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A particle is describing circular motion in a horizontal plane in contact with the smooth 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 semi-vertical angle of the cone is $α$ . The period of revolution of the particle

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decreases as h decreases

decreases as $α$ increases

increases as $α$ increases

increases as h increases

answer is A, D.

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Detailed Solution

$R = h \cot α N \cos α = mg (i) N \sin α = \frac{{mv}^{2}}{R} (ii)$

Solving these two equations, we get

$v = \sqrt{Rgtanα} = \sqrt{(hcotα) (gtanα)} = \sqrt{gh} Now, T = \frac{2 πR}{v} = \frac{2 πh cotα}{\sqrt{gh}}$

$= 2 π \sqrt{\frac{h}{g}} cotα ∴ T \propto \sqrt{h} and T \propto cotα$

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