In the diagram,BAC is a rigid fixed rough wir | vedclass

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Question

(A) The angle $\angle BAC = 60^o$. Let $	heta$ be the angle the wire makes with the vertical. Since the system is symmetric,$	heta = 30^o$.The exten

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$2$

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(A) The angle $\angle BAC = 60^o$. Let $	heta$ be the angle the wire makes with the vertical. Since the system is symmetric,$	heta = 30^o$.The extension in the string is $x = PQ - 2a$. In $	riangle APQ$,$PQ = 2(3a) \sin(30^o) = 3a$. So,$x = 3a - 2a = a$.The spring force is $F_s = kx = (\frac{mg}{a}) \cdot a = mg$.For equilibrium of ring $P$ along the wire: $mg \cos(30^o) - f - F_s \sin(30^o) = 0$.$f = mg \frac{\sqrt{3}}{2} - mg \cdot \frac{1}{2} = mg \frac{\sqrt{3}-1}{2}$.For equilibrium perpendicular to the wire: $N - mg \sin(30^o) - F_s \cos(30^o) = 0$.$N = mg \cdot \frac{1}{2} + mg \cdot \frac{\sqrt{3}}{2} = mg \frac{1+\sqrt{3}}{2}$.For limiting friction,$f = \mu N$,so $\mu = \frac{f}{N} = \frac{\sqrt{3}-1}{\sqrt{3}+1} = \frac{(\sqrt{3}-1)^2}{3-1} = \frac{4-2\sqrt{3}}{2} = 2 - \sqrt{3}$.Thus,$\mu + \sqrt{3} = (2 - \sqrt{3}) + \sqrt{3} = 2$.

Column-$I$	Column-$II$
$(1)$ Definition of force	$(a)$ Newton's third law of motion
$(2)$ Measurement of force	$(b)$ Newton's second law of motion
	$(c)$ Newton's first law of motion

Similar Questions

An object of mass $10 \,kg$ is projected from the ground with a speed of $40 \,m/s$ at an angle of $60^{\circ}$ with the horizontal. The rate of change of momentum of the object one second after projection in $SI$ units is ........ [Take $g = 9.8 \,m/s^2$]

Match the items in Column-$I$ with those in Column-$II$.

Column-$I$ Column-$II$

$(1)$ Definition of force $(a)$ Newton's third law of motion

$(2)$ Measurement of force $(b)$ Newton's second law of motion

$(c)$ Newton's first law of motion

For the given system,which of the following relations holds between $T_1$,$T_2$,and $T_3$?

Explain why
$(a)$ a horse cannot pull a cart and run in empty space,
$(b)$ passengers are thrown forward from their seats when a speeding bus stops suddenly,
$(c)$ it is easier to pull a lawn mower than to push it,
$(d)$ a cricketer moves his hands backwards while holding a catch.

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