The moment of inertia of the following structure (written as PHYSICS) about axis AB made of thin uniform rods of mass per unit length λ is . Find the value of n.

Engineering

Physics

Moment of Inertia

Question

The moment of inertia of the following structure (written as PHYSICS) about axis AB made of thin uniform rods of mass per unit length λ is $\frac{78}{n} λ ℓ^{3}$ . Find the value of n.

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The moment of inertia of all seven rods parallel to AB and not lying on AB is

= 7 × (λℓ) ℓ² = 7 λ ℓ³

the moment of inertia of all five rods lying on AB = 0

The moment of inertia of all 18 rods perpendicular to AB is

= 18 (λℓ) $\frac{ℓ^{2}}{3}$ = 6 λ ℓ³

Hence net MI of rod about AB = 7 λ ℓ³ + 6 λ ℓ³ = 13 λ ℓ³ Ans.