First slide

Mechanical equilibrium force cases

Question

A bead of mass m is located on a parabolic wire with its axis vertical and vertex directed downward as in figure and whose equation is x² = ay. If the coefficient of friction is µ , the highest distance above the x-axis at which the particle will be in equilibrium is

Moderate

A

µa
B

µ²a
C

(1/4) µ²a
D

(1/2) µ²a

Solution

For sliding not to occur when
$\large \tan \theta = \frac{{dy}}{{dx}} = \frac{{2x}}{a} = \frac{{2\sqrt {ya} }}{a} = 2\sqrt {\frac{y}{a}}$
$\large \therefore 2\sqrt {\frac{y}{a}} \leqslant \mu \;or\;y\, \leqslant \,\,\frac{{a{\mu ^2}}}{4}$

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