Question

Three balls of masses $2 \, \text{kg}$, $4 \, \text{kg}$, and $6 \, \text{kg}$ respectively are arranged at the centre of the edges of an equilateral triangle of side $2 \, \text{m}$.The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of the triangle, will be ____ $\text{kg} \, \text{m}^2$

Answer 1

Correct Answer: 4

For an equilateral triangle, the distance \( r \) from the center of each edge to the centroid \( C \) is:
\[r = \frac{1}{\sqrt{3}}\]
The moment of inertia \( I \) about point \( C \) and perpendicular to the plane is given by:
\[I = r^2 \left[ 2 + 4 + 6 \right]\]
Substitute \( r = \frac{1}{\sqrt{3}} \):
\[I = \left( \frac{1}{\sqrt{3}} \right)^2 \times 12\]
\[I = \frac{1}{3} \times 12 = 4 \, \text{kg} \cdot \text{m}^2\]