Interference Fringe Shift from Buoyancy and Elastic Support

Problem Statement

A glass plate B with a flat horizontal upper surface is fixed in place. On top of it lies a uniform rectangular glass plate A of length $L$L. The left end of plate A (at $x = 0$x=0) is in contact with the upper surface of plate B, and this joint acts like a smooth hinge that can rotate — plate A can tilt by a small angle in the vertical plane.

Between the lower surface of the right end of plate A (at $x = L$x=L) and the upper surface of plate B, a small cylindrical elastic spacer of natural length $l_0$l0​ and unknown spring constant is inserted, supporting plate A. The spacer deformation follows Hooke's law; its mass and volume are negligible.

Let the mass of plate A be $M$M, its density $\rho$ρ, and the magnitude of gravitational acceleration be $g$g.

Between plates A and B, a very thin wedge-shaped gap is formed.

[Experiment 1] The entire apparatus is placed in air. Monochromatic light of wavelength $\lambda$λ is shone vertically downward onto the apparatus, and bright and dark interference fringes are observed due to interference between reflections from the lower surface of plate A and the upper surface of plate B.

The spacing between adjacent bright fringes is measured to be $\Delta x_1$Δx1​.

The refractive index of air is $1$1; the density and buoyancy of air are negligible.

[Experiment 2] The entire apparatus is fully submerged in a transparent unknown liquid of density $\rho_0$ρ0​ and refractive index $n$n, filling the wedge-shaped gap with the liquid as well.

After sufficient time and the plate A reaches equilibrium, monochromatic light of wavelength $\lambda$λ is again shone vertically downward. The spacing between adjacent bright fringes is now $\Delta x_2$Δx2​.

Assume the refractive indices of plates A and B are both greater than $n$n. The spring constant of the spacer does not change due to the liquid.

From these results, find the refractive index $n$n of the liquid.

Constraints

• Mass of plate A: $M = 0.40 \mathrm{\ kg}$M=0.40 kg
• Density of plate A: $\rho = 2.0 \times 10^3 \mathrm{\ kg/m^3}$ρ=2.0×103 kg/m3
• Length of plate A: $L = 0.20 \mathrm{\ m}$L=0.20 m
• Natural length of elastic spacer: $l_0 = 1.0 \times 10^{-4} \mathrm{\ m}$l0​=1.0×10−4 m
• Gravitational acceleration: $g = 10 \mathrm{\ m/s^2}$g=10 m/s2
• Wavelength of monochromatic light: $\lambda = 6.0 \times 10^{-7} \mathrm{\ m}$λ=6.0×10−7 m
• Density of liquid: $\rho_0 = 1.0 \times 10^3 \mathrm{\ kg/m^3}$ρ0​=1.0×103 kg/m3
• Fringe spacing in air: $\Delta x_1 = 0.75 \times 10^{-3} \mathrm{\ m}$Δx1​=0.75×10−3 m
• Fringe spacing in liquid: $\Delta x_2 = 0.50 \times 10^{-3} \mathrm{\ m}$Δx2​=0.50×10−3 m

Input Format

The refractive index $n$n is expressed as an irreducible fraction $\frac{A}{B}$BA​ with $A, B$A,B coprime positive integers. Compute $100A + B$100A+B and give the answer as a positive integer.