Moment Of Inertia - Practice Questions & MCQ

1. Definition

• Moment of inertia  (I) of a body is a measure of its ability to resist change in its rotational state of motion.

• Moment of inertia play the same role in rotatory motion as is played by mass in translatory motion .

2. Formula

• Moment of inertia of a particle

$
I=m r^2
$

Where $m$ is the mass of particle and $r$ is the perpendicular distance of particle from rotational axis.
- Moment of inertia for system of particle

$
\begin{aligned}
I & =m_1 r_1^2+m_2 r_2^2+\ldots \ldots \ldots m_n r_n^2 \\
& =\sum_{i=1}^n m_i r_i^2
\end{aligned}
$

(This is Applied when masses are placed discreetly)
- Moment of inertia for continuous body

$
I=\int r^2 d m
$

Where r is the perpendicular distance of a particle of mass dm of rigid body from axis of rotation
3. Dimension $=\left[M L^2\right]$
4. S.I. unit $=k g-m^2$

3. It  depends on mass, distribution of mass and on the position of the axis of rotation.

4. It does not depend on angular velocity, angular acceleration, torque, angular momentum and rotational kinetic energy.

5. It is a tensor quantity.

6. Radius of gyration (K)-

• Radius of Gyration of a body about an axis is the effective distance from the axis where the whole mass can be assumed to be concentrated so that moment of inertia remains the same.

• - Formula- $K=\sqrt{\frac{I}{M}}$
  or, $I=M K^2$
  - It does not depend on the mass of body
  - It depends on the shape and size of the body, distribution of mass of the body w.r.t. the axis of rotation etc.
  - Dimension- $M^o L^1 T^o$
  - S.I. unit : Meter.