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Q.

A thin wire is bent into the shape of a parabola in the vertical plane as shown. A small bead ‘P’ can freely slide along the wire. The coefficient of friction between the bead and the wire is $' μ'$ . The condition on y so that the bead does not slip on the wire is :

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a

$y \leq 2 a μ$

b

$y \leq \frac{2 a}{μ}$

c

$y \geq \frac{2 a}{μ}$

d

$y \geq 2 a μ$

answer is B.

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Detailed Solution

Given
Coefficient of friction= $μ$
From diagram $y^{2} = 4 a x$
Differentiating on both sides
$2 y \frac{d y}{d x} = 4 a$
$\Rightarrow \frac{d y}{d x} = \frac{2 a}{y}$
For bead to be stationary, slope of tangent on the curve $\frac{d y}{d x} = \tan θ \leq μ$
$\Rightarrow \frac{2 a}{y} \leq μ$

$\Rightarrow y \geq \frac{2 a}{μ}$

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