Definition of Moment of Inertia
Moment of Inertia is one of the frequent topics in Mechanics. We use this term for the rotational motion and works like a mass in a translational motion. But how?? Let’s see.
Inertia is the inability of a body to change its state by itself. for. eg A object at rest cannot come to a motion by itself or an object moving within a uniform motion in a straight line in a space cannot come to rest by itself, a child who kept a doll in one place will come again in that place to find the doll because doll will remain at rest in the same place.
According to Newton’s first law of motion, An external agent i.e. force is required to change the state of an object. But the force to change the state of an object depends upon the mass of an object. The greater mass means, the greater force is required to change the state, and hence the higher inability to change its state by itself, which means higher inertia. So, mass is the measurement of inertia.
In linear motion, the measurement of mass is the measurement of inertia
The rotational motion also has rotational inertia. Torque is required to produce an angular acceleration to a rigid body rotating about an axis with a uniform angular velocity. But the torque required doesn’t depend only upon the mass of a rigid body but also on the distribution of masses in a rigid body. So inertia for the rotational body depends upon both masses as well as the distribution of masses. That inertia is termed a moment of inertia for a rotational body.
Moment of inertia in rotational motion is analogous to mass in linear motion.
Expression for the moment of inertia
let us consider the rigid body is made up of 
number of particles of masses 
are at a distance of 
as shown in the figure.
The moment of inertia of a particle rotating about an axis can be defined as the product of the mass and the square of the perpendicular distance of a particle from an axis of rotation.
Hence, the moment of inertia of all particles about the given axis of rotation becomes 
.
Then Moment of inertia of a rigid body about an axis can be found by adding the moment of inertia of all particles about that axis.
moment of inertia = sum of moment of inertia of all particles
![\[I = m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + ……. + m_nr_n^2 \]](https://anylearnnepal.com/wp-content/ql-cache/quicklatex.com-21242b7fe070eee09e244ca1f1efd574_l3.png "Rendered by QuickLaTeX.com")
![\[\boxed{I = \sum_{i=1}^n m_ir_i^2}\]](https://anylearnnepal.com/wp-content/ql-cache/quicklatex.com-d5497cd36f50d1f0672c8a5c5c1fb975_l3.png "Rendered by QuickLaTeX.com")
Points to remember about Moment of Inertia
- The moment of inertia is a scalar quantity.
- The moment of inertia in rotatory motion is analogous to mass in translational motion.
- The moment of inertia depends upon the mass as well as the perpendicular distance of the mass from an axis of rotation.
- It’s unit is
. - It’s dimension is
![\left[ M^1L^2T^0 \right]](https://anylearnnepal.com/wp-content/ql-cache/quicklatex.com-6b39c32d04628e11d450af23a2359ba2_l3.png "Rendered by QuickLaTeX.com")
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