Question

A solid 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?

Answer 1

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):
$F_{//}= F_a$

The parallel component of the weight is:
$F_{//} = mg \sin \alpha$
where m is the object mass and $\alpha$ is the angle of the inclined plane.

The frictional force is
$F_a = \mu m g \cos \alpha$
where $\mu$ is the coefficient of static friction.

Equalizing the two forces, we have
$mg \sin \alpha = \mu m g \cos \alpha$
from which we find
$\mu = \tan \alpha$

and so, in our problem the coefficient of static friction must be
$\mu=\tan 51^{\circ} =1.23$