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?


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.

  • $\begingroup$ @gov no problem, just in case the problem asks you to justify why the mass will remain on the inner surface, I updated a little bit more argument. $\endgroup$
    – Shuhao Cao
    Aug 20 '13 at 22:45
  • $\begingroup$ Great, I appreciate it! :) $\endgroup$
    – gov
    Aug 20 '13 at 22:56

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