A smooth hollow cone whose vertical angle is 2α, with it axis vertical and vertex downwards, revolves about its axis n times per second. Find where a particle may be placed on the inner surface of cone so that it rotates with same speed

Let the particle be placed at P. The forces on particle P are 

(a) Weight m g acting downward 

(b) Reaction R, perpendicular to OP 

(c) Normal force

$\mathrm{N}=4{\mathrm{π}}^2{\mathrm{n}}^2\left(\mathrm{BP}\right)\mathrm{m}=4{\mathrm{π}}^2{\mathrm{n}}^2\mathrm{rm}$

Resolving and equating the forces, we have

$\begin{array}{l}\mathrm{Rsin}⁡\mathrm{Î\pm }=\mathrm{mg}\\ \mathrm{Rcos}⁡\mathrm{Î\pm }=\mathrm{m}\left(4{\mathrm{π}}^2{\mathrm{n}}^2\right)\mathrm{r}\\ ∴ \tan ⁡\mathrm{Î\pm }=\frac{\mathrm{g}}{4{\mathrm{π}}^2{\mathrm{n}}^2\mathrm{r}}\\ \mathrm{r}=\frac{\mathrm{gcot}⁡\mathrm{Î\pm }}{4{\mathrm{π}}^2{\mathrm{n}}^2}\end{array}$