Question

Consider a solid sphere of radius $R$ and mass density $\rho\left(r\right) - \rho \left(1-\frac{r^{2}}{R^{2}}\right),$ $0 < r \le R.$ The minimum density of a liquid in which it will float is :

• A. $\frac{2\rho_{0}}{5}$
• B. $\frac{\rho_{0}}{5}$
• C. $\frac{2\rho_{0}}{3}$
• D. $\frac{\rho_{0}}{3}$

Answer 1

Answer: (A)

In case of minimum density of liqued, sphere will be floating while completely submerged So $mg = B$
$\int\limits^{R}_{{0}}$$\rho\left(4\pi r^{2}\,dr\right)=B$
$=\rho_{0}$$\int\limits^{R}_{{0}}$$\left(1-\frac{r^{2}}{R^{2}}\right)4\pi r^{2}\,dr=\frac{4}{3}\pi R^{3}\,\rho_{\ell}g$
On Solving
$\rho_{\ell}=\frac{2\rho_{0}}{5}$