I've asked this question before, but it was closed down as I didn't show any working. I have now completed all of the question apart from (bii). I think that the polar coordinates are:
$$x=l \sin(\alpha)\cos(\omega)\\ y=l \sin(\alpha)\sin(\omega)\\ z=l\cos(\alpha)$$
but I'm unsure. Could someone please check these polar coordinates and then guide me where to go from there.
A rectilinear tube of length $2l$ rotates with a constant angular speed $\omega$ around the vertical axis through the middle of the tube at a constant angle $\alpha$ (not equal to $\theta$) with the tube. The tube does not move up or down. A bead slides along the tube.
(a) At some moment of time the bead is at a distance $b$ from the middle of the tube, and has a speed $V_r$ relative to the tube. What is the absolute speed $V_a$ of the bead at this moment of time? (Hint: The absolute velocity is the vector sum of the relative and transport velocities.)
(b) Suppose that the distance $r$ of the bead from the middle of the tube changes according to the equation $\ddot r = −A$, $A =$ const > 0 (i.e. the bead moves with a constant acceleration relative to the tube). Suppose that at the initial moment of time $t = 0$ the bead is at the upper endpoint of the tube and has the speed $0$ relative to the tube.
(i) Determine the dependence of $r$ on time $t$ for $0 \le t \le t_∗$, where $t_∗$ is the instant when the bead reaches the middle of the tube. Find $t_∗$.
(ii) Consider an absolute Cartesian coordinate frame $O_{xyz}$ with the origin $O$ at the middle of the tube and the vertical direction of the axis $O_z$. Assume that at $t = 0$ the tube is in the plane $y = 0$ of this system. Determine the dependence of coordinates $x, y, z$ of the bead on time for $0 \le t \le t_∗$. (Hint: Determine the dependence of spherical polar coordinates of the bead on time. Then pass to Cartesian coordinates.)