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 d

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-a 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)`