A point particle of mass m, moves along the uniformly rough track PQR… | JEE Main 2016

Q32JEE Main 2016Difficult

Conservation of Energy

A point particle of mass $m,$ moves along 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 $QR$ .

The value of the coefficient of friction $\mu$ and the distance $x\ (= QR),$ are, respectively close to:

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Answer: (a)

Length $PQ = \frac{h}{\sin 30^\circ} = 4\text{ m}$ . Loss on PQ $= \mu mg\cos 30^\circ \times 4 = 2\sqrt{3}\mu mg$ .
Loss on QR $= \mu mg x$ . Equal losses $\Rightarrow x = 2\sqrt{3} \approx 3.5\text{ m}$ .
Total loss = $mgh$ : $2 \times 2\sqrt{3}\mu mg = 2mg \Rightarrow \mu = \frac{1}{2\sqrt{3}} \approx 0.29$
So the answer is (a) $0.29$ and $3.5\text{ m}$ .

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