Spring Compressed on Parabola

Classical Mechanics Level 3

Two \(\SI{1}{\kilo\gram}\) beads are at the ends of a compressed spring resting on a smooth parabolic wire \(\big(y = x^2\big).\) The spring has a natural length of \(\SI{1}{\meter}\) and a force constant of \(\SI[per-mode=symbol]{5}{\newton\per\meter}.\)

If the system is in static equilibrium, how far away from the \(y\)-axis is each bead (in meters)?

Note: The spring is horizontal and gravity is \(\SI[per-mode=symbol]{10}{\meter\per\second\squared}\) in the \(-y\) direction.

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