A block slides down a perfectly smooth inclined plane of angle in time t . If the same inclined plane is made rough with a coefficient of kinetic friction given by _k = 8 9 , the time taken by the block to slide down the same distance from rest is nt . Find the value of n .

Step-by-step solution

Let the length of the inclined plane be s . For the smooth inclined plane, the acceleration of the block is: a_S = g The time taken is t , so: s = 1 2 a_S t^2 For the rough inclined plane, the acceleration is: a_R = g - _k g Substituting _k = 8 9 : a_R = g - ( 8 9 ) g a_R = g - 8 9 g = 1 9 g The time taken on the rough incline is nt , so: s = 1 2 a_R (nt)^2 Equating the two expressions for s : 1 2 a_S t^2 = 1 2 a_R (nt)^2 n^2 = a_S a_R n^2 = g 1 9 g = 9 n = 3 Thus, the value of n is 3 . Answer: 3