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

Step-by-step solution

For the smooth upper half, using v 2 − u 2 = 2 as , we get v 2 − 0 2 = 2 g ( sinϕ ) × l v = 2 g   ( sinϕ ) × l For the lower half, using v 2 − u 2 = 2 as , we get 0 2 − 2 g   sin   ϕ × l = − 2 ( μ k   cosϕ − sinϕ ) g × l or sinϕ = μ k cosϕ − sinϕ μ k cosϕ = 2 sinϕ or μ k = 2 tanϕ