Fringe Shift in a Michelson Interferometer Combined with Mechanics

Problem Statement

A Michelson interferometer uses a monochromatic light source with vacuum wavelength $\lambda$λ. Light from the source is split into two paths by a half-mirror (beam splitter). Path 1 has a fixed mirror $M_1$M1​, and Path 2 has a movable mirror $M_2$M2​ placed on a smooth horizontal floor with mass $M$M. The movable mirror $M_2$M2​ is connected to a light spring with spring constant $k$k and can oscillate horizontally.

Furthermore, the movable mirror $M_2$M2​ and the space it moves in are completely immersed in a long narrow tank filled with a transparent liquid of absolute refractive index $n$n. (The half-mirror and fixed mirror $M_1$M1​ are in air, with refractive index $1$1.)

In the initial state, the movable mirror $M_2$M2​ is at rest near the natural length of the spring, and the detector observes interference fringes of a certain brightness.

A bullet of mass $m$m flies horizontally at speed $v_0$v0​ and embeds completely into the back of the movable mirror $M_2$M2​ (perfectly inelastic collision). The collision time is extremely short, so immediately after the collision, $M_2$M2​ (with the bullet) acquires velocity $V$V and begins oscillating in the liquid. The resistance of the liquid is negligible.

When the movable mirror $M_2$M2​ oscillates, the optical path length of Path 2 changes continuously, causing the interference fringes at the detector to change periodically (fringe shift). Find the maximum frequency $f_{max}$fmax​ (number of fringe changes per second) observed at the detector during this oscillation.

Assume the speed of light is incomparably larger than the speeds of the bullet and mirror, so Doppler wavelength changes are negligible. Only changes in optical path difference need to be considered.

Constraints

• Vacuum wavelength of light source: $\lambda = 5.60 \times 10^{-7} \text{ m}$λ=5.60×10−7 m
• Absolute refractive index of liquid: $n = 1.25$n=1.25
• Mass of movable mirror: $M = 0.480 \text{ kg}$M=0.480 kg
• Spring constant: $k = 200 \text{ N/m}$k=200 N/m
• Mass of bullet: $m = 0.020 \text{ kg}$m=0.020 kg
• Initial speed of bullet: $v_0 = 175 \text{ m/s}$v0​=175 m/s

Input Format

Compute $f_{max}$fmax​ in units of $\text{MHz}$MHz ($10^6 \text{ Hz}$106 Hz) and give the answer as a positive integer equal to the value multiplied by $100$100.