Question

A sphere of solid material of relative density 9 has a concentric spherical cavity and just floats in water. If the radius of the sphere be R, then the radius of the cavity (r) will be related to R as:
A. \[{{r}^{3}}=\dfrac{8}{9}{{R}^{3}}\]
B. \[{{r}^{3}}=\dfrac{2}{9}{{R}^{3}}\]
C. \[{{r}^{3}}=\dfrac{\sqrt{8}}{9}{{R}^{3}}\]
D. \[{{r}^{3}}=\sqrt{\dfrac{2}{3}}{{R}^{3}}\]

Answer 1

Hint:When a body is placed in water it will either float or sink and that is governed by the Archimedes principle. Also, there is another law called the law of floatation which determines how the body will float on water. There may be conditions that float but completely sink. Also, there is loss in the weight of the body.

Complete step by step answer:
The law of floatation states that a body floats if the weight of the body is equal to the weight of the water displaced. Archimedes principle states that the upward buoyant force that is exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaced.
${{W}_{s}}={{W}_{w}} \\
\Rightarrow {{V}_{s}}{{\rho }_{s}}g={{V}_{w}}{{\rho }_{w}}g \\
\Rightarrow \dfrac{4}{3}\pi ({{R}^{3}}-{{r}^{3}})\times 9{{\rho }_{w}}=\dfrac{4}{3}\pi {{R}^{3}}{{\rho }_{w}} \\
\Rightarrow 9({{R}^{3}}-{{r}^{3}})={{R}^{3}} \\
\therefore {{r}^{3}}=\dfrac{8}{9}{{R}^{3}} \\ $
So, the correct option is A.

Additional Information: -
In everyday language, weight and mass are used interchangeably widely. But in physics, they have distinct meanings. Mass is a constant quantity and has a constant value everywhere whereas weight is the gravitational force with which the body is attracted towards itself.

Note:Relative density is defined as the density of a body divided by the density of water. Also, density is defined as the ratio of mass of the body divided by the total volume of that body. For a sphere volume is given by \[\dfrac{4}{3}\pi {{R}^{3}}\], where R is the radius of the sphere. Since, in this problem there was a cavity and it was also spherical in shape, so we have subtracted that volume from the total volume of the sphere.