Question

Four thin uniform rods each of length L and mass m are joined to form a square. The moment of inertia of square about an axis along its one diagonal is?

Answer 1

Answer:

Moment of inertia about the centre of one rod = I₁ = ml²/12

moment of inertia through the centre of the square, will be through an axis parallel to moment of inertia perpendicular to the rod

I₂ = I₁ + m(l/2)²

I₂ = 4ml²/12

For four rods,

I₂ = (4ml²/12)×4

From perpendicular axis theorem,

I₂ = I₃ + I₄

where I₃ and I₄ are moment of inertia through the diagonals. For a square, both diagonals are equal

I₃ = I₄

I₂ = 2 I₃

(4ml²/12)×4 = 2 I₃

I₃ = 2ml²/3

Hope it helps.

Answer 2

From Perpendicular axis theorem

$I_z = I_y + I_x \\I_z = 2I_x$

From parallel axis theorem.

$I_z = I_(com) + md^2\\I_z = ML^2/12 + M(L/2)^2 \\ = ML^2/12 + ML^2/4\\ = 1/3 ML^2$

For single rod,

For 4 rods,

$4/3 ML^2$,

I_x = I_z/2 = (4/3)ML^2 /2

   = 2/3 ML^2