Angle of Friction:
Let us consider a body placed on a rough surface. A normal force R acts on the body such that R is equal and opposite to its weight W. Now if a horizontal force is applied to the body, a frictional force say f1, comes into play.
Under this condition, there are two forces f1 and R acting on the body. The resultant reaction force Q1 and it makes some angle a with the normal reaction. Now, if the force applied to the body is increased, the friction also increases until it becomes equal to the limiting friction f. Then, let the angle made by the resultant reaction force Q with R Be λ. This angle λ is called Angle of Friction or Limiting Angle of Friction.
Angle of Repose:
Consider a body of weight W, placed on a surface inclined at an angle θ with the horizontal. The normal force of the inclined plane on the body is R=cosθ. In this case, the component Wsinθ along the plane tries to set up a downward motion of the body. Consequently, a frictional force develops upwards along the surface. If the inclination θ is such that the surface just prevents the downward motion of the body, then the frictional force f is the limiting friction expressed as f = Wsinθ.
Hence, the coefficient of friction,
µ = f/R = Wsinθ/Wcosθ = tanθ
Therefore, when the tangent of the angle of inclination equals the coefficient of friction, the body remains in limiting equilibrium on the inclined plane. If the angle of inclination θ is increased further, Wsinθ also increases. Then the limiting friction f can’t balance the body and it starts to slide down the inclined plane. The angle of inclination θ is called the Angle of Repose of the inclined plane.