Precision Spring Scale for Pharmacy Use

Problem Statement

A pharmacy uses a precise scale consisting of a spring $S$S hung vertically to prepare small doses of medication. The upper end of spring $S$S is fixed to a point $P$P on the ceiling, and the natural length of the spring (when nothing is attached) is $L_0$L0​. The spring obeys Hooke's law, so its extension is proportional to the mass of the object hung from it.

First, a standard weight $A$A of mass $m_1$m1​ is gently hung from the lower end of the spring as a calibration test. The spring stretches and reaches equilibrium, and the total length of the spring at that point is $L_1$L1​.

Next, weight $A$A is removed and replaced by a reagent container $B$B of mass $m_2$m2​. Find the total length $L_2$L2​ of the spring when it reaches equilibrium with container $B$B hanging from it. The magnitude of gravitational acceleration $g$g is constant. The mass of the spring and air resistance can be neglected.

Constraints

• Natural length of spring: $L_0 = 0.112\text{ m}$L0​=0.112 m
• Mass of standard weight $A$A: $m_1 = 0.200\text{ kg}$m1​=0.200 kg
• Total length with weight $A$A hung: $L_1 = 0.152\text{ m}$L1​=0.152 m
• Mass of reagent container $B$B: $m_2 = 0.635\text{ kg}$m2​=0.635 kg

Input Format

Find the total length $L_2$L2​ in units of $\text{m}$m and give the answer as a positive integer equal to the value multiplied by $1000$1000.