Question

Three balls of masses $4kg,3kg$ and $5kg$ are arranged at the corners of an equilateral triangle $\Delta ABC$ 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 $\Delta ABC$ .
Then, $CD=\sqrt{A{C}^2-A{D}^2}=\sqrt{2^2-1^2}$
$=\sqrt{4-1}=\sqrt{3}$

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