Watt Governor: Non-Linear Small Oscillations in a Rotating System

Problem Statement

Let $g$g be the magnitude of gravitational acceleration on a certain planet.

A thin vertical rotating axis is fixed. At a smooth hinge at point O on this axis, two massless rigid rods of length $l$l connect to two balls A and B (both of mass $m$m, treated as point masses).

From each ball, another massless rigid rod of length $l$l extends, and the other ends of these rods are connected to a common smooth hinge at sleeve C (treated as a point mass of mass $M$M).

Sleeve C can slide up and down the rotating axis without friction. Balls A and B are always in symmetric positions on opposite sides of the axis. The quadrilateral O-A-C-B always forms a rhombus, and these four points always lie in the same plane (which contains the axis) and rotate together around the axis.

The entire device is forced to rotate at constant angular velocity $\omega$ω around the axis. Sleeve C reaches a steady position at distance $z_0$z0​ from O (as seen in the rotating frame). At this point, let $\theta_0$θ0​ ($0 < \theta_0 < \pi/2$0<θ0​<π/2) be the angle between each rod and the rotation axis.

Then, in the rotating frame, sleeve C is given a small vertical displacement and released, after which it undergoes small oscillations about the steady-state position.

Find the period $T$T of these small oscillations.

Air resistance and friction at all hinges are negligible.

Constraints

• Gravitational acceleration: $g = 12 \text{ m/s}^2$g=12 m/s2
• Mass of balls A and B: $m = 4 \text{ kg}$m=4 kg
• Mass of sleeve C: $M = 3 \text{ kg}$M=3 kg
• Length of rigid rods: $l = 1.4 \text{ m}$l=1.4 m
• Angular velocity of rotation: $\omega = 5 \text{ rad/s}$ω=5 rad/s

Input Format

The period $T$T of small oscillations can be expressed as $T = \frac{A}{B}\pi \text{ [s]}$T=BA​π [s] with coprime positive integers $A, B$A,B. Find the value of $100A + B$100A+B and give the answer as a positive integer.