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(D) The distance $r$ from the centroid to the midpoint of each side of an equilateral triangle of side length $a = 2 \,m$ is given by $r = \frac{a}{2\

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$4$

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(D) The distance $r$ from the centroid to the midpoint of each side of an equilateral triangle of side length $a = 2 \,m$ is given by $r = \frac{a}{2\sqrt{3}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} \,m$.The moment of inertia $I$ of the system about an axis passing through the centroid and perpendicular to the plane is the sum of the moments of inertia of individual masses: $I = \sum m_i r_i^2$.Since all masses are at the same distance $r$ from the centroid,$I = (m_1 + m_2 + m_3) r^2$.Substituting the values: $I = (2 + 4 + 6) 	imes (\frac{1}{\sqrt{3}})^2$.$I = 12 	imes \frac{1}{3} = 4 \,kg \,m^2$.

Three balls of masses $2 \,kg$,$4 \,kg$,and $6 \,kg$ are placed at the midpoints of the sides of an equilateral triangle of side length $2 \,m$. The moment of inertia of the system about an axis passing through the centroid and perpendicular to the plane of the triangle is . . . . . .$kg \,m^2$.

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