Angular Acceleration

Opening the Idea of Angular Acceleration

When a rigid body rotates, its angular velocity $\omega$ can change with time — just as a car’s linear velocity changes when it accelerates. The quantity that measures how fast $\omega$ changes is called angular acceleration. It is the rotational analogue of linear acceleration.

For a particle moving along a circle of radius $r$, the linear velocity $v$ and angular velocity $\omega$ are related by $v = r\omega$. If $\omega$ changes, $v$ changes too, and the linear acceleration $a$ has two parts: one due to the change in speed (tangential) and one due to the change in direction (centripetal). Angular acceleration directly governs the tangential part.

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Definition of Angular Acceleration

Consider a rigid body rotating about a fixed axis. Let its angular velocity at time $t$ be $\omega(t)$ and at a slightly later time $t + \Delta t$ be $\omega(t + \Delta t)$. The average angular acceleration over the interval $\Delta t$ is

The instantaneous angular acceleration is the limit of this ratio as $\Delta t \to 0$:

Angular acceleration is a vector quantity. Its direction is along the axis of rotation, given by the right-hand rule: if $\omega$ increases, $\alpha$ points in the same direction as $\omega$; if $\omega$ decreases, $\alpha$ points opposite to $\omega$. For fixed-axis rotation, we often treat it as a signed scalar (positive if $\omega$ increases in the chosen sense, negative if it decreases). The SI unit is $\text{rad/s}^2$.

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Relation Between Angular Acceleration and Linear Acceleration

For a particle at a perpendicular distance $r$ from the axis, its linear speed is $v = r\omega$. Differentiating with respect to time:

Here $a_t$ is the tangential acceleration — the component of linear acceleration that changes the magnitude of velocity (the speed). It is tangent to the circular path.

But the particle also has a centripetal (radial) acceleration $a_c = \frac{v^2}{r} = r\omega^2$, which changes the direction of velocity. The total linear acceleration vector $\vec{a}$ is the vector sum:

with magnitudes $a_t = r\alpha$ and $a_c = r\omega^2$. Since these two components are perpendicular (tangential and radial), the magnitude of total acceleration is

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Properties of Angular Acceleration (as listed in the textbook)

The textbook presents three key properties that follow from the definition and the relation $v = r\omega$. Each is derived below.

›Proof

Property 1: If the angular velocity is constant, angular acceleration is zero.

If $\omega = \text{constant}$, then $\frac{d\omega}{dt} = 0$, so $\alpha = 0$. This is the rotational analogue of constant linear velocity implying zero linear acceleration. In uniform circular motion ($\omega$ constant), the tangential acceleration is zero, but the centripetal acceleration $r\omega^2$ is non-zero — the particle still accelerates because its direction changes.

›Proof

Property 2: The direction of angular acceleration is the same as the direction of the change in angular velocity.

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