A thin circular plate of mass M and radius R has its density varying as ρ ( r ) =ρ 0 r with ρ 0 as constant and r is the distance from its centre. The moment of Inertia of the circular plate about an

Question

A thin circular plate of mass M and radius R has its density varying as ρ ( r ) =ρ 0 r with ρ 0 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 = a M R 2 . The value of the coefficient a is:

TestHub · Accepted Answer

d m = 2 π r d r × ρ = 2 π ρ 0 r 2 d r M = 2 π ρ 0 ∫ 0 R r 2 d r = 2 3 π ρ 0 R 3 I CM = ρ d m r 2 = 2 π ρ 0 ∫ r 4 d r = 2 5 π ρ 0 R 5 I c m = 3 5 M R 2 I = I c m + M R 2 = 8 5 M R 2

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