A board of mass M is placed on a rough inclined plane and a man of mass m walks down the board. If the coefficient of friction between the board and inclined plane is μ μ , the acceleration of the man, such that plank does not slip, is given by
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Let F 1 be the force between the man and the board and F 2 be the force of friction between the inclined plane and the board.
Here F 1 can have a value between Mg sin θ θ - μ μ (M + m) g cos θ θ and Mg sin θ θ + μ μ (M + m) g cos θ θ … (i)
Limiting value of F 2 = μ μ N 2
= μ μ (M + m) g cos θ θ
the force equations are
F 1 + m sin θ θ = ma
i.e., F 1 = ma – mg sin θ θ
From (i)
Mg sin θ θ - μ μ (M + m) g cos θ θ ≤ F 1 ≤ Mg sin θ θ + μ μ (M + m) g cos θ θ
i.e., Mg sin θ θ - μ μ (M + m) g cos θ θ ≤ ma – mg sin θ θ ≤ mg sin θ θ + μ μ (M + m) g cos θ θ
i.e., 
(sin θ θ - μ μ cos θ θ )g ≤ a ≤ 
(sin θ θ + μ μ cos θ θ ) g
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