The particle $P$ is released at time $t=0$ from the position $r=r_{0}$ inside the smooth tube with no velocity relative to the tube, which is driven at the constant angular velocity $\omega_{0}$ about a vertical axis. Determine the radial velocity $v_{r},$ the radial position $r,$ and the transverse velocity $v_{\theta}$ as functions of time $t .$ Explain why the radial velocity increases with time in the absence of radial forces. Plot the absolute path of the particle during the time it is inside the tube for $r_{0}=0.1 \mathrm{m}, l=1 \mathrm{m},$ and $\omega_{0}=1 \mathrm{rad} / \mathrm{s}$.