JEE MAIN - Physics (2016 (Offline) - No. 22)
A point particle of mass $$m,$$ moves long the uniformly rough track $$PQR$$ as shown in the figure. The coefficient of friction, between the particle and the rough track equals $$\mu .$$ The particle is released, from rest from the point $$P$$ and it comes to rest at point $$R.$$ The energies, lost by the ball, over the parts, $$PQ$$ and $$QR$$, of the track, are equal to each other , and no energy is lost when particle changes direction from $$PQ$$ to $$QR$$.
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The value of the coefficient of friction $$\mu $$ and the distance $$x$$ $$(=QR),$$ are, respectively close to:
$$0.29$$ and $$3.5$$ $$m$$
$$0.29$$ and $$6.5$$ $$m$$
$$0.2$$ and $$6.5$$ $$m$$
$$0.2$$ and $$3.5$$ $$m$$
Explanation
Using work energy theorem for the motion of the particle,
Loss in $$P.E.=$$ Work done against friction from $$p \to Q$$
$$ + $$ work done against friction from $$Q \to R$$
$$mgh = \mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$
$$h = \mu \,\cos \,\theta \times PQ + \mu \left( {QR} \right)$$
$$2 = \mu \times {{\sqrt 3 } \over 2} \times {2 \over {\sin \,{{30}^ \circ }}} + \mu x$$
$$2 = 2\sqrt 3 \mu + \mu x$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....(i)$$
[ $$\sin \,\,{30^ \circ } = {2 \over {PQ}}$$ ]
According to the question, work done $$P \to Q = $$ work done $$Q \to R$$
$$ \Rightarrow $$$$\mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$
$$\therefore$$ $$2\sqrt 3 \,\mu = \mu x$$
$$\therefore$$ $$x \approx 3.5m$$
From $$(i)$$
$$2 = 2\sqrt 3 \mu + 2\sqrt 3 \mu = 4\sqrt 3 \mu $$
$$\therefore$$ $$\mu = {2 \over {4\sqrt 3 }} = {1 \over {2 \times 1.732}} = 0.29$$
Loss in $$P.E.=$$ Work done against friction from $$p \to Q$$
$$ + $$ work done against friction from $$Q \to R$$
$$mgh = \mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$
$$h = \mu \,\cos \,\theta \times PQ + \mu \left( {QR} \right)$$
$$2 = \mu \times {{\sqrt 3 } \over 2} \times {2 \over {\sin \,{{30}^ \circ }}} + \mu x$$
$$2 = 2\sqrt 3 \mu + \mu x$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....(i)$$
[ $$\sin \,\,{30^ \circ } = {2 \over {PQ}}$$ ]
According to the question, work done $$P \to Q = $$ work done $$Q \to R$$
$$ \Rightarrow $$$$\mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$
$$\therefore$$ $$2\sqrt 3 \,\mu = \mu x$$
$$\therefore$$ $$x \approx 3.5m$$
From $$(i)$$
$$2 = 2\sqrt 3 \mu + 2\sqrt 3 \mu = 4\sqrt 3 \mu $$
$$\therefore$$ $$\mu = {2 \over {4\sqrt 3 }} = {1 \over {2 \times 1.732}} = 0.29$$
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