3.2 Angular acceleration $\left( \alpha \right)$

Physics > Basics of Rotational Motion > 3.0 Kinematics of a plane motion

  Basics of Rotational Motion
    1.0 Rigid body
    2.0 Motion of rigid body
    3.0 Kinematics of a plane motion
    4.0 Moment of inertia
    5.0 Radius of gyration $(K)$
    6.0 Theorems of moment of inertia
    7.0 Moment of inertia of uniform continious rigid bodies
3.2 Angular acceleration $\left( \alpha \right)$
It is defined as the time rate of angular velocity.

If a body rotates through unequal angles in equal interval of time, then the motion is known as accelerated angular motion.

Mathematically, $$\overrightarrow \alpha = \frac{{d\overrightarrow \omega }}{{dt}}$$
During the rotation of the rigid body, angular velocity $\left( \omega \right)$ increases by $\Delta \omega $ during the time interval $\Delta t$.

So, the average angular acceleration $\left( {{{\overrightarrow a }_{avg}}} \right)$ during the time interval $\Delta t$ can be written as, $${\overrightarrow \alpha _{avg}} = \frac{{\Delta \overrightarrow \omega }}{{\Delta t}}$$
Also, the instantaneous angular acceleration $\left( {\overrightarrow \alpha } \right)$ at any time $t$ is given by, $$\begin{equation} \begin{aligned} \overrightarrow \alpha = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \overrightarrow \omega }}{{\Delta t}} \\ \overrightarrow \alpha = \frac{{d\overrightarrow \omega }}{{dt}} \\\end{aligned} \end{equation} $$
Let us assume two different cases,

Case 1: Accelerated angular motion
Case 2: Deaccelerated angular motion
Case 1:

In case of accelerated angular motion, angular velocity increases. Mathematically, $${\omega _f} > {\omega _i}$$ where,
${\omega _f}$: final angular velocity
${\omega _i}$: initial angular velocity

So, $${\omega _f} - {\omega _i} > 0$$ or $$d\omega > 0\quad ...(i)$$
We can write angular acceleration as, $$\overrightarrow \alpha = \frac{{d\overrightarrow \omega }}{{dt}}\quad ...(ii)$$ Therefore from equation $(i)$ & $(ii)$ we get, $$\begin{equation} \begin{aligned} \overrightarrow \alpha = \left( { + \frac{{d\omega }}{{dt}}} \right)\widehat k \\ \overrightarrow \alpha = \frac{{d\omega }}{{dt}}\left( { + \widehat k} \right) \\\end{aligned} \end{equation} $$
Angular acceleration $\left( {\overrightarrow \alpha } \right)$ is in the anti-clockwise direction as shown in the figure.

  • Case 2:

    In the case of de-accelerated angular motion, angular velocity decreases. Mathematically, $${\omega _f} < {\omega _i}$$ where,
    ${\omega _f}$: final angular velocity
    ${\omega _i}$: initial angular velocity

    So, $${\omega _f} - {\omega _i} < 0$$ or $$d\omega < 0\quad ...(i)$$
    We can write angular acceleration as, $$\overrightarrow \alpha = \frac{{d\overrightarrow \omega }}{{dt}}\quad ...(ii)$$ Therefore from equation $(i)$ & $(ii)$ we get, $$\begin{equation} \begin{aligned} \overrightarrow \alpha = \left( { - \frac{{d\omega }}{{dt}}} \right)\widehat k \\ \overrightarrow \alpha = \frac{{d\omega }}{{dt}}\left( { - \widehat k} \right) \\\end{aligned} \end{equation} $$
    Angular acceleration $\left( {\overrightarrow \alpha } \right)$ is in the clockwise direction as shown in the figure.

    Note:


    • Angular acceleration is a vector quantity
    • Direction of angular acceleration is given by right hand thumb rule
      • If the angular acceleration is in anti-clockwise direction then angular acceleration is towards positive $z$-axis i.e. $\left( {\widehat k} \right)$
      • If the angular acceleration is in clockwise direction then angular acceleration is towards negative $z$-axis i.e. $\left( { - \widehat k} \right)$
    • Direction of angular acceleration is perpendicular to the plane in which the particle moves
    • Unit of angular acceleration is $radian/sec^2$ or $rad/s^2$
    • Dimensional formula of angular acceleration is $\left[ {{M^0}{L^0}{T^{ - 2}}} \right]$
    • All the particles on the rigid body moves in a circle with same angular acceleration
    • For a particle moving in a circle on a rigid body, angular velocity $\left( {{{\overrightarrow \omega }_0}} \right)$ or angular accleration$\left( {{{\overrightarrow \alpha }_0}} \right)$ about centre of a circle is double the angular velocity $\left( {{{\overrightarrow \omega }_C}} \right)$ or angular acceleration $\left( {{{\vec \alpha }_C}} \right)$ about any point on the circumference of the circle $${\overrightarrow \omega _0} = 2{\overrightarrow \omega _C}\quad \& \quad {\overrightarrow \alpha _0} = 2{\overrightarrow \alpha _C}$$


    • When the angular acceleration $\left( {\overrightarrow \alpha } \right)$ and the angular velocity $\left( {\overrightarrow \omega } \right)$ is in same direction, then the motion is said to be accelerated rotational motion
    • When the angular acceleration $\left( {\overrightarrow \alpha } \right)$ and the angular velocity $\left( {\overrightarrow \omega } \right)$ is in opposite direction, then the motion is said to be de-accelerated rotational motion
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