We next consider the motion of a spring that is subject to a frictional force (in the case of
the horizontal spring of Figure 2) or a damping force (in the case where a vertical spring
moves through a fluid as in Figure 3). An example is the damping force supplied by a
shock absorber in a car or a bicycle.
We assume that the damping force is proportional to the velocity of the mass and acts
in the direction opposite to the motion. (This has been confirmed, at least approximately,
by some physical experiments.) Thus