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Consider a solid sphere of radius R and mass density ρ(r)=ρ₀(1-(r²)/(R²)), 0<r≤ R. The minimum density of a liquid in which it will float is

Asked in JEE Main 8th Jan 1st Shift 2020 · Two liquids, cavities and relative density

• (1) (2ρ₀)/5
• (2) (ρ₀)/5
• (3) (2ρ₀)/3
• (4) (ρ₀)/3

Step-by-step solution

Why the other options are wrong

• (2) comes from no consistent slip lands here; it is half the key, and nothing in the integration calls for a second halving. The two checkable wrong routes give (2ρ₀)/3 (option 3) and (3ρ₀)/5 (keeping only the (r²)/(R²) piece of the density).
• (3) comes from averages the density over the radius instead of over the volume: 1/R∫₀^Rρ₀(1-(r²)/(R²))dr=(2ρ₀)/3. Each shell must be weighted by its own volume 4π r²dr, which shifts the average towards the sparse outer layers.
• (4) comes from a filler paired with option 2; the only fractions the integral produces are 1/3 and 1/5 inside 1/3-1/5=2/(15), and 1/3 standing alone is that subtraction never carried out combined with the normalisation never applied.

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