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

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a

$\frac{4 π^{2} n^{2}}{g}$

b

$\frac{g s i n α}{4 π^{2} n^{2}}$

c

$\frac{g s i n α}{4 π^{2} n^{2}}$

d

$\frac{g c o t α}{4 π^{2} n^{2}}$

answer is A.

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Detailed Solution

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$N s i n a = m g$ ..... (i)
$N c o s a = \frac{m v^{2}}{6}$ .....(ii)
$\frac{\cos α}{\sin α} = \frac{v^{2}}{r g} = \frac{ω^{2} r^{2}}{r g} = \frac{ω^{2} r^{2}}{g}$
$c o t α = \frac{ω^{2} r}{g}; r = \frac{g \cot α}{ω^{2}}$
$r = \frac{g \cot α}{4 π^{2} n^{2}}$

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