The smooth particle $P$ moves in the slot of a rotating arm $OB$ and is attached to a spring extending from the fixed point $A$ to $P$. The mass of $P$ is $m$. The spring's stiffness is $k_{sp}$. The unstretched length of the spring is $L_0$. The radial position and radial velocity of $P$ are $r$ and $\dot{r}$, respectively. At the instant shown in Figure A1: the angle $\phi$ is known, and the length of the spring is $1.5L_0$. The arm $OB$ is rotating with a constant angular speed $\omega$ (clockwise) to the horizontal position. $\mathbf{u}_r$ and $\mathbf{u}_\theta$ represent the unit vectors of radial and transverse directions, respectively.

Note: $\angle APB = \phi = 60^\circ$, $m = 3$ kg, $k_{sp} = 10$ N/m, $L_0 = 1.6$ m, $\omega = 2$ rad/s.