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3-2.Motion in Plane
hard
A particle is describing circular motion in a horizontal plane in contact with the smooth inside surface of a fixed right circular cone with its axis vertical and vertex down. The height of the plane of motion above the vertex is $h$ and the semivertical angle of the cone is $\alpha $ . The period of revolution of the particle
Aincreases as $h$ increases keeping $\alpha $ same
Bdecreases as $h$ increases keeping $\alpha $ same
Cdecreases as $\alpha $ increases keeping $h$ same
DNone of these
Solution
As $\mathrm{N} \sin \alpha=\mathrm{mg}$
$\mathrm{N} \cos \alpha=\mathrm{m} \omega^{2} \mathrm{r}$
$\tan \alpha=\frac{g}{\omega^{2} r}$
$\therefore \mathrm{T}^{2} \propto \mathrm{r} \tan \alpha$
$\therefore \mathrm{T}^{2} \propto \mathrm{h} \tan ^{2} \alpha$
for constant $\alpha$
$\mathrm{T}^{2} \propto \mathrm{h}$
Thus when $h$ increases $T$ also increases
$\mathrm{N} \cos \alpha=\mathrm{m} \omega^{2} \mathrm{r}$
$\tan \alpha=\frac{g}{\omega^{2} r}$
$\therefore \mathrm{T}^{2} \propto \mathrm{r} \tan \alpha$
$\therefore \mathrm{T}^{2} \propto \mathrm{h} \tan ^{2} \alpha$
for constant $\alpha$
$\mathrm{T}^{2} \propto \mathrm{h}$
Thus when $h$ increases $T$ also increases
Standard 11
Physics
