The time taken by a block of mass m to slide down from the highest point to the lowest point on a rough inclined plane is 50 % more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at 45^ with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is _____.

Step-by-step solution

Let the length of the inclined plane be s . For the smooth inclined plane, the acceleration of the block is a₁ = g . The time taken to slide down is t₁ = 2s a₁ = 2s g . For the rough inclined plane, the acceleration of the block is a₂ = g - _k g . The time taken to slide down is t₂ = 2s a₂ = 2s g ( - _k ) . Given that t₂ is 50 % more than t₁ , we have t₂ = t₁ + 0.5 t₁ = 3 2 t₁ . Squaring both sides gives t₂^2 = 9 4 t₁^2 . Substituting the expressions for t₁^2 and t₂^2 : 2s g ( - _k ) = 9 4 ( 2s g ) 1 - _k = 9 4 4 =