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  4. An object with uniform density ρ is attached to a spring that is known to stretch linearly with applied force as shown below. <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/physics/89bed1b9e3d4c920b105a51a4c7800db-.png /> When the spring object system is immersed in a liquid of density ρ1 as shown in the above figure, the spring stretches by an amount x1(ρ>ρ1). When the experiment is repeated in a liquid of density (ρ2<ρ1), the spring stretches by an amount x2. Neglecting any buoyant force on the spring, the density of the object is

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Q. An object with uniform density $\rho$ is attached to a spring that is known to stretch linearly with applied force as shown below.
image
When the spring object system is immersed in a liquid of density $\rho_{1}$ as shown in the above figure, the spring stretches by an amount $x_{1}\left(\rho>\rho_{1}\right)$. When the experiment is repeated in a liquid of density $\left(\rho_{2}<\rho_{1}\right)$, the spring stretches by an amount $x_{2}$. Neglecting any buoyant force on the spring, the density of the object is

KVPYKVPY 2011

Solution:

For equilibrium of block hung from string,
Spring force + Buoyant force $=$ Weight of block
image
So, we have
$k x_{1}+\rho_{1} V g=\rho V g $....(i)
$k x_{2}+\rho_{2} V g=\rho V g$ ....(ii)
Eliminating $k$, we get
$\rho=\frac{\rho_{1} x_{2}-\rho_{2} x_{1}}{x_{2}-x_{1}}$