Q: The upper half of an inclined plane with inclination of φ is perfectly smooth while the lower half is rough. A body at the top starts from rest will again come to rest at the bottom, the coefficient of friction for the lower half if given by

(a) 2 tanφ

(b) tanφ

(c) 2 sinφ

(d) 2 cosφ

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Ans: (a)

Sol: For upper half :

$v^2 = u^2 + 2 a s $

$v^2 = 0 + 2 (g sin\phi) (\frac{d}{2}) $ …(i)

For Lower half :

$0 = v^2 + 2(g sin\phi – \mu g cos\phi) (\frac{d}{2}) $ …(ii)

From (i) ,

$ 0 = 2 (g sin\phi) (\frac{d}{2}) + 2(g sin\phi – \mu g cos\phi) (\frac{d}{2}) $

$ 0 = 2 sin\phi – \mu cos\phi $

$\mu = 2 tan\phi $