Angle of Friction and Angle of Repose

Angle of friction:
The angle made by the resultant of limiting friction with the normal reaction is called angle of friction. In the figure below, θ is the angle of friction.

In triangle OAR,

Tanθ=AROR=OFOR=FR

Here,FR=μis the coefficient of friction.

∴Tanθ=μ

Hence, the coefficient of friction is equal to the tangent of angle of friction.
Angle of Repose: The angle made by the inclined plane with horizontal such that the body just begin to slide is called angle of repose. In figure below, α is the angle of repose.

Angle of repose

Let us consider a body of mass ‘m’ placed over the surface of inclined plane OB. Here, mg is the weight of the body acting vertically downward. R is the normal reaction. mgsinθ and mgcosθ are the rectangular components of ‘mg’ as shown in figure. F is the limiting friction acting upward the plane. As the body just begin to slide, then from figure, we can write:
R = mgcosα……(i)
F = mgsinα……(ii)
Dividing equation (ii) by (i), we get:

FR=mgsinαmgcosα

or,μ=tanα[∵μ=FR]

Here, α is the angle of repose. So, we can conclude that coefficient of friction is equal to the tangent of angle of repose.
Since tanθ = tanα
∴ θ = α
Hence it is proved that the angle of friction is equal to the angle of repose.