A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less spring of force constant $k$. The other end of the spring is connected to another solid sphere of radius $R$ and density $3 \rho$. The complete arrangement is placed in a liquid of density $2 \rho$ and is allowed to reach equilibrium. The correct statement(s) is (are) (A) the net elongation of the spring is $\frac{4 \pi R^{3} \rho g}{3 k}$. (B) the net elongation of the spring is $\frac{8 \pi R^{3} \rho g}{3 k}$ (C) the light sphere is partially submerged. (D) the light sphere is completely submerged.

(A) the net elongation of the spring is $\frac{4 \pi R^{3} \rho g}{3 k}$.; (D) the light sphere is completely submerged.

(B) the net elongation of the spring is $\frac{8 \pi R^{3} \rho g}{3 k}$

(C) the light sphere is partially submerged.

JEE Advanced · 2013 · 02 Jun 2013 · Paper 1 · Fluid Mechanics

WhatsApp Email

A solid sphere of radius

R

and density

ρ

is attached to one end of a mass-less spring of force constant

k

. The other end of the spring is connected to another solid sphere of radius

R

and density

3 ρ

. The complete arrangement is placed in a liquid of density

2 ρ

and is allowed to reach equilibrium. The correct statement(s) is (are)
(A) the net elongation of the spring is

\frac{4 π R ^{3} ρ g}{3 k}

.
(B) the net elongation of the spring is

\frac{8 π R ^{3} ρ g}{3 k}

(A)the net elongation of the spring is $\frac{4 π R ^{3} ρ g}{3 k}$ .✓ Correct
(B)the net elongation of the spring is $\frac{8 π R ^{3} ρ g}{3 k}$
(C)the light sphere is partially submerged.
(D)the light sphere is completely submerged.✓ Correct

Answer: (A), (D)

Solution

Total weight of the system:

W = [ρ \times \frac{4}{3} π R^{3} + 3 ρ \times \frac{4}{3} π R^{3}] g = \frac{16 π R ^{3} ρ g}{3}

For equilibrium, buoyancy force equals weight:

V \times 2 ρ g = \frac{16 π R ^{3} ρ g}{3}

V = \frac{8 π R ^{3}}{3}

Total volume of both spheres =

2 \times \frac{4}{3} π R^{3} = \frac{8 π R ^{3}}{3}

Since the submerged volume equals the total volume, both spheres are completely submerged. ✓ (D is correct)

For spring elongation, considering forces on the denser sphere:

k x + \frac{4}{3} π R^{3} \times 2 ρ g = 3 ρ \times \frac{4}{3} π R^{3} \times g

k x = \frac{4 π R ^{3} ρ g}{3}

x = \frac{4 π R ^{3} ρ g}{3 k}

✓ (A is correct)

Hence the correct answers are (A) and (D).

JEE Advanced · 2013 · 02 Jun 2013 · Paper 1 · One or more correct options

Practise more: all JEE Advanced PYQs on Fluid Mechanics →

JEE Advanced · 2013 · 02 Jun 2013 · Paper 1 · Fluid Mechanics

Solution

More JEE Advanced questions on Fluid Mechanics