1. Your trifilar pendulum strings aren't vertical, as they are in every article I've seen on the trifilar pendulum method.

I did in this way because the pendulum oscillates more slowly in this way and timings are more accurate. It was calibrated with various cylinders of known weight and radius of gyration. The pendulum works well, this is not the cause of the large difference we are facing.

better to work directly in moments, as the calculations get a lot messier when trying to work in radii of gyration — especially when *M*_{top} and *M*_{plate} are of comparable size.

The oscillation of the trifilar pendulum is proportional to the radius of gyration, so if you use a trifilar pendulum the radius of gyration can't be ignored, because it stays necessarily at the core of the calculations. The explanation you read is a bit lenghty because I tried to explain the logicality of the calculations step by step.

Anyway, I am far from to be perfect, and it could be that somewhere I made an error, so I want to estimate the moment of inertia of my top Nr. 22 with a different, simpler approach, for to see if there is really a large error in my previous calculations.

This is the design of the top:

The central parts of the top are light parts with very little influence on the total moment of inertia of the top, so I ignore them, (it is not the case to observe the single leaves when we don't see the forest).

Nearly all the moment of inertia is in the flywheel. So I consider just it and nothing else.

For to make the calculations easier, instead of the torus I consider an approximately equivalent holed cylinder, having the same inner and outer diameters, (43 and 80 mm), and the same weight, (600 grams).

The height of such a holed cylinder is 19 mm, (the flywheel is made of copper, density 8.96).

The orange rectangle is the section of this holed cylinder. It is superposed to the section of the flywheel. It has its same diameters, the height is a bit littler, but some parts are larger, (the corners), so the area is similar, and certainly the moment of inertia of this holed cylinder will be not very different from that of the top, but much easier to calculate.

First I calculate the data of the WHOLE CYLINDER without the hole:

VOLUME: 0.04 x 0.04 x 3.14 x 0.019 = 0.0000955 m

^{3}WEIGHT: 0.0000955 x 8.96 = 0.000855 ton = 0.855 kg

RADIUS OF GYRATION: 0.04 x 0.707 =0.02828 m

MOMENT OF INERTIA: 0.02828 x 0.02828 x 0.855 = 0.000684 kg m

^{2}Then I calculate the data of the inner cylinder ,(THE HOLE),to be subtracted from the data above:

VOLUME: 0.0215 x 0.0215 x 3.14 x 0.019 = 0.0000276 m

^{3} WEIGHT: 0.0000276 x 8.96 = 0.000247 ton = 0.247 kg

RADIUS OF GYRATION : 0.0215 x 0.707 = 0.0152 m

MOMENT OF INERTIA : 0.0152 x 0.0152 x 0.247 = 0.000057 kg m

^{2}Then I calculate the MOMENT OF INERTIA OF THE HOLED CYLINDER:

0.000684 - 0.000057 = 0.000627 kg m

^{2}This is not far from the data I obtained with the trifilar pendulum, (0.000636).

Considering that this is an approximative estimate, and that the little moment of inertia of the core of the top has to be added, the matching is good.

**Jeremy**, do you find something wrong in these simple calculations ? I believe that you messed with something, maybe you used the diameter instead of the radius in your calculations.