A) \[\frac{{{7}^{1/3}}}{2}\]
B) \[\frac{{{5}^{1/3}}}{2}\]
C) \[\frac{{{9}^{1/3}}}{2}\]
D) \[\frac{{{3}^{1/3}}}{2}\]
Correct Answer: A
Solution :
[a] Let \[\rho \] be the density of the material, \[{{\rho }_{0}}\] be the density of water when the sphere has just started sinking, the weight of the sphere = weight of water displaced (approx.) \[\Rightarrow \,\,\,\frac{4}{3}\pi ({{R}^{3}}-{{r}^{3}})\rho g=\frac{4}{3}\pi {{R}^{3}}{{\rho }_{0}}g\] \[\Rightarrow \,\,({{R}^{3}}-{{r}^{3}})\rho ={{R}^{3}}{{\rho }_{0}}\Rightarrow \frac{({{R}^{3}}-{{r}^{3}})}{{{R}^{3}}}=\frac{{{\rho }_{0}}}{\rho }\] \[\Rightarrow \,\,\frac{r}{R}=\frac{{{(7)}^{1/3}}}{2}\]
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