Q: A solid sphere of radius ‘R’ has a concentric cavity of radius R/3 inside it. The sphere is found to just float in water with the highest point of it touching the water surface. Find the specific gravity of the material of the sphere.

Sol: $\large \frac{V_{cavity}}{V_S} $

$\large = \frac{V_S – V_{metal}}{V_S} = 1 – \frac{V_{metal}}{V_S}$ …(i)

(V_{S} = Total volume of the sphere)

According to Archimedes’s principle

Weight of body = Weight of displaced liquid

$\large m g = V_S d_w g $

$\large V_{metal}d_S g = V_S d_w g $

(d_{S} = Density of Solid material, d_{w} = Density of Water)

$\large \frac{V_{metal}}{V_S} = \frac{d_w}{d_S} $ ….(ii)

From equation (i) & (ii)

$\large \frac{V_{cavity}}{V_S} = 1 – \frac{d_w}{d} $

$\large \frac{V_{cavity}}{V_S} = 1 – \frac{1}{d/d_w} = 1 – \frac{1}{Sp.\; gravity} $

$\large \frac{\frac{4}{3}\pi (R/3)^3}{\frac{4}{3}\pi R^3} = 1 – \frac{1}{Sp.\; gravity}$

$\large \frac{1}{27} = 1 – \frac{1}{Sp.\; gravity}$

$\large Sp. \;gravity = \frac{27}{26} $