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  4. A thin circular plate of mass M and radius R has its density varying as ρ (r) = ρ0 r 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 :

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Q. A thin circular plate of mass $M$ and radius $R$ has its density varying as $\rho (r) = \rho_0 r$ with $\rho_0$ 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 $I = aMR^2$. The value of the coefficient $a$ is :

JEE MainJEE Main 2019System of Particles and Rotational Motion

Solution:

$M= \int\limits^{R}_{0} \rho_{0} r \left(2\pi rdr\right) = \frac{\rho_{0} \times2 \pi \times R^{3}}{3} $
$\underset{\text{ (MOI about COM) }{ I_0 }$ = \int\limits^{R}_{0} \rho_{0} r (2 \pi rdr) \times r^{2} = \frac{\rho_{0} \times2\pi R^{5}}{5} } $
by parallel axis theorem
$ I = I_{0} + MR^{2} $
$ = \frac{\rho_{0} \times2 \pi R^{5}}{5} + \frac{\rho_{0} \times2\pi R^{3}}{3} \times R^{2} = \rho_{0} 2\pi R^{5} \times\frac{8}{15} $
$ = MR^{2} \times\frac{8}{5} $