Let us consider a block moving with speed v on a horizontal, friction-free surface. Now the block encounters a plane inclined upward at some angle q with the horizontal. The coefficient of friction between the block and the incline is u. Determine the acceleration of the block on the incline a)going up the plane; b) going down the plane.

block moving up the plane

block moving down the plane

Green - friction orange - Normal red mgsinq -x dir blue mgcos q-y dir black = mg

Going up the plane we find in the y - direction that FN = mg cos q.

Going up the plane we find in the x - direction that

Check the accompanying problem with no friction. Common sense dictates that the block will not slide as far up the plane when friction is introduced. The mathematics bears out that sentiment.

Now gravity pulls the block down the plane (see the drawing in the middle panel above). If the hill is steep enough, then we have in the x-direction

This calculation shows that the acceleration will point down the hill but will be smaller than with no friction.

Does the block slide down the hill at all?

About three lines above this one, wew showed that

Again, common sense suggests that the block could slide up the incline and stop and then not slide down again if either friction were verylarge or if the angle of elevation of the plane were reduced. In this equation if the first part is bigger than the second part, the block does not move. Let's figure what angle sshould cause this sort of thing to happen. If a = + ugcosq - g sinq. = 0, then u cos q = sin q. That means that u = tan q or q = arctan u

Page revised 12/13/06