Ignoring a component in DE

Question

A bead of mass $m$ is moving on the inner surface of a smooth circular cylinder of radius $R$ whose $Oz$-axis is directed vertically downwards.
A bead started to move from $x$-axis with velocity $\vec V$ parallel to $y$-axis.

I used cylindrical coordinates.
After calculation, using Conservation of energy law and Conservation of angular momentum law, I've got this equation $$\dot r^2+\dot z^2-2mgz=0$$

I have to show that $z$ changes as $$z(t)=\frac{gt^2}{2}$$

If I ignore $\dot r^2$, I get it. But I am not sure if I may do that.

Can anyone help me a bit?

Answer 1

The mass is moving on the surface of a cylinder, this is to say, in cylindrical coordinates you used, $r = \sqrt{x^2+y^2}$ does not change along with time, it remains a constant, namely the radius of the cylinder.

Remark: the reason why the mass will be always on the inner surface of this cylinder follows from physics, determined by the initial condition. Ignoring the velocity component in the $z$-direction, the inner surface can provide arbitrary large centripetal radially inward force to sustain the circular motion perpendicular to $z$-axis. Initially the mass is moving tangentially w.r.t. the surface, the magnitude of $x,y$-components of the velocity will remain the same. The trajectory will be a spiral, stretched more and more in the $z$-direction as time goes on.