A block starting from rest slides down a rough inclined plane of angle 30^ . The time taken by the block to slide down a certain distance on this rough incline is twice the time it would take to slide down the same distance on a perfectly smooth incline of the same angle. The coefficient of kinetic friction between the block and the rough incline is :

Step-by-step solution

Let the distance be d . For a block starting from rest, the distance covered is given by d = 1 2 at^2 . This implies that the acceleration a is inversely proportional to the square of the time taken, so a_ smooth a_ rough = ( t_ rough t_ smooth )^2 . Given t_ rough = 2 t_ smooth , we have a_ smooth = 4 a_ rough . The acceleration on the smooth incline is: a_ smooth = g 30^ = g 2 The acceleration on the rough incline is: a_ rough = g( 30^ - _k 30^ ) = g ( 1 2 - _k 3 2 ) Substituting these into the relation a_ smooth