A sphere of relative density σ and diameter D has concentric cavity of diameter d. The ratio of D/d, if it just floats on water in a tank is: (a) ((σ - 2)/(σ + 2))1/3 (b) ((σ + 1)/(σ - 1))1/3 (c) ((σ - 1)/(σ))1/3 (d) ((σ)/(σ - 1))1/3

Answer: (d). Just floats: weight of shell = upthrust on the whole sphere (fully submerged). σ ρw π/6(D³ - d³) g = ρw π/6 D³ g σ D³ - σ d³ = D³ ⇒ D³(σ - 1) = σ d³ D/d = ((σ)/(σ - 1))1/3, so the answer is (d).

A sphere of relative density σ and diameter D has concentric cavity… | JEE Main 2024 (9 Apr, Shift 1)

Q8JEE Main 2024 · 9 APR · SHIFT 1Medium

Archimedes' Principle & Floatation

A sphere of relative density $\sigma$ 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:

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Answer: (d)

Just floats: weight of shell = upthrust on the whole sphere (fully submerged).
$\sigma \rho_w \dfrac{\pi}{6}(D^3 - d^3) g = \rho_w \dfrac{\pi}{6} D^3 g$
$\sigma D^3 - \sigma d^3 = D^3 \Rightarrow D^3(\sigma - 1) = \sigma d^3$
$\dfrac{D}{d} = \left(\dfrac{\sigma}{\sigma - 1}\right)^{1/3}$ , so the answer is (d).

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Archimedes' Principle & Floatation question from JEE Main 2024 (9 Apr, Shift 1)

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