Basics of Rotational Motion
4.0 Moment of inertia
4.0 Moment of inertia
Moment of inertia of a rigid body about an axis of rotation is defined as the sum of product of mass $\left( {{m_i}} \right)$ and square of the perpendicular distance $\left( {{r_i}} \right)$ from an axis of rotation for all the particles of the body.
It is denoted by symbol $I$.
Moment of inertia for a single particle $$I = m{r^2}$$
Moment of inertia $(I)$ for a system of $N$ particles $$I = \sum\limits_{i = 1}^N {{m_i}r_i^2} $$
Moment of inertia $(I)$ for a continuous rigid body $$I = \int_m {{r^2}dm} $$
where,
$\int_m : $ Integrating over the entire mass
$r$: perpendicular distance from an axis of rotation (AOR)
$dm$: infinitesimally small section of mass
Moment of inertia $(I)$ about an axis for a system of particles is equal to the sum of the moment of inertia of individual particles about that axis of rotation.
Mathematically, $$I = {I_1} + {I_2} + ... + {I_n}$$
Moment of inertia depends upon,
1. Position of an axis of rotation
2. Orientation of an axis of rotation
3. Shape of the body
4. Size of the body
5. Distribution of mass of the body about an axis of rotation
- Significance of moment of inertia
The role of moment of inertia in the study of rotational motion is same as that of mass in the study of linear motion
It is more difficult to rotate a body at rest having larger moment of inertia
Similarly, it is more difficult to stop a rotating body which has larger moment of inertia
Note:- Moment of inertia is a scalar quantity
- Its SI unit is $kg-m^2$
- Dimensional formula is $\left[ {{M^1}{L^2}{T^0}} \right]$
