Small Oscillations of a Mass Point on a Rotating Ring

Problem Statement

Consider a fixed coordinate system with the positive $z$z-axis pointing vertically upward. The magnitude of gravitational acceleration is $g$g, and gravity acts in the $-z$−z direction.

A thin, smooth ring of radius $R$R is fixed with its diameter aligned along the $z$z-axis and rotates at constant angular velocity $\omega$ω about the $z$z-axis.

A mass point of mass $m$m is threaded onto the ring and can slide along it without friction. The position of the mass point is uniquely described by the central angle $\theta$θ ($0 \le \theta \le \pi$0≤θ≤π) measured from the lowest point of the ring (on the $z$z-axis).

When the angular velocity satisfies $\omega > \sqrt{g/R}$ω>g/R​, there exists a stable equilibrium position $\theta = \theta_0$θ=θ0​ ($0 < \theta_0 < \pi/2$0<θ0​<π/2) other than the lowest point.

Starting from this stable equilibrium position, the mass point is displaced slightly along the ring and released from rest, after which it undergoes small oscillations about $\theta_0$θ0​.

Let $\Omega$Ω be the angular frequency of this small oscillation. Find the value of $\Omega^2$Ω2.

Constraints

• Magnitude of gravitational acceleration: $g = 9.80 \text{ m/s}^2$g=9.80 m/s2
• Radius of ring: $R = 0.50 \text{ m}$R=0.50 m
• Angular velocity of ring: $\omega = 7.00 \text{ rad/s}$ω=7.00 rad/s
• Mass of mass point: $m = 0.100 \text{ kg}$m=0.100 kg

Input Format

Compute $\Omega^2$Ω2 in units of $\text{rad}^2/\text{s}^2$rad2/s2 and give the answer as a positive integer equal to the value multiplied by $100$100.