10.1. Simple Harmonic Motion http://www.ck12.org
is proportional to the displacement of the mass (recall Hooke’s Law). The mass will pass through the equilibrium
position and as it does, the spring will begin to slow the mass down until it is brought to rest at position–x, opposite
to where the mass started.
After the mass passes its equilibrium position, forceFbof the spring upon the mass changes direction so that it keeps
pointing toward the equilibrium position. In other words, the force on the mass by the spring is always directed
toward the equilibrium position of the mass. We call this force a restoring force.
These observations lead us to the conditions for simple harmonic motion:
An object performing SHM must have a restoring force acting upon it that seeks to return it to its equilibrium
position. The magnitude of that force is directly proportional to the object’s displacement. That is,F=−kxwhere
the negative sign indicates that the restoring force and displacement are oppositely directed andkis a constant of
proportionality.
Motivating the Conditions for Simple Harmonic Motion
Let us return to uniform circular motion.Figure10.5 shows an objectPin uniform circular motion along with its
accelerationaand radiusrindicating the location of the objectPalong the circle. We have dropped the subscript on
the centripetal accelerationacwe used earlier.
FIGURE 10.5
We see inFigure10.5, the accelerationaand position vectorrhave components in thex−direction andy−direction.
We have not bothered to label they−components, though everything that is true for thex−components is equally
true for they−components.