**The Swinging Test for the Moments of Inertia I1 and I2**

**This test was performed as follows:**

**In order to test the relative values of I1 and I2 for one of the pendulums in practice, for performing the timing, we rigged up a wire contact to start and stop a timer as the pendulum passed its bottom center point.**

**Then, to get an estimate of I1, we set this pendulum swinging in the I1 mode (to and fro in the plane of the ring) with a relatively small amplitude (about 2 cm) so that the motion was effectively SHM (i.e., the elliptic function effect was negligible). First, we measured the time taken for 11 oscillations (actually counted), which was 21.36 seconds - so that the period was initially determined as being approximately 1.9418 seconds. Then we swung the pendulum for about 50 oscillations (judged), which took 1:37.19, i.e. 97.19 seconds. Clearly therefore there had been in fact exactly 50 oscillations, so that the period was determined (now to a higher accuracy) as being 1.9438 seconds. Then we swung the pendulum for about 100 oscillations (judged), which took 3:16.33, i.e. 196.33 seconds. Clearly therefore there had been in fact exactly 101 oscillations, so that the period was determined (to a further accuracy) as being 1.94386 seconds. Finally, we swung the pendulum for about 500 oscillations (judged), which took 16:25.5, i.e. 985.5 seconds. Clearly therefore there had been in fact exactly 507 oscillations, so that the period was determined (to a final accuracy) as being 1.94379 seconds.
**

**Next, to get an estimate of I2, we set this pendulum swinging in the I2 mode (to and fro perpendicular to the plane of the ring), again with a relatively small amplitude. Although in principle this mode of oscillation is unstable, in practice it continued reasonably well over the time of the experiment. First, we swung the pendulum for 50 oscillations (counted), which took 1:37.11, i.e. 97.11 seconds. Thus the period was determined (to a first lower accuracy) as being 1.9422 seconds. Then we swung the pendulum for about 100 oscillations (judged), which took 3:14.22, i.e. 194.22 seconds. Clearly therefore there had been in fact exactly 100 oscillations, so that the period was again determined (to a further accuracy) as being 1.9422 seconds. Finally, we swung the pendulum for about 500 oscillations (judged), which took 16:14.95, i.e. 974.95 seconds. Clearly therefore there had been in fact exactly 502 oscillations, so that the period was determined (to a final accuracy) as being 1.94213 seconds. **

**The ratio between these periods is 1.00085. Now the ratio between the calculated values for I1 and I2 is 1.00184. Since the period is proportional to the square root of the moment of inertia (the restoring torque being the same in both modes), we have for the relation between calculation and experiment: 1.00092 to 1.00085. IMHO, this represents reasonable, but not fantastic, agreement; it's within 10%, which is as good as we could realistically expect from our primitive timing equipment, and bearing in mind other possible inaccuracies which could affect this result (elliptic function effect, precession, amplitude decay, air resistance, etc.).**

-o0o-

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