The coefficient of static friction along an inclined plane is equal to tan alpha. What about the coefficient of kinetic friction?

Answer 1

It's not. The coefficient of static friction is only equal to the tangent of the angle of incline at the maximum angle before the object begins to slide. At this point static friction equals the component of the weight along the incline (weight X sin alpha). Static friction is given by the coefficient of static friction times the normal force (weight X cos alpha) fs = us N = us mg cos(alpha) Wx =mg sin(alpha) fs = Wx us mg cos(alpha) = mg sin(alpha) us = [sin(alpha)] / [cos(alpha)] = tan(alpha) Similarly, the coefficient of kinetic friction equals the tangent of the angle of incline only if the object is sliding down the incline at constant velocity (net force equals zero). If the object is accelerating along the incline (make this the x axis): Fnet, x = Wx - f max = mg sin(alpha) - uk mg cos(alpha)

uk = [g sin(alpha) - ax] / [g cos(alpha)]