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

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 $r = \frac{1}{\sqrt{3}}$:
$I = \left( \frac{1}{\sqrt{3}} \right)^2 \times 12$
$I = \frac{1}{3} \times 12 = 4 \, \text{kg} \cdot \text{m}^2$