A sphere of radius R , made from material of specific gravity S G , is submerged in a tank of water. The sphere is placed over a hole, of radius a , in the tank bottom. For the dimensions given, det

Question

A sphere of radius R , made from material of specific gravity S G ​ , is submerged in a tank of water. The sphere is placed over a hole, of radius a , in the tank bottom. For the dimensions given, determine the minimum S G ​ required for the sphere to remain in the position shown.

TestHub · Accepted Answer

The sphere's bottom over the hole is not wetted, so buoyancy = weight of displaced water minus pressure force on the hole area:

$F_{B} = ρ_{w} g \cdot \frac{4}{3} π R^{3} - ρ_{w} g H \cdot π a^{2}$

For minimum $S_{G}$ : weight $= F_{B}$

$S_{G} ρ_{w} \frac{4}{3} π R^{3} g = ρ_{w} g [\frac{4}{3} π R^{3} - H π a^{2}]$

$S_{G} = 1 - \frac{3 H a ^{2}}{4 R ^{3}} = 1 - \frac{3 \times 800 \times ( 2 ) ^{2}}{4 \times ( 20 ) ^{3}} = 1 - 0.30 = 0.70$

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