A thin circular plate of mass M and radius R has its density varying… | JEE Main 2019 (8 Apr, Shift 1)

Q165JEE Main 2019 · 8 APR · SHIFT 1Medium

Moment of Inertia & Radius of Gyration

A thin circular plate of mass M and radius R has its density varying as $\rho$ (r) = $\rho$ 0r 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 = aMR2. The value of the coefficient a is:

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Answer: (c)

$M = \int_0^R \rho_0 r \cdot 2\pi r\,dr = \frac{2\pi\rho_0 R^3}{3}$
$I_c = \int_0^R \rho_0 r \cdot r^2 \cdot 2\pi r\,dr = \frac{2\pi\rho_0 R^5}{5} = \frac{3}{5}MR^2$
Edge axis (parallel axis): $I = \frac{3}{5}MR^2 + MR^2 = \frac{8}{5}MR^2$
So $a = \frac{8}{5}$ , answer (c).

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