When a body slides down from rest along a smooth inclined plane making an angle of `45^(@)` with the horizontal, it takes time T. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time pT, where p is some number greater than 1. Calculate the coefficient of friction between the body and the rough plane.

Explanation

When slope is smooth as shown in figure boyd slides on slope
`F = ma`
`F= mg sin theta `
`a= g sin theta `
`theta = 45^(@)`

`a= g (1)/(sqrt(2))`
Let distance covered by body be d d = mt + `(1)/(2 ) at^(2)`
` d= (1)/(2) ( g)/(sqrt(2)) t^(2)`
`d= (gt^(2))/( 2sqrt(2))`
Here t = pt
d= - (p T ) `+(1)/(2) ` a (p T )
Comparing (1) and (2)
`(1-mu ) p^(2) =1`
`1-mu = (1)/( p^(2))`
`mu= 1= (1)/(p^(2))`