Angular acceleration

Angular acceleration $\alpha$ measures how rapidly the angular velocity of a rotating body changes. It is the exact rotational analogue of linear acceleration: where in rectilinear motion the acceleration tells how quickly $v$ varies, in rotational motion $\alpha$ tells how quickly $\omega$ varies.

$\alpha = \frac{\Delta\omega}{\Delta t}$

It is measured in radians per second squared ($\mathrm{rad/s^2}$). If $\alpha$ is constant, the rotational motion follows kinematic equations identical to those of uniformly accelerated motion, with $\theta$ in place of $s$, $\omega$ in place of $v$ and $\alpha$ in place of $a$.

Translation / rotation analogies

$v \leftrightarrow \omega \qquad a \leftrightarrow \alpha \qquad m \leftrightarrow I \qquad F \leftrightarrow M \qquad p \leftrightarrow L$

This correspondence table is the key to the whole chapter: every translational formula has a rotational twin obtained by replacing each symbol with its counterpart.

The two accelerations of a point on the body

A point mass located at distance $r$ from the axis of rotation does not have a single acceleration, but two distinct contributions, because its velocity is a vector that can change both in magnitude and in direction:

• tangential acceleration $a_t = \alpha\,r$ — changes the magnitude of $\vec{v}$ (the point goes faster or slower along the circumference);
• centripetal acceleration $a_c = \omega^2 r$ — changes the direction of $\vec{v}$ (the point curves, staying on the circumference).

Key formula

$a_t = \alpha\,r \qquad\qquad a_c = \omega^2 r$

The intuition is that the farther the point is from the axis, the faster it is and the more it accelerates tangentially for the same $\alpha$: it’s the same reason the tip of a helicopter blade moves much faster than the hub, despite having the same angular velocity.

Links

Topics: Dinamica rotazionale Concepts: Accelerazione angolare · Velocità angolare Methods: Analogia traslazione-rotazione

Related exercises: Problem — Which ω(t) for a braked disc · Problem — Wheel at 40 rpm · Problem — Double-radius pulley