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A bead of mass m rests on a vertical smooth parabola $x^{2} = c y at (- a, b)$ and rotating with angular speed $ω$ as shown in the figure. The required value of angular velocity $ω$ is $\sqrt{\frac{n g}{c}}$ . Find $n$ .

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answer is B.

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Detailed Solution

Here $N cosθ=mg$

and ${N sinθ=maω}^{2}$

or $tanθ= \frac{{aω}^{2}}{g}$

$x^{2} =cy$

$\Rightarrow 2x=c \frac{dy}{dx}$

Or $\frac{dy}{dx} = \frac{2x}{c}$

$∴ tanθ= \frac{2a}{c}$

or $\frac{{aω}^{2}}{g} = \frac{2a}{c}$

$∴ ω = \sqrt{\frac{2 g}{c}}$

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