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A sphere of relative density σ and diameter D has a concentric cavity of diameter d. The ratio D/d, if it just floats on water in a tank, is

Asked in JEE Main 9th April 1st Shift 2024 · Two liquids, cavities and relative density

• (1) ((σ-1)/σ)^1/3
• (2) (σ/(σ-1))^1/3
• (3) ((σ+1)/(σ-1))^1/3
• (4) ((σ-2)/(σ+2))^1/3

Step-by-step solution

Why the other options are wrong

• (1) comes from is the ratio the other way up, d/D; the working ends at (d³)/(D³)=(σ-1)/σ and the last step of inverting it is skipped. It is also always less than 1, while D must exceed d.
• (3) comes from counting the cavity twice on the buoyancy side: solving σ(D³-d³)=D³+d³ gives exactly ((σ+1)/(σ-1))^1/3. The cavity is inside the sphere, so the water displaced is the outer volume (π D³)/6 alone.
• (4) comes from no consistent slip lands here; no step in the working produces a factor 2, and the expression is smaller than 1 for every σ>2, which would need the cavity to be bigger than the sphere holding it.

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