We can obtain a general idea of the rotation of rigid bodies and compare it with real situations or we can consider some real bodies as nearly rigid to obtain our results. We first derive an expression of kinetic energy of a rotating rigid body and define moment of inertia later.
It means we first find the kinetic energy of a rotating rigid body and compare it with the kinetic energy of a particle in linear motion - where we can find the measure of mass of a rotating body called moment of inertia.
What is kinetic energy of a rotating rigid body?
Consider a rigid body which rotates about y-axis as shown in Figure 1. The body is made up of large number of particles of masses \(m_1\), \(m_2\), \(m_3\) and so on. The mass \(m_1\) is at a distance \(r_1\) and the mass \(m_2\) is at a distance \(r_2\) from the axis of rotation and so on. When the body rotates, each particle of the body moves in its own circle of a particular radius centred on the axis.
The angular velocity \(\omega\) of all particles is the same but the magnitude of linear velocity (linear speed) is not the same for all particles. That's because a particle at larger distance from the axis of rotation of the body needs to move along a circle of larger radius but the time is the same for all particles to complete one rotation that is, the angular velocity for all particles is constant.
Let the particle of mass \(m_1\) rotates with linear speed \(v_1\) and the particle of mass \(m_2\) rotates with linear speed \(v_2\), and so on. Now the kinetic energy of any particle of mass \(m_i\) rotating with its linear speed \(v_i\) is \(\frac{1}{2}{m_i}{v_i}^2\).
So the total kinetic energy of the rotating rigid body is the sum of the kinetic energies of all particles rotating in the body. Suppose there are \(n\) particles and \(K_i\) is the kinetic energy of any particle where \(i\) can vary from 1 to \(n\). The total kinetic energy \(K\) of the rotating body is
\[K = \sum\limits_{i = 1}^n {{K_i}} = \sum\limits_{i = 1}^n {\frac{1}{2}{m_i}{v_i}^2} \tag{1} \label{1}\]
You know that \(v = \omega r\). Since angular velocity is constant, the linear speed \(v_i\) is \(v_i = \omega r_i\). So the above expression becomes,
\[K = \frac{1}{2}\left( {\sum\limits_{i = 1}^n {{m_i}{r_i}^2} } \right){\omega ^2} \tag{2} \label{2}\]
You know the kinetic energy of a particle of mass \(m\) moving with linear speed \(v\) is \(\frac{1}{2}m{v^2}\). So by comparison of \(\frac{1}{2}m{v^2}\) for linear motion with Eq. \eqref{2} for rotational motion, the quantity \({\sum\limits_{i = 1}^n {{m_i}{r_i}^2} }\) is the measure of mass of the rotating body called moment of inertia. The moment of inertia \(I\) of the rotating rigid body is
\[I = \sum\limits_{i = 1}^n {{m_i}{r_i}^2} \]
And we can write Eq. \eqref{2} as,
\[K = I{\omega ^2} \tag{3} \label{3}\]
The kinetic energy of a rotating rigid body about an axis is defined in terms of the moment of inertia and the angular speed. If you think a single particle rotating along a circle of radius \(r\) with linear speed \(v\), you can easily write the kinetic energy of the particle as \(\frac{1}{2}m{v^2}\) but you know \(v = \omega r\) and the kinetic energy becomes \(\frac{1}{2}m{r^2}{\omega ^2} = \frac{1}{2}I{\omega ^2}\) where \(I\) for the single particle is \(m{r^2}\).
In words you can define the moment of inertia of a body rotating about an axis as the product of its mass and square of its distance from the axis of rotation.
If a body is not only a point mass but a rigid body of uniform mass distribution, you can not simply take the body as a single point mass but instead you have to define a particular element of the body and use the integration to find the moment of inertia of the whole body.
In fact you can not tell exactly how much moment of inertia a particular body has. It means you can make as many axes as possible about which a rigid body can rotate. So the moment of inertia of the same body is different for different rotation axis. That's the reason why you have to tell about which axis the rigid body rotates.
Moment of inertia is not constant for non-rigid bodies even if the rotation axis is the same and it is because the distance of particles of non-rigid body from the axis of rotation varies.
