Smooth block is released at rest on a 45^@incline and then slides a distance d. If the time taken of slide on rough incline is n times as large as that to slide than on a smooth incline. Show that coefficient of friction. mu = (1- (1)/(n^2))

Smooth block is released at rest on a 45^@incline and then slides a di

1.	Smooth block is released at rest on a 45^@incline and then slides a distance d. If the time taken of slide on rough incline is n times as large as that to slide than on a smooth incline. Show that coefficient of friction. mu = (1- (1)/(n^2))
Answer» Solution :When there is no friction, the block slides down the inclined plane with ACCELERATION. ` a = g SIN theta ` when there is friction, the downward acceleration of the block is ` a. =g (sin theta - mu COS theta)` As the block slides a distance d in each case so ` d = 1/2 at^2 = 1/2 a.t.^2` `(a)/(a.) = (t.^2)/(t^2) = ((nt)^2)/(t^2) = n^2 " or"(g sin theta)/(g (sin theta -mu cos theta) )=n^2` Solving , we get (using ` theta =45^@` ) `mu = 1 - (1)/(n^2)`