## Physical Pendulum

### simple harmonic motion - oscillations - stationarity

** What it shows: **

The period of a physical pendulum is measured and compared to theory. The pivot point, and thus
the period, is adjustable along the length of the pendulum making it possible to demonstrate that
there is a pivot point where the period is a minimum (stationary point).

** How it works: **

The physical pendulum is a 1/2" diameter x 100cm long brass rod. A collar with a "knife edge" can
be fixed anywhere along the length of the pendulum and serves as the pivot point. The period of
this pendulum is given by
^{
1
}

where L is the length of the rod, x is the distance from the pivot point to the center of mass, and
g is the acceleration due to gravity. For very small and very large values of x the period is large, but
between these two extremes there is a minimum where x is

With L = 100cm, x_{min} = 28.9cm. A large analog timer
^{
2
}
is used to verify the period.

** Setting it up: **

The pendulum knife-edge support is a U-shaped piece of aluminum that is clamped onto a standard
lab bench rod. Use a 3/4" dia. iron rod, as rigidity is important.

** Comments: **

A clean simple application of calculus and finding minimum or stationary points. Rating **

1
This result is found in many textbooks. R. Guglielmino and T. Boyce, The Physics Teacher
**27**, 361 (1989) give a straightforward derivation.

2
Sargent-Welch Scientific model 812

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