Consider a block placed on a rough floor. Now, the reaction force is because it is equal and opposite to the weight . Now apply a horizontal force so that the block just begins to slide, i.e., the frictional force is equal to the limiting friction. When this condition is satisfied, the angle which the resultant (between the normal and external force) makes with the vertical is called the angle of friction.
The tangent of the angle of friction is equal to the coefficient of static friction.
Let us see the mathematical aspect of it. Consider the figure .
The weight W can be resolved into two rectangular components. W cos f and W sin f. The component W cos f balances the normal reaction R while the component W sin f is equal to the limiting friction flimiting.
We notice that the angle of repose or sliding and the angle of friction are both equal.
Consider the body to be sliding down the plane. You must take all force components along and perpendicular to the surface of the block.
Balancing components along the perpendicular of the inclined plane
Balancing components along the surface of the inclined plane
[a is the acceleration of the block along the inclined surface]
This equation gives the acceleration of a body sliding down a rough inclined plane.
Similarly, when any body is pushed on a horizontal floor, work is done against friction.
Work done against friction = force . displacement= fkinetic . S
Component parallel to the surface
(Note that the body is being pulled with a constant velocity)
Let the body be pulled through a distance S.Hence, the work done will be P.S
In the case of moving a block down, mg sin q will help it to come down, but friction will oppose it. Hence, work done in moving a block down the incline:
Work = mg (m cos q - sin q)S.