Fluid Mechanics | Surface Tension And Viscosity | JEE Main previous year questions with Solutions

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Topic Question Set

Q 1 :
A sphere of relative density $σ$ and diameter D has concentric cavity of diameter d. The ratio of $\frac{D}{d}$ , if it just floats on water in a tank is: [2024]

${(\frac{σ}{σ - 1})}^{\frac{1}{3}}$
${(\frac{σ + 1}{σ - 1})}^{\frac{1}{3}}$
${(\frac{σ - 1}{σ})}^{\frac{1}{3}}$
${(\frac{σ - 2}{σ + 2})}^{\frac{1}{3}}$

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(A) Weight of sphere, $W = \frac{4}{3} π (\frac{D^{3} - d^{3}}{8}) σ g$

Buoyant force $F_{b} = \frac{4}{3} π (\frac{D^{3}}{8}) ρ_{w} g$

For Just Float, $W = F_{b}$

$\frac{4}{3} π (\frac{D^{3} - d^{3}}{8}) σg = \frac{4}{3} π (\frac{D^{3}}{8}) ρ_{w} g$

$\Rightarrow 1 - \frac{d^{3}}{D^{3}} = \frac{ρ_{w}}{σ} (ρ_{w} = 1)$

$1 - \frac{1}{σ} = {(\frac{d}{D})}^{3} \Rightarrow \frac{D}{d} = {(\frac{σ}{σ - 1})}^{1 / 3}$

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