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  4. A ring of mass M and radius R lies in x-y plane with its centre at origin as shown. The mass distribution of ring is non uniform such that, at any point P on the ring, the mass per unit length is given by λ=λ0 cos 2 θ (where λ0 is a positive constant). Then the moment of inertia of the ring about z -axis is:

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Q. A ring of mass $M$ and radius $R$ lies in $x-y$ plane with its centre at origin as shown. The mass distribution of ring is non uniform such that, at any point $P$ on the ring, the mass per unit length is given by $\lambda=\lambda_{0} \cos ^{2} \theta$ (where $\lambda_{0}$ is a positive constant). Then the moment of inertia of the ring about $z$ -axis is:Physics Question Image

System of Particles and Rotational Motion

Solution:

Divide the ring into infinitely small lengths of mass $d m_{i}$. Even though mass distribution is non-uniform, each mass $d m_{i}$ is at same distance $R$ from origin.
$\therefore $ MI of ring about $z$ -axis is
$=d m_{1} R^{2}+d m_{2} R^{2}+\ldots+d m_{n} R^{2}=M R^{2}$