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 $mg=B$
$\underset{0}{\overset{R}{\int }}$$\rho \left(4\pi r^{2}dr\right)=B$
$={\rho }_0$$\underset{0}{\overset{R}{\int }}$$\left(1-\frac{r^2}{R^2}\right)4\pi r^{2}dr=\frac{4}{3}\pi R^3{\rho }_{\ell }g$
On Solving
${\rho }_{\ell }=\frac{2{\rho }_0}{5}$