A bead of mass m stays at point P(a, b) on a wire bent in the shape...

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 bead of mass m stays at point P(a, b) on a wire bent in the shape of a parabola y = 4C x ^ 2 and rotating with

Options

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

Correct answer

A. $2\sqrt{2gC}$

Step-by-step solution

A bead of mass m stays at point P(a, b) on a wire bent in the shape of a parabola y = 4C x ^ 2 and rotating with

$m{\omega}^{2}acos\theta = mgsin\theta$

$\omega = \sqrt{\dfrac{gtan\theta}{a}}$

$y = 4c{x}^{2}$

$tan\theta = \dfrac{dy}{dx} = 8xC$

$( tan\theta{)}_{a, b} = 8aC$

$\omega = \sqrt{\dfrac{g \times 8ac}{a}} = 2\sqrt{2gC}$

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