Fluid Mechanics

JEE Main level, single correct, medium

36 of 192

Question 36 of 192

A sphere of relative density σ\sigma and diameter DD has concentric cavity of diameter dd. The ratio of Dd\frac{D}{d}, if it just floats on water in a tank is :

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Answer: (B) (σσ−1)1/3\left(\frac{\sigma}{\sigma-1}\right)^{1 / 3}

weight (w)=43π(D3−d38)σg(w)=\frac{4}{3} \pi\left(\frac{\mathrm{D}^3-\mathrm{d}^3}{8}\right) \sigma \mathrm{g}
Buoyant force (Fb)=1×43π(D38)⋅g\left(\mathrm{F}_{\mathrm{b}}\right)=1 \times \frac{4}{3} \pi\left(\frac{\mathrm{D}^3}{8}\right) \cdot \mathrm{g}
For Just Float ⇒w=Fb\Rightarrow \mathrm{w}=\mathrm{F}_{\mathrm{b}}
⇒(D3−d3)σ=D3⇒1−d3D3=1σ⇒1−1σ=(dD)3⇒(σσ−1)13=(Dd)\begin{aligned} & \Rightarrow\left(D^3-d^3\right) \sigma=D^3 \\ & \Rightarrow 1-\frac{d^3}{D^3}=\frac{1}{\sigma} \\ & \Rightarrow 1-\frac{1}{\sigma}=\left(\frac{d}{D}\right)^3 \\ & \Rightarrow\left(\frac{\sigma}{\sigma-1}\right)^{\frac{1}{3}}=\left(\frac{D}{d}\right)\end{aligned}

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