Check Your Understanding A block of mass 1.0 kg rests on a horizontal surface. The frictional coefficients for the block and surface are
and
(a) What is the minimum horizontal force required to move the block? (b) What is the block’s acceleration when this force is applied?
One situation where friction plays an obvious role is that of an object on a slope. It might be a crate being pushed up a ramp to a loading dock or a skateboarder coasting down a mountain, but the basic physics is the same. We usually generalize the sloping surface and call it an
inclined plane but then pretend that the surface is flat. Let’s look at an example of analyzing motion on an inclined plane with friction.
Downhill skier
A skier with a mass of 62 kg is sliding down a snowy slope at a constant velocity. Find the coefficient of kinetic friction for the skier if friction is known to be 45.0 N.
Strategy
The magnitude of kinetic friction is given as 45.0 N. Kinetic friction is related to the normal force
by
; thus, we can find the coefficient of kinetic friction if we can find the normal force on the skier. The normal force is always perpendicular to the surface, and since there is no motion perpendicular to the surface, the normal force should equal the component of the skier’s weight perpendicular to the slope. (See
[link] , which repeats a figure from the chapter on Newton’s laws of motion.)
The motion of the skier and friction are parallel to the slope, so it is most convenient to project all forces onto a coordinate system where one axis is parallel to the slope and the other is perpendicular (axes shown to left of skier). The normal force
is perpendicular to the slope, and friction
is parallel to the slope, but the skier’s weight
has components along both axes, namely
and
The normal force
is equal in magnitude to
so there is no motion perpendicular to the slope. However,
is less than
in magnitude, so there is acceleration down the slope (along the
x -axis).
We have
Substituting this into our expression for kinetic friction, we obtain
which can now be solved for the coefficient of kinetic friction
Solution
Solving for
gives
Substituting known values on the right-hand side of the equation,
Significance
This result is a little smaller than the coefficient listed in
[link] for waxed wood on snow, but it is still reasonable since values of the coefficients of friction can vary greatly. In situations like this, where an object of mass
m slides down a slope that makes an angle
with the horizontal, friction is given by
All objects slide down a slope with constant acceleration under these circumstances.
We have discussed that when an object rests on a horizontal surface, the normal force supporting it is equal in magnitude to its weight. Furthermore, simple friction is always proportional to the normal force. When an object is not on a horizontal surface, as with the inclined plane, we must find the force acting on the object that is directed perpendicular to the surface; it is a component of the weight.
