Define angle of friction and angle of repose. Find the relationship between them.

Explanation

Angle of friction: Angle between normal reaction and resultant of limiting friction and normal is known as angle of friction
The resultant of limiting friction (F) and normal reaction (R) is represented by R. in the figure.
Angle `theta` between R and R. is called angle of friction. from figure
`tantheta= (F)/(R)` ,
But `(F)/(R)= mu_(s) :. tan theta= mu_(s)`

Angle of repose: It is the minimum angle made by the inclined plane with the horizontal plane such that the body placed on inclined plane just begins to slide.
Let `alpha` be the angle made by inclined plane with horizontal plane.
From figure, we have
`R= mg cosalpha` and `F= mg sinalpha`
`:. (F)/(R)= (mg sinalpha)/(mgcosalpha)= tanalpha` But `(F)/(R)= mu_(x)= tan alpha`......(ii)
From equation (i) and (ii), we get `tantheta = tanalpha implies theta= alpha` .....(iii)

Thus, angle of friction is equal to angle of repose.