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Q.

In the given figure, a smooth parabolic wire track lies in the $x y -$ plane (vertical). The shape of track is defined by the equation $y = x^{2}$ . A ring of mass $m$ which can slide freely on the wire track, is placed at the position $A (1, 1)$ . The track is rotated with constant angular speed $ω$ such that there is no relative slipping between the ring and the track. The value of $ω$ is

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a

$\sqrt{2 g}$

b

$g$

c

$2 g$

d

$\frac{g}{2}$

answer is A.

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Detailed Solution

Question Image

$N \cos θ = m g$ and $N \sin θ = m ω^{2} r$

$∴ \tan θ = \frac{ω^{2} r}{g}$ …… (i)

Given $y = x^{2}; ∴ \frac{d y}{d x} = 2 x$

or $\tan θ = 2 \times 1 = 2$ …… (ii)

From above equations we get

$ω = \sqrt{2 g} (r = 1 m)$

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In the given figure, a smooth parabolic wire track lies in the xy-plane (vertical). The shape of track is defined by the equation y=x2 . A ring of mass m which can slide freely on the wire track, is placed at the position A (1,1). The track is rotated with constant angular speed ω such that there is no relative slipping between the ring and the track. The value of ω is