Question

A given object takes n times more time to slide down 45° rough inclined plane as it takes to slide down a perfectly smooth 45° incline. The coefficient of kinetic friction between the object and the incline is 

(A) [1 / (2 – n²)] 

(B) [1 – (1 / n²)] 

(C) √[1 – (1 / n²)] 

(D) √[1 / (1 – n²)]

Answer 1

Correct option: (B) [1 – (1 / n²)]

Explanation:

Let time to slide down rough plane = t¹ 

time to slide down smooth place = t

 t¹ = nt

smooth surface, a = g sin θ

rough surface, a¹ = g(sin θ – u cos θ) – since it is sliding down 

also    s = v_o t + (1 / 2) at² 

as       v_o = o,

s = (1 / 2) at² 

t = √(2s / a)

t¹ = nt

√(2s / a₁) = n √(2s / a)

i.e. (1 / a¹) = n²(1 / a)

i.e. (a / a₁) = n² 

{(g sin θ) / (g(sin θ – μ cos θ))} = n² 

sin θ = n² (sin θ – μ cos θ)

θ = 45°

(1 / √2) = n² [(1 / √2) – (μ / √2)]

(1 / √2) = n² {(1 – μ) / (√2)} 

1 = n² (1 – μ) 

(1 / n²) = 1 – μ 

μ = [1 – (1 / n²)]