Consider a solid sphere of radius $\mathrm{R}$ and mass density $\rho(\mathrm{r})=\rho_{0}\left(1-\frac{\mathrm{r}^{2}}{\mathrm{R}^{2}}\right), 0<\mathrm{r} \leq \mathrm{R} .$ The minimum density of a liquid in which it will float is
JEE MAIN 2020, Diffcult
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In case of minimum density of liqued, sphere will be floating while completely submerged So $\mathrm{mg}=\mathrm{B}$
$\mathrm{m}=\int_{0}^{\mathrm{R}} \rho\left(4 \pi \mathrm{r}^{2} \mathrm{dr}\right)=\mathrm{B}$
$=\rho_{0} \int_{0}^{R}\left(1-\frac{r^{2}}{R^{2}}\right) 4 \pi r^{2} d r=\frac{4}{3} \pi R^{3} \rho_{\ell} g$
On Solving
$\rho_{\ell}=\frac{2 \rho_{0}}{5}$
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