Asolid ball is released from rest and slides down a hillside that slopes downward at an angle 51.0 ∘ from the horizontal. what minimum value must the coefficient of static friction between the hill and ball surfaces have for no slipping to occur?

In order for the object not to slip, the component of the weight parallel to the surface must be equal to the frictional force (which acts in the opposite direction):


The parallel component of the weight is:

where m is the object mass and  is the angle of the inclined plane.

The frictional force is

where  is the coefficient of static friction.

Equalizing the two forces, we have

from which we find


and so, in our problem the coefficient of static friction must be