# A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as $\left ( \frac{\sigma 0}{r} \right )$, then the radius of gyration of the disc about its axis passing through the centre is :Option 1)$\frac{a+b}{2}$Option 2)  $\sqrt{\frac{a^{2}+b^{2}+ab}{3}}$Option 3)$\sqrt{\frac{a^{2}+b^{2}+ab}{2}}$Option 4)$\frac{a+b}{3}$

$I=\int dm r^{2}$

$\int_{a}^{b}\frac{\sigma_0}{r}2\pi r dr.r^2=\sigma_0 2\pi [\frac{b^3-a^3}{3}]$

$I=mK^2 \\m=\int_{a}^{b}\frac{\sigma_0}{r}2\pi r dr=2\pi \sigma_0 (b-a) \\\therefore K=\sqrt{\frac{b^3-a^3}{3(b-a)}}=\sqrt{\frac{a^2+b^2+ab}{3}}$

Option 1)

$\frac{a+b}{2}$

Option 2)

$\sqrt{\frac{a^{2}+b^{2}+ab}{3}}$

Option 3)

$\sqrt{\frac{a^{2}+b^{2}+ab}{2}}$

Option 4)

$\frac{a+b}{3}$

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