A uniform cubical block of density ρ b (in g/cc) & side length a is released from rest from the position where one of its horizontal surface (lower) just touches the water (density ρ w = 1 gm / cc

Question

A uniform cubical block of density ρ b ​ (in g/cc) & side length a is released from rest from the position where one of its horizontal surface (lower) just touches the water (density ρ w ​ = 1 gm / cc ). Various values of ρ b ​ are given in List-I. Neglect viscosity and surface tension. In List-II, expression for the time period of block are given. Match the correct expression of the time period in List-II with the density given in List-I. List-I List-II (I) 5 1 ​ (P) 2 π ρ w ​ g ρ b ​ a ​ ​ (II) 5 2 ​ (Q) 2 π ρ a ​ g ρ w ​ a ​ ​ (III) 2 1 ​ (R) 3 4 π ​ ρ w ​ ρ b ​ ​ g a ​ ​ + ρ w ​ − ρ b ​ ρ b ​ ​ 3 ρ b ​ g 4 ρ w ​ a ​ ​ (IV) 3 2 ​ (S) 3 2 π ​ ρ w ​ ρ b ​ ​ g a ​ ​ + ρ w ​ − ρ b ​ ρ b ​ ​ 3 2 ​ ρ b ​ g ρ w ​ a ​ ​ (T) None of these

TestHub · Accepted Answer

For extra immersion x of the floating block, restoring force = $F = - ρ_{w} g a^{2} x$ and mass = ρ_b a³, so $T = 2 π \frac{ρ _{b} a}{ρ _{w} g}$

Released from rest at the surface, the block dips to twice the equilibrium immersion: $2 x_{e q} = \frac{2 ρ _{b}}{ρ _{w}} a$

(I) ρ_b = 1/5, (II) 2/5, (III) 1/2: maximum immersion 2ρ_b/ρ_w × a ≤ a, so the block never goes fully under and performs pure SHM $T = 2 π \frac{ρ _{b} a}{ρ _{w} g} \to P$

(IV) ρ_b = 2/3: the block submerges completely for a part of the motion (partly SHM, partly uniform acceleration under constant buoyancy) → R.

I → P; II → P; III → P; IV → R, so option (C) is correct.

List-I	List-II
(I) $\frac{1}{5}$	(P) $2 π \frac{ρ _{b} a}{ρ _{w} g}$
(II) $\frac{2}{5}$	(Q) $2 π \frac{ρ _{w} a}{ρ _{a} g}$
(III) $\frac{1}{2}$	(R) $\frac{4 π}{3} \frac{ρ _{b}}{ρ _{w}} \frac{a}{g} + \frac{ρ _{b}}{ρ _{w} - ρ _{b}} \frac{4 ρ _{w} a}{3 ρ _{b} g}$
(IV) $\frac{2}{3}$	(S) $\frac{2 π}{3} \frac{ρ _{b}}{ρ _{w}} \frac{a}{g} + \frac{ρ _{b}}{ρ _{w} - ρ _{b}} \frac{2}{3} \frac{ρ _{w} a}{ρ _{b} g}$
	(T) None of these

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