Motion in helix-shaped wire 
A bead of mass $m$ is moving on a vertical axis helix shaped wire of uniform radius $R$ under force of gravity.

Can we apply centripetal force = $(mv^2)/R$ (at any instant when horizontal velocity is v)? I couldn't quite understand how we can or can't as the radius of curvature of its trajectory isn't $R$ (as this isn't a circular motion). But again the vertical and horizontal velocities are independent, so maybe we can. Any clarification would be helpful.
 A: Sure. If you only consider the "polar part" of the motion, then for Newton's law in cylindrical coordinates we have
$$\mathbf F=m(\ddot r-r\dot\theta^2)\,\hat r+m(r\ddot\theta+2\dot r\dot\theta)\,\hat\theta+m\ddot z\,\hat z$$
Since the helix has a constant radius, the centripetal component of the force just becomes
$$F_c=-mr\dot\theta^2=-\frac{mv^2}{r}$$
where $v=r\dot\theta$ is the speed parallel to the x-y plane.
Of course, there is also a tangential component acting on the bead from the wire, and there is also the vertical force of gravity to consider (the other terms in the above equation) but that doesn't change how to look at the centripetal force here. 
Remember, "centripetal force" is just the force component in some direction (in this case the center of the circle). It is just like saying "vertical force", "horizontal force" etc. 
Also, note that to do this analysis you actually need to know what $\theta(t)$ is. Using Newtonian mechanics this seems like a backwards task, because we need to know the forces to determine $\theta(t)$. This is where Lagrangian mechanics would be useful, as you can determine $\theta(t)$ just from knowing about the constraints and energies of your system. Then from there you can apply Newtonian mechanics to actually determine the centripetal force acting on the bead.
