Question

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 :

• A. $\frac{3}{2}$
• B. $\frac{1}{2}$
• C. $\frac{3}{5}$
• D. $\frac{8}{5}$

Answer 1

Answer: (D)

$\frac{8}{5}$

Answer 2

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