As shown in the figure, a small ball with gravity g is placed vertically on a smooth ring with radius R. the stiffness coefficient is K and the natural length is l (L)

As shown in the figure, a small ball with gravity g is placed vertically on a smooth ring with radius R. the stiffness coefficient is K and the natural length is l (L)

When the angle is θ, l '= 2R * cos θ. T = (2R * cos θ - L) * K
The stress analysis shows that t * sin θ = g * sin2 θ
That is t * sin θ = g * 2Sin θ cos θ, then 2G * cos θ = t = (2R * cos θ - L) * K
We obtain that θ = arccos (LK / (2rk-2g))