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Question:

A thin circular plate of mass M and radius R has its density varying as ρ(r) = ρ₀r with ρ₀ as constant and r is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = aMR². The value of the coefficient a is?

35

32

12

85

Solution:

Correct option is D. 85
M = ∫₀ᴿ ρ₀r(2πrdr) = ρ₀ × 2π × R³/3
I₀ (MOI about COM) = ∫₀ᴿ ρ₀r(2πrdr) × r² = ρ₀ × 2πR⁵/5
By parallel axis theorem
I = I₀ + MR² = ρ₀ × 2πR⁵/5 + ρ₀ × 2πR³/3 × R² = ρ₀2πR⁵ × 8/15 = MR² × 8/5 = 8/5MR²