Friction

Friction

ANGLE OF REPOSE
The angle of repose (\(\alpha\))is defined as the angle of inclination of a plane with respect to horizontal for which the body will be in limiting equilibrium on the inclined plane
As the angle of inclination \(\theta\) is gradually increased,at a particular value of  \(\theta\) say \(\alpha\) ,the body will be just ready to slide down the plane.This angle of inclination  is called the angle of repose.
                                        

1.If\(\theta\) <\(\alpha\);then \( mg\sin \theta < f_L \) \( = \left( {\mu _s mg\cos \theta } \right) \) and body is at rest.noe friction is static.
The static fricitional force (f)= \( mg\sin \theta \)
2.The angle of inclination \(\theta\) < \(\alpha\);the bdy is ready to slide .The body is in limiting equilibrium,the net force on it should be equal to zero.That means mg sin\(\alpha\) becomes equal to the limiting friction(f_L)
i.e., f_L (or) \( f_L = \mu _s mg\cos \alpha = mg\sin \alpha \) 
              \( \therefore \mu _s = Tan\alpha (or)\alpha = Tan^{ - 1} (\mu _s ) \)
The angle of repose is equal to angle of friction 
3.If angle of inclination\(\theta\) > \(\alpha\);
then\( mg\sin \theta \) > f_L \( = \left( {\mu _s mg\cos \theta } \right) \) and the body slides down on a rough inclined plane under kinetic friction.
The kinetic friction \( f_k = \mu _k N = \mu _k mg\cos \theta \)