A load of mass M kg is suspended from a steel wire of length 2m and radius 1.0 mm in Searle's apparatus experiment. The increase in length produced in the wire is 4.0 mm. Now the load is fully immersed in a liquid of relative density 2. The relative density of the material of the load is 8. The new value of the increase in the length of the steel wire is:

Let the mass of the load be M kg.
The initial increase in length of the steel wire is 4.0 mm.
The relative density of the liquid is 2.
The relative density of the material of the load is 8.
When the load is suspended in air, the weight of the load is Mg, where g is the acceleration due to gravity.
The force acting on the wire is Mg.
Let the Young's modulus of the steel wire be Y.
The cross-sectional area of the wire is A = πr² = π(1.0 x 10⁻³)² m².
The initial increase in length is ΔL₁ = 4.0 x 10⁻³ m.
The stress in the wire is σ = Mg/A.
The strain in the wire is ε = ΔL₁/L = (4.0 x 10⁻³)/2 = 2.0 x 10⁻³.
Young's modulus is Y = σ/ε = (Mg/A)/(ΔL₁/L) = MgL/(AΔL₁).
When the load is immersed in the liquid, the buoyant force acting on the load is equal to the weight of the liquid displaced by the load.
Let V be the volume of the load.
The mass of the liquid displaced is ρ_liquid * V, where ρ_liquid is the density of the liquid.
The weight of the liquid displaced is ρ_liquid * V * g.
The buoyant force is ρ_liquid * V * g = 2ρ_water * V * g.
The apparent weight of the load in the liquid is Mg - 2ρ_water * V * g.
The density of the material of the load is 8ρ_water.
The mass of the load is M = 8ρ_water * V.
The apparent weight of the load in the liquid is (8ρ_water * V * g) - (2ρ_water * V * g) = 6ρ_water * V * g = (3/4)Mg.
The new force acting on the wire is (3/4)Mg.
The new increase in length is ΔL₂.
The new stress is (3/4)Mg/A.
The new strain is ΔL₂/L.
Y = ((3/4)Mg/A)/(ΔL₂/L) = (3/4)MgL/(AΔL₂).
Since Y is constant, we have MgL/(AΔL₁) = (3/4)MgL/(AΔL₂).
ΔL₂ = (3/4)ΔL₁ = (3/4)(4.0 mm) = 3.0 mm.
Therefore, the new value of the increase in the length of the steel wire is 3.0 mm.