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A bead under the influence of gravity, slides down a frictionless wire whose y coordinate is changing with x co-ordinate as shown in the figure. Assume that at position O, the wire is vertical and the bead passes this point with a given speed $v_{0}$ downward. If the shape of the wire is such that the vertical component of velocity remains $v_{0}$ at all time, find (a + b + c) in the shape function of wire given by $y = \frac{{(a g v_{0} x)}^{\frac{b}{c}}}{2 g}$ , where g is gravitational acceleration.

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answer is 8.

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Detailed Solution

$v = \sqrt{v_{0}^{2} + 2 g y}$
and $v \sin θ = v_{0}$ (as given in question)

Question Image

$∴ \sin θ = \frac{v_{0}}{\sqrt{v_{0}^{2} + 2 g y}}$

$∴ \tan θ = \frac{v_{0}}{\sqrt{2 g y}}$
$\frac{d y}{d x} = \frac{v_{0}}{\sqrt{2 g y}}$

$∴ y = \frac{{(3 g v_{0} x)}^{2 / 3}}{2 g}$

$∴ a + b + c = 8$

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