A solid sphere of radius R and density is attached to one end of a mass-less spring of force constant . The other end of the spring is connected to another solid sphere of radius R and density . The complete arrangement is placed in a liquid of density

Question

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) [2013]

Smart Achievers · Accepted Answer

(1, 4)

$The complete system is as shown in figure.$

$Let x be the net elongation of the spring.$

$At equilibrium, for upper sphere$

$W + F_{S} = F_{B}$

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

$\Rightarrow k x = \frac{4}{3} π R^{3} (2 ρ) g - \frac{4}{3} π R^{3} ρ g$

$\Rightarrow k x = \frac{4 π R^{3} ρ g}{3}$ or $x = \frac{4 π R^{3} ρ g}{3 k}$

Solution

1 The net elongation of the spring is $\frac{4 π R^{3} ρ g}{3 k}$

2 The net elongation of the spring is $\frac{8 π R^{3} ρ g}{3 k}$

3 The light sphere is partially submerged

4 The light sphere is completely submerged

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Solution

1 The net elongation of the spring is 4πR3ρg3k

2 The net elongation of the spring is 8πR3ρg3k

3 The light sphere is partially submerged

4 The light sphere is completely submerged

Ans. (1, 4) The complete system is as shown in figure. Let x be the net elongation of the spring. At equilibrium, for upper sphere W+FS=FB 43πR3ρg+kx=43πR3(2ρ)g ⇒kx=43πR3(2ρ)g-43πR3ρg ⇒kx=4πR3ρg3 or x=4πR3ρg3k

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1 The net elongation of the spring is $\frac{4 π R^{3} ρ g}{3 k}$

2 The net elongation of the spring is $\frac{8 π R^{3} ρ g}{3 k}$