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Q.

A piece of wire is bent in the shape of a parabola y=kx2  (yaxis  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 x-axis with an acceleration ‘a’. The distance of the new equilibrium position of the bead is  axgk 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

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There will be pseudo force in the -ve x direction on the bead

At equilibrium F=0

Therefore in the tangential line at the equilibrium position

macosθ=mgsinθ

At that point, slope of the curve is 

dydx=tanθ

ddx(kx2)=tanθ

Therefore from the above equations

 2kx=ag

x = a2gk

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