ \ \ \ \ \ \left(rad/s\right)}

where such a limit is, by definition, equal to the derivative of $\theta $ with respect to $t$ and is denoted by $\displaystyle{\frac{d\theta }{dt}}$.

Note: it is common to simply call the instantaneous angular velocity of a particle its angular velocity.

Example: consider a particle moving along a circular path with an angular position given by $\theta \left(t\right)=-t^2+3t+2$. Its instantaneous angular velocity is then given by

\omega \left(t\right)=-2t+3\ \ \ \ \ \left(rad/s\right)

This yields a function of time for $\omega \left(t\right)$ which will output the instantaneous angular velocity of the particle at any instant $t$ of its motion.