The velocity-time graphs of a block sliding from rest down a perfectly smooth incline and a rough incline (both having an angle of inclination of 30^ and the same length) are plotted. Both graphs are straight lines passing through the origin. If the ratio of the slope of the graph for the smooth incline to that of the rough incline is 2 , find the coefficient of kinetic friction between the block and the rough inclin

Step-by-step solution

The slope of a velocity-time graph represents the acceleration of the object. Let a_S be the acceleration on the smooth incline and a_R be the acceleration on the rough incline. We are given: a_S a_R = 2 The acceleration of a block on a smooth incline of angle is: a_S = g For = 30^ : a_S = g 30^ = g 2 The acceleration on a rough incline is: a_R = g( - _k ) For = 30^ : a_R = g ( 30^ - _k 30^ ) = g ( 1 2 - _k 3 2 ) Substitute these into the ratio: g 2 g ( 1 2 - _k 3 2 ) = 2 Cancel g and simplify the denominator: 1 2