A steel wire is suspended vertically from a rigid support. When loaded with a weight in air, it extends by lₐ and when the weight is immersed completely in water, the extension is reduced to lᵥᵥ. Then the relative density of the material of the weight is

To find the relative density of the material of the weight, let us analyze the forces acting on the weight and the resulting extension in the steel wire.

According to Hooke's Law, the extension $l$ produced in a wire of length $L$, area of cross-section $A$, and Young's modulus $Y$ when subjected to a tension $T$ is given by:
$l=\frac{TL}{AY}$
This shows that the extension is directly proportional to the tension in the wire ($l\propto T$).

When the weight is loaded in air, the tension in the wire is equal to the actual weight of the object. Let $V$ be the volume of the weight, $\rho$ be the density of the material of the weight, and $g$ be the acceleration due to gravity. The weight in air is:
$W_{\text{air}}=V\rho g$
Thus, the extension $l_a$ is proportional to this weight:
$l_a\propto V\rho g$     — (Equation 1)

When the weight is completely immersed in water, it experiences an upward buoyant force (upthrust) equal to the weight of the displaced water. Let ${\rho }_w$ be the density of water. The buoyant force is:
$F_b=V{\rho }_{w}g$
The effective weight of the object in water (apparent weight) becomes:
$W_{\text{water}}=W_{\text{air}}-F_b=V\rho g-V{\rho }_{w}g=Vg(\rho -{\rho }_w)$
The new extension $l_{vv}$ is proportional to this apparent weight:
$l_{vv}\propto Vg(\rho -{\rho }_w)$     — (Equation 2)

To find the relation, we divide Equation 1 by Equation 2:
$\frac{l_a}{l_{vv}}=\frac{V\rho g}{Vg(\rho -{\rho }_w)}=\frac{\rho }{\rho -{\rho }_w}$

Taking the reciprocal of both sides gives:
$\frac{l_{vv}}{l_a}=\frac{\rho -{\rho }_w}{\rho }=1-\frac{{\rho }_w}{\rho }$

Rearranging the equation to solve for the density ratio:
$\frac{{\rho }_w}{\rho }=1-\frac{l_{vv}}{l_a}=\frac{l_a-l_{vv}}{l_a}$

The relative density of the material of the weight is defined as the ratio of the density of the material to the density of water ($\frac{\rho }{{\rho }_w}$). Taking the reciprocal:
$\text{Relative Density}=\frac{\rho }{{\rho }_w}=\frac{l_a}{l_a-l_{vv}}$

Therefore, the relative density of the material of the weight is represented by $\frac{l_a}{l_a-l_{vv}}$.