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 $$=$$ λ$$0cos2$$θ (where$$ λ$$0$$ is a positive $$constant). Then the moment of inertia of the ring about $$z-axis$$ is:

Question

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 $$=$$ λ$$0cos2$$θ (where$$ λ$$0$$ is a positive $$constant). Then the moment of inertia of the ring about $$z-axis$$ is:

Infinity Learn · Accepted Answer

Divide the ring into infinitely small lengths of mass dmi. Even though mass distribution is $$non-uniform$$, each mass dmi is at same distance R from origin.∴  MI of ring about $$z-axis$$ $$is=$$ $$dm1R2+dm2R2+$$....$$+dmnR2$$ $$=$$ $$MR2$$

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