Kinetic Energy in rotational motion

Table Of Contents

Kinetic Energy Introduction
Formula of Kinetic energy in translational and rotational motion
Expression of Kinetic energy in rotational motion

Kinetic Energy Introduction

Kinetic energy represents the energy possessed by the body by virtue of its motion. Whenever a body is in motion, it possesses kinetic energy. The kinetic energy of a body in linear motion is given by: $K.E = \frac{1}{2}mv^2$ where $m$ and $v$ are the mass and velocity of a body respectively.

In a similar way, the motion of a body in rotational motion also has kinetic energy. The kinetic energy in rotational motion is given by: $K.E = \frac{1}{2}I \omega^2$ where $I$ and $\omega$ are the moment of inertia and angular velocity of a body respectively.

Formula of Kinetic energy in translational and rotational motion

The formula of kinetic energy in translational and rotational motion is shown in the below table.

Expression of Kinetic energy in rotational motion

Consider a rigid body in a rotational motion with constant angular velocity $\omega$ . Let’s say it is made up of ‘n’ number of particles of masses $m_1, m_2, m_3, \ldots, m_n$ all have the same angular velocity $\omega$ as they are compactly arranged in a rigid body. But they have different linear velocities as $v_1, v_2, v_3, \ldots, v_n$ .

Since, the kinetic energy of first particle is = $\frac{1}{2} m_1 v_1^2$

The total kinetic energy in rotational motion of a rigid body can be calculated by adding the kinetic energy of all particles that form a rigid body.

$\text{Kinetic Energy of rigid body} = \frac{1}{2}m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \frac{1}{2} m_3 v_3^2 + \ldots + \frac{1}{2} m_n v_n^2$

But $v = \omega r$ So,

$\text{K.E} = \frac{1}{2}m_1 (\omega r_1)^2 + \frac{1}{2} m_2 (\omega r_2)^2 + \frac{1}{2} m_3 (\omega r_3)^2 + \ldots + \frac{1}{2} m_n (\omega r_n)^2$

$\text{K.E} = \frac{1}{2}m_1 \omega^2 r_1^2 + \frac{1}{2} m_2 \omega^2 r_2^2 + \frac{1}{2} m_3 \omega^2 r_3^2 + \ldots + \frac{1}{2} m_n \omega^2 r_n^2$

$\text{K.E} = \omega^2 \left( \frac{1}{2}m_1 r_1^2 + \frac{1}{2} m_2 r_2^2 + \frac{1}{2} m_3 r_3^2 + \ldots + \frac{1}{2} m_n r_n^2 \right)$

$\text{K.E} = \omega^2 \sum_{i=1}^n m_i r_i^$

$\text{K.E} = \omega^2 I$

where $I = \sum_{i=1}^n m_i r_i^2$ is a moment of inertia of a rigid body.

$\boxed{\text{K.E} = I \omega^2}$

This is the formula of Kinetic Energy in rotational motion.

Sharing is Caring

Translational Motion	Rotational Motion
$\boxed{K.E = \frac{1}{2} m v^2}$ $m$ = mass of a body $v$ = linear velocity of a body	$\boxed{K.E = \frac{1}{2} I \omega^2}$ $I$ = Moment of Inertia $\omega$ = angular velocity

Kinetic Energy Introduction

Formula of Kinetic energy in translational and rotational motion

Expression of Kinetic energy in rotational motion

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