question_answer

A hollow sphere of external radius R and thickness \[t(t<<R)\]is made of a metal of density p. The sphere will float in water, if

A) \[t\le \frac{R}{\rho }\]

B)  \[t\le \frac{R}{3\rho }\]

C)  \[t\le \frac{R}{2\rho }\]

D)  \[t\ge \frac{R}{3\rho }\]

Correct Answer: B

Solution :

 The density of material is p The hollow sphere will float if its weight is less than the weight of the water displaced by the volume of the sphere This implies mass of the sphere is less than that for the same volume of water. Now. mass of spherical cell \[{{m}_{1}}=4\pi {{R}^{2}}\times t\times \rho \] While the mass of water having same volume \[{{m}_{2}}=\frac{4}{3}\pi {{R}^{3}}\times {{\rho }_{g}}=\frac{4}{3}\rho {{R}^{3}}\] where, \[{{\rho }_{g}}=\]density of water \[=\frac{1\,kg}{{{m}^{3}}}\] For the floatation of sphere, \[{{m}_{1}}\le {{m}_{2}}\] \[4\pi {{R}^{2}}\times t\times \rho \le \frac{4}{3}\pi {{R}^{3}}\] \[\Rightarrow \] \[t\rho \le \frac{R}{3}\] \[\Rightarrow \] \[t\le \frac{R}{3\rho }\]