question_answer

A particle describes a horizontal circle on the smooth inner surface of a conical funnel as shown in Fig. 5(HT).7. If the height of the plane of the circle above the vertex is 9-8 cm, find the speed of the particle.

Answer:

                The forces acting on the particle are: weight (mg) of particle acting vertically downwards. Reaction (R) of funnel is acting perpendicular to wall of the funnel. As is clear from Fig. 5(HT).7,       \[R\,\cos \theta =mg\]                                                                                  ?..(i) \[R\,\sin \text{ }\!\!\theta\!\!\text{ }=\frac{m{{\upsilon }^{2}}}{r}\]                                                                                        ?.. (ii) Dividing (ii) by (i), we get \[\tan \text{ }\!\!\theta\!\!\text{ =}\frac{{{\upsilon }^{2}}}{rg}\] \[\upsilon =\sqrt{r\,g\,\tan \text{ }\!\!\theta\!\!\text{ }}\] From Fig. 5(HT).7, \[\tan \text{ }\!\!\theta\!\!\text{ =}\frac{h}{r}\] \[\therefore \]  \[\upsilon =\sqrt{rg\times \frac{h}{r}}=\sqrt{gh}=\sqrt{9.8\times 9.8\times {{10}^{-2}}}=0.98\,m\,{{s}^{-1}}\]