Question

Which of the following is true for moment of inertia (I) ?
(This question has multiple correct options)
A) If $I$ about an axis is minimum, then it must pass through the center of mass.
B) Moments of inertia about all axes passing through the center of mass have the same $I$.
C) Perpendicular axis theorem can’t be applied for $3$ dimensional body.
D) Parallel axis theorem can be applied for$\;3$ dimensional body.

Answer 1

To solve this question we need to know the perpendicular axis theorem, parallel axis theorem and moment of inertia about the center of mass .By applying the above theorem’s we will be able to get more than one option correct.

Complete step by step solution:
B) Moment of inertia is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation.
Moment of inertia about all axis passing through center of mass have different $I$

Therefore, option (B) is INCORRECT.

C) Perpendicular Axis Theorem: The moment of inertia of a $2$ dimensional object about an axis passing perpendicularly from it, is equal to the sum of the moment of inertia of the object about two mutually perpendicular axes lying in the plane of the object.
$I_{zz}{\text{ }} = {\text{ }}I_{xx}{\text{ }} + {\text{ }}I_{yy}$
Here,
$I_{zz}$ is the moment of inertia along $z{\text{ }}axis$
$I_{xx}$ is the moment of inertia along $x{\text{ }}axis$
$I_{yy}$ is the moment of inertia along $y{\text{ }}axis$
The perpendicular axis theorem stated above can be applied to laminar bodies only; these are the bodies whose mass is concentrated in a single plane, that is why this theorem can’t be applied for $3$ dimensional body.

Therefore, option (C) is CORRECT.

D) Parallel Axis Theorem: The moment of inertia of a body about an axis parallel to the body passing to through its center is equal to the sum of moment of inertia of body about the axis passing through the center and product of mass of the body times the square of distance between the two axis.
$I{\text{ }} = {\text{ }}I_C{\text{ }} + {\text{ }}M{h^2}$
Here,
$I$ is the moment of inertia of the body
$I_C$is the moment of inertia about the center
$M$ is the mass of the body
${h^2}$ is the square of the distance between the two axis
The parallel axis theorem stated above can be applied to$\;3D$ bodies this is a general expression.

Therefore, option (D) is CORRECT.

A) According to the parallel axis theorem if $\;I$ about an axis is minimum, then $I$about any other axis parallel to the axis passing through the center of mass is more. Hence, it must pass through the center of mass.

Therefore, option (A) is CORRECT.

Final answer $\Rightarrow$ Option (A), (C) and (D) are CORRECT.

Note: Do not get confused with parallel axis, perpendicular axis theorem. Read the question carefully so that you do not miss the fact that more than one option is correct.