Question

A solid sphere of uniform density starts from rest and rolls without slipping down an inclined plane with angle ? = 30o. the sphere has mass m = 8 kg and radius r = 0.19 m . the coefficient of static friction between the sphere and the plane is ? = 0.64. what is the magnitude of the frictional force on the sphere?

Answer 1

Let a be the linear acceleration of the center of mass, fs the magnitued of the frictional force, I be the rotational inertia of the ball, M be the mass, R the moment arm, and g be the gravity.

For any body rolling along an incline of angle O with the horizontal, a is :
   a = - g sin O / ( 1 + (I / MR^2))

and 
   fs  = - I (a / R^2)

Thus, substituting the given value to the formulas above, we get (assuming  g = 9.8 m/s^2 and I = 2/5 MR^2 for a solid sphere)
 
a = -(9.8 m/s^2)sin30/(1 + ((2/5MR^2) / (MR^2)))
   = -(9.8 m/s^2) sin 30/ (1 + (7/5))
   = -3.5 m/s^2

and

fs = - (2/5MR^2)a / R^2
    = - (2/5M)a
    = - (2 * 8 / 5) ( -3 .5)
    = 11.2 N