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, xmin = 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.

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