Question #42827

A particle of mass $m$ and charge $+q$ is constrained to move along a frictionless, non-conducting straight wire aligned with the $x$-axis. The setup is permeated by an external electric field given by $\vec{E} = E_0 \left( \frac{2x}{L} - \frac{x^2}{L^2} \right) \hat{i}$, where $E_0$ and $L$ are positive constants. The entire wire assembly is placed inside a rocket that accelerates along the positive $x$-axis with a constant acceleration $a_0 = \frac{3 q E_0}{4 m}$. In the reference frame of the rocket, the particle is released from rest at a position slightly displaced from its stable equilibrium point. Determine the position of the stable equilibrium point $x_{eq}$ and the angular frequency $\omega$ of its small oscillations.