A piece of wire is bent in the shape of a parabola $y = k x^{2}$ (y-axis vertical) with a bead of mass ‘ $m$ ’ on it. The bead can slide on the wire without friction. The wire is now accelerated parallel to the axis with an acceleration ‘ $a$ ’ . The distance of the equilibrium position of the bead is $\frac{a}{x g k}$ where the bead can stay at rest with respect to the wire from y-axis. The value of ‘ $x$ ’ is

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Detailed Solution

$a = g T a n θ = g (\frac{d y}{d x})$

$y = k x^{2}$

$\Rightarrow \frac{d y}{d x} = 2 k x$

$∴ a = g 2 k x$

$x = \frac{a}{2 g k}$

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