A particle of mass m moves under the influence of a force derived from a potential \(V = \frac{1}{2}m(\omega_x^2x^2 + \omega_y^2y^2 + \omega_z^2z^2)\ \) in the coordinates of a non-inertial frame F. The frame F is rotating with respect to an inertial frame with an angular velocity \( \Omega \hat{i} \ \), where \( \hat{i} \ \) is the unit vector along their common x-axis. The motion of the particle is unstable for all angular frequencies satisfying:
1
\(\Omega^2 - \omega_y^2)(\Omega^2 - \omega_z^2) > 0\ \)
2
\(\Omega^2 - (\omega_y + \omega_z)^2)(\Omega^2 - |\omega_y - \omega_z|^2) > 0 \ \)
3
\(\Omega^2 - (\omega_y + \omega_z)^2)(\Omega^2 - |\omega_y - \omega_z|^2) < 0\ \)
4
\(\Omega^2 - \omega_y^2)(\Omega^2 - \omega_z^2) < 0 \ \)
5
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