JEE Main · 2005 · 08 May 2005 · Offline · Work, Power & Energy

The upper half of an inclined plane with inclination

ϕ

is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by

(A) $2 sin ϕ$
(B) $2 cos ϕ$
(C) $2 tan ϕ$ ✓ Correct
(D) $tan ϕ$

Answer: (C)

Solution

Take each half to have length

s

. The block begins and ends at rest, so over the whole run the loss in gravitational potential energy is spent entirely against friction, which acts only on the rough lower half. The drop over the full length

2 s

supplies

m g (2 s) sin ϕ

, while friction on the lower half does

μ m g cos ϕ \cdot s

. Equating the two,

2 sin ϕ = μ cos ϕ

, so

μ = 2 tan ϕ

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option

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