Chapter 8
The Pendulum
Asimple plane pendulum(Fig. 8.1) consists of a massmattached to one end a light rod of lengthL; the other
end of the rod is attached to a frictionless pivot. The pendulum is initially displaced from the vertical by an
angle 0 and released, causing it to swing back and forth. Is the pendulum a simple harmonic oscillator?
Figure 8.1: A simple plane pendulum.Analyzing the geometry of the pendulum shows that the restoring force—the force acting on the pendulum
directing it back to its equilibrium position (vertical)—ismgsin, whereis the angle from the vertical,
gis the acceleration due to gravity, and the minus sign indicates that the restoring force acts opposite the
direction of angular displacement. We can write the restoring force as
FDmgsin: (8.1)But for a simple harmonic oscillator, the restoring force must be in the formFDkx, so the pendulum is
nota simple harmonic oscillator.
Suppose, however, that we restrict the pendulum tosmalloscillations. For small angles, we can make the
approximation sin, whereis in radians. Under this approximation, Eq. (8.1) becomes
Fmg; (8.2)whichisthe form of equation of a simple harmonic oscillator. So while the pendulum is not strictly a simple
harmonic oscillator, it isapproximatelya simple harmonic oscillator when the oscillations are small.