3.1 Angular velocity $\omega $

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
3.1 Angular velocity $\omega $
Consider a rigid body rotating about an axis of rotation (AOR). As the rigid body rotates in the $x-y$ plane, so the axis of rotation is parallel or along the $z-$ axis.

Assume a particle $P$ on the rigid body at a perpendicular distance $r$ from the axis of rotation.

The position of the particle at any time $t$ is described by an angle $\theta $. So, the angle $\theta $ is known as the angular position of the particle about the point $O$.

As the rigid body rotates about an axis of rotation. The particle $P$ moves in a circle with center at the point $O$ in the $x-y$ plane.

If a rigid body rotates through an angle $\Delta \theta $ during the time interval $\Delta t$. Then the average angular velocity $\left( {{\omega _{avg}}} \right)$ during the time interval $\Delta t$ is given by, $${\overrightarrow \omega _{avg}} = \frac{{\Delta \overrightarrow \theta }}{{\Delta t}}$$ or $${\overrightarrow \omega _{avg}} = \frac{{\Delta \theta }}{{\Delta t}}\widehat k$$

Note: According to the right-hand thumb rule, anti-clockwise direction is towards $\widehat k$.

Also, the instantaneous angular velocity $\left( {\overrightarrow \omega } \right)$ at any time $t$ is given by, $$\overrightarrow \omega = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \overrightarrow \theta }}{{\Delta t}}$$ or $$\overrightarrow \omega = \frac{{d\overrightarrow \theta }}{{dt}}$$
Therefore, the angular velocity $\left( {\overrightarrow \omega } \right)$ is defined as the time rate change of angular displacement.

The instantaneous angular velocity is defined as the time rate change of angular displacement at any instant of time.

The angular velocity $\left( {\overrightarrow \omega } \right)$ is about the axis of rotation (AOR) and is same for all the particles.

Note:
  • Angular velocity is a vector quantity
  • Direction of angular velocity is given by right-hand thumb rule
    • If a body rotates in anti-clock wise direction, the angular velocity is towards positive $z$-axis i.e. $\left( {\widehat k} \right)$
    • If a body rotates in clock wise direction, the angular velocity is towards negative $z$-axis i.e. $\left( { - \widehat k} \right)$
  • The magnitude of angular velocity is known as angular speed
  • Direction of angular velocity is perpendicular to the plane in which the particle moves
  • Unit of angular velocity is $radian/sec$ or $rad/s$
  • Dimensional formula of angular velocity is $\left[ {{M^0}{L^0}{T^{ - 1}}} \right]$
  • All the particles on the rigid body moves in a circle with same angular velocity
If the body rotates through equal angles in equal interval of time, then the angular velocity is known as uniform angular velocity.

Mathematical equation of uniform angular velocity, $$\overrightarrow \omega = \frac{{d\overrightarrow \theta }}{{dt}} = {\text{constant}}$$ or $$\overrightarrow \omega dt = d\overrightarrow \theta $$
Integrating the above equation for proper limits we get, $$\begin{equation} \begin{aligned} \overrightarrow \omega \int\limits_0^t {dt} = \int\limits_0^\theta {d\overrightarrow \theta } \\ \overrightarrow \omega \left[ t \right]_0^t = \left[ {\overrightarrow \theta } \right]_0^\theta \\ \overrightarrow \omega (t - 0) = \left( {\overrightarrow \theta - 0} \right) \\\end{aligned} \end{equation} $$ So, $$\overrightarrow \theta = \overrightarrow \omega t$$
Improve your JEE MAINS score
10 Mock Test
Increase JEE score
by 20 marks
Detailed Explanation results in better understanding
Exclusively for
JEE MAINS and ADVANCED
9 out of 10 got
selected in JEE MAINS
Lets start preparing
Read Complete Chapter
DIFFICULTY IN UNDERSTANDING CONCEPTS?
TAKE HELP FROM THINKMERIT DETAILED EXPLANATION..!!!
9 OUT OF 10 STUDENTS UNDERSTOOD
START LEARNING