Question 304 of 308

A bead of mass $m$ stays at point $P\left(a,b\right)$ on a wire bent in the shape of a parabola $y=4C{x}^{2}$ and rotating with angular speed $\omega$ (see figure). The value of $\omega$ is (neglect friction)

A$2\sqrt{2gC}$B$\sqrt{2gC}$C$\sqrt{\dfrac{2gC}{ab}}$D$\dfrac{\sqrt{2g}}{C}$

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Answer: (A) $2\sqrt{2gC}$

$m{\omega }^{2}acos\theta =mgsin\theta$
$\omega =\sqrt{\dfrac{g\tan \theta }{a}}$
$y=4c{x}^{2}$
$\tan \theta =\dfrac{dy}{dx}=8xC$
$(\tan \theta {)}_{a,b}=8aC$
$\omega =\sqrt{\dfrac{g\times 8ac}{a}}=2\sqrt{2gC}$