Acceleration due to gravity (idea)

An elementary proof that acceleration due to gravity is independent of mass

We've all heard this from friends, physics teachers, science museum demonstrators, and the rest of the popular science mob. But it's not very convincing, is it? (Well, I think the Smithsonian Institute has a device that drops lead and feathers down 2 tubes in vacuum which shows you this; that's also pretty convincing...)

So here's a simple argument. It also works to show the period of a pendulum is independent of its mass.

Take your falling object and drop it. It falls at some acceleration. OK, take it and drop it again. It falls with the same acceleration as it did last time, right? OK, so take an identical copy of your falling object; it still takes the same time to fall.

Right. Take the original and the copy, hold them together in one hand, and drop both. Each takes the same time to fall. Now do the same, but instead of calling them "two identical objects", pretend you've joined them and they're now a single object. Of course, if they accelerate at different speeds the "single" object would fall apart, but we know that if we drop this "single" object both halves fall at the same rate.

We've just shown that the acceleration of a falling object is the same as the acceleration of a falling object with twice the mass!

To generalise, take any two objects (the ratio of whose masses is a rational number) and pretend each one is made up of some number of very small objects with identical mass. The above argument shows that the small objects all fall together, so the two objects must fall at the same rate.

For objects with irrational mass ratios, it doesn't work. But any continuity argument will help you out.