Three identical particles, each of mass 2 kg , are placed at the vertices of an equilateral triangle of side 3 m . The moment of inertia of the system about an axis perpendicular to the plane of the triangle and passing through its centroid is ________ kg m ^2 .

Step-by-step solution

Let the side of the equilateral triangle be a = 3 m and the mass of each particle be m = 2 kg . The distance r of any vertex from the centroid of an equilateral triangle is given by: r = a 3 The moment of inertia of the system about an axis passing through the centroid and perpendicular to the plane is the sum of the moments of inertia of the three particles. I = m₁ r₁^2 + m₂ r₂^2 + m₃ r₃^2 Since all particles have the same mass and are at the same distance from the centroid: I = 3 m ( a 3 )^2 I = 3 m a^2 3 = m a^2