W.1.1 Simple Harmonic Motion
c. Simple pendulum: A simple pendulum is
on the same device. You could also use the "Faith in Physics" pendulum, M.11.3,
or track a pendulum with a sonic ranger and plot out the sine waves of its motion (see
M.2.3).

d. Spring compared to Pendulum: A pendulum
whose length is equal to the displacement of a spring from equilibrium when the weight is
attached has the same period of oscillation as the spring. 


e. Physical pendula: Each physical pendulum
is compared to a simple pendulum with the same period. The bar can be reversed as shown,
and has the same period in either position. 
f. Torsion pendulum: the weights can be
moved as shown to change the rotational inertia, and therefore the period.
g. Coupled pendula: Three varieties are available.
 Two pendula coupled by a spring (shown to the right) will show normal modes, and
transfer of energy between the single pendula swinging modes.
 A mass on a spring has the vibrating spring mode resonantly coupled to the pendulum
mode. This is an example of parametric oscillation (see W. 1.8). The spring frequency is
approximately twice the swinging frequency (pendulum mode).. The spring mode
parametrically drives the pendulum mode, but the pendulum motion causes the tension in the
spring to vary at twice the pendulum frequency, and therefore resonantly drives the spring
mode. The transfer of energy between these two modes is impressive.
 A Wilberforce pendulum has the vibrating spring mode coupled to a torsion
pendulum mode.
h. Coupled gliders on an air track: Two gliders with three springs,
two running to the fixed ends and the third between show normal modes and exchange of
energy.
i. Some Unusual Pendula: Suggested by Bruce Denardo
 An inverted pendulum; that is, a vertical stick, free at the top, whose support
is vertically vibrated, can be stabilized about a vertical position.
 Precession of a spherical pendulum a spherical pendulum is a conical
pendulum set swinging in an ellipse. For small amplitudes the periods along the major and
minor axes are almost equal and the ellipse remains stationary in space. But for large
amplitudes the ellipse precesses forward because the period for planar pendulum motion on
the major axis is somewhat larger than that along the minor axis. (The period of a planar
pendulum increases somewhat with amplitude as the small angle small oscillation
approximation fails.)
 The Mystery Spot Pendulum is named after a supposed site of gravitational anomaly
in Northern California. A conical pendulum first swings in a circle one way, then
gradually reverses to circle the other way. (Demystification courtesy of Bob Keolian.)

