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

Step-by-step solution

s = u t + 1 2 a t 2 , a = g sin θ , u = 0 Then,    s = 0 + 1 2 a t 2 or   t = 2 s a = 2 s g sin θ [for smooth plane] For the rough plane. the effective value of acceleration along the incline is a ' = g sin θ - μ k g cos θ and t ' = 2 s a = 2 s g sin θ - μ k g cos θ Now,    t ' t = n = g sin θ g sin θ - μ k g cos θ = sin 45 ° sin 45 ° - μ k cos 45 ° = 1 1 - μ k ⇒   1 n 2 = 1 - μ