IIT-JEE 2012 · Paper 2, Physics question 3

Set up. Let the shell's wall material have volume $v$ and the whole closed cylinder (outer) volume $V$, so it holds at most $V-v$ of water. Let a fraction $f$ of that capacity be filled. Half-submerged, the displaced volume is $V/2$:
$$\rho_w\frac V2=\rho_c\rho_wv+\rho_wf(V-v).$$

More than half-filled means $f>\frac12$:
$$\frac V2-\rho_cv>\frac{V-v}2\iff\rho_cv<\frac v2\iff\rho_c<0.5.$$

So the shell is more than half-filled exactly when $\rho_c<0.5$ (and less than half-filled when $\rho_c>0.5$).

• (A) is true; (B), (C) and (D) contradict this.

(A)