Question

Three balls of masses $4kg,3kg$ and $5kg$ are arranged at the corners of an equilateral triangle $ΔABC$ of side $2m.$ The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of the triangle is

• A. $16kg-m^{2}$
• B. $4\sqrt{3}kg-m^{2}$
• C. $4kg-m^{2}$
• D. $16\sqrt{3}kg-m^{2}$

Answer 1

Answer: (A)

Let $G$ be the centroid of $ΔABC$ .
Then, $CD=\sqrt{AC^{2} - AD^{2}}=\sqrt{2^{2} - 1^{2}}$
$=\sqrt{4 - 1}=\sqrt{3}$

$AG=BG=CG=\frac{2}{3}CD=\frac{2}{3}\sqrt{3}=\frac{2}{\sqrt{3}}$
Moment of inertia about an axis through centroid $G$ and perpendicular to plane of $ΔABC$ is
$I=4\times AG^{2}+3\times BG^{2}+5\times CG^{2}$
$=\left(\right.4+3+5\left.\right)\left(CG\right)^{2}=12\times \frac{4}{3}=16kg-m^{2}$