PNC-01-230 – Instantaneous Angular Velocity

Instantaneous Angular Velocity

The average angular velocity describes 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 because the information between time $t_i$ and $t_f=t_i+\mathrm{\Delta }t$ is not recorded. To define the instantaneous angular velocity, the time interval $\mathrm{\Delta }t$ between two consecutives measurements must shrink to an infinitely small amount thus providing, in a sense, a video of the particle’s motion rather than a sequence of snapshots.

We define the instantaneous angular velocity as the angular velocity of an object at any instant $t$ and we denote it by $\omega \left(t\right)$. Mathematically, we achieve this by making the time interval $\mathrm{\Delta }t$ in the average angular velocity formula infinitely small (shrinking it to zero without every reaching zero) which turns the average angular velocity into the instantaneous angular velocity.

Properties of instantaneous angular velocity:

• A positive instantaneous angular velocity at an instant $t$ denotes motion in the positive direction of rotation at that given instant.

• A negative instantaneous angular velocity at an instant $t$ denotes motion in the negative direction of rotation at that given instant.