Question

Moment of inertia of uniform triangular plate about axis passing through sides $A B$, $A C, B C$ are $I_{P}, I_{B}$ and $I_{H}$ respectively and about an axis perpendicular to the plane and passing through point $C$ is $I_{C}$ Then:

• A. $I_{C} > I_{P} > I_{B} > I_{H}$
• B. $I_{H} > I_{B} > I_{C} > I_{P}$
• C. $I_{P} > I_{H} > I_{B} > I_{C}$
• D. none of these

Answer 1

Answer: (A)

Moment of inertia is more when mass is farther from the axis. In case of axis $B C$, mass distribution is closest to it and in case of axis $A B$ mass distribution is farthest. Hence
$I_{B C} < I_{A C} < I_{A B} \Rightarrow I_{P} > I_{B} > I_{H}$
$I_{C} = I_{C M}+m y^{2}=I_{B}^{\prime}-m x^{2}+m y^{2}$
Here $I_{B}'$ is moment of inertia of the plate about an axis perpendicular to it and passing through $B$.
$\Rightarrow I_{C}=I_{B}'+m\left(y^{2}-x^{2}\right)=I_{P}+I_{B}+m\left(y^{2}-x^{2}\right)$
It means $I_{C} > I_{P}+I_{B}$ also $I_{C} > I_{P}$
$\therefore I_{C} > I_{P} > I_{B} > I_{H}$