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Question

# A ring of mass M and radius R lies in the x−y plane with its centre at the origin as shown. The mass distribution of the ring is non-uniform such that at any point P on the ring, the mass per unit length is given by λ=λ0cos2θ (where λ0 is a positive constant). Then the moment of inertia of the ring about z− axis is:

A
MR2
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B
12MR2
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C
12Mλ0R
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D
1πMλ0R
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Solution

## The correct option is A MR2Given, λ=λ0cos2θ Let us consider an elemental mass ′dm′ of length ′ds′. Even though mass distribution is non-uniform, each elemental mass chosen is at same distance ′R′ from the centre. ∴ Moment of inertia of the ring about z− axis is (dm1)R2+(dm2)R2+(dm3)R2+...+(dmn)R2 ⇒I=[dm1+dm2+dm3+...+dmn]r2 ⇒I=MR2 i.e MOI of ring depends only on the total mass M and radius R

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