Three sphere of masses `m_,m andm_` are located at the vertices of an equilateral triangle having side of same length. Find the moment of inertia of the system about one of side of triangle.

Options

A. `(3)/(4)ml^(2)`
B. `(4)/(3)ml^(2)`
C. `(3)/(2)ml^(2)`
D. `(2)/(3)ML^(2)`

Answer

A

Explanation

Suppose equilateral triangle `DeltaABC` has length l and mass m are arranged on each of its vertices. The moment of inertia about an axis passing one of side `bar(BC)` of triangle is I, then `I=mr^(2)`, where r = distance of sphere A from BC From right angle `DeltaAPB`
`l^(2)=r^(2)+((l)/(2))^(2)`
`therefore r^(2)=l^(2)-(l^(2))/(4)=(3l^(2))/(4)`
`therefore I=mxx(3l^(2))/(4)`
`=(3)/(4)ml^(2)`