Q 11-09-074JEE MainJEE Main 2024 (9 Apr, Shift 1)Medium
A sphere of relative density $\sigma$ and diameter $D$ has a concentric cavity of diameter $d$. The ratio of $\dfrac Dd$, if it just floats on water in a tank, is:
Answer: (B) $\left(\dfrac{\sigma}{\sigma-1}\right)^{1/3}$
"Just floats" means fully submerged with weight equal to buoyancy (the cavity is empty):
$$\sigma\rho_w\frac\pi6(D^3 - d^3)g = \rho_w\frac\pi6D^3g \;\Rightarrow\; 1 - \frac{d^3}{D^3} = \frac1\sigma$$
$$\frac{d^3}{D^3} = \frac{\sigma - 1}{\sigma} \;\Rightarrow\; \frac Dd = \left(\frac{\sigma}{\sigma - 1}\right)^{1/3}$$
Solution by Sreeraj P, M.Sc Physics