Monday, May 16, 2016

Galileo's balls

Isn't there something curious about the thought experiment described here:

"But he did devise a simple thought experiment that told us something profound about gravity. Take two weights, one light, one heavy. If heavier objects fall faster than light ones, as Aristotle said, then the lighter weight will lag behind. That implies that when the two are tied together, they will fall more slowly than the heavy weight alone. But together, they weigh more than the heavy alone, so they should fall faster. Wait, so is it faster or slower?"

This supposedly "demonstrates" that heavier objects can't really fall faster than light ones, because it creates a paradox: tied together, the conglomerate must fall both faster and slower than the heavy object alone.

My problem is this: in a fluid, light objects, especially those with relatively  large surface areas, really do fall faster than heavy ones, especially if the latter have relatively small surface areas. So if I drop a rock and a feather, the rock hits the ground first. Now, if I tie the rock to the feather...

My point is that there really isn't any paradox: if we have tied the objects together, we just have to evaluate the new conglomerate as a single object, and ask what governs its fall. A giant feather tied to a small rock will fall slower than the rock alone, while a small feather tied to a big rock will fall at the speed of the big rock. No paradox!


  1. Galileo's thought experiment was certainly flawed. What it can show is that a model in which mass is the only determinant of acceleration is a flawed one, especially if you have some embedded idea of inertia in there (e.g. the larger ball has to exert a pull on the smaller one but the smaller one is resisting it).

    I blogged something about this last summer.

  2. A certain intellectual type just loves thought experiments. But there are about 99 ways to do bad thought experiments. Here, you've skewered a good example.

  3. Doesn't Galileo assume the two objects in his thought experiment have equal air resistance (for example: they are both balls as implied in the title of the post) ?

    1. rob, Alex (above) is a professional physicist: see his post given in his comment.

    2. And, the smaller ball will have less air resistance, not equal: they will have less drag, because a lower Reynold's number, which varies directly with their diameter.

    3. I've put up another post on it. I think I now understand why he got the right result: He was considering a model that has both the idea of a "natural" speed for an object of a given mass AND an implicit idea of inertia. These ideas clash violently, and if the assumption of a "natural" speed or tendency is explicit while inertia is implicit then you'll reject the only assumption that you made explicit.

    4. Actually, the Reynold's number measures the ratio of inertial effects to viscous drag. There is both inertial drag (air resisting you because you are changing its velocity when you push on it) and viscous drag (essentially microscopic friction between the surface and the air molecules). The smaller ball will have less inertial drag AND less viscous drag, but the ratio of inertial drag to viscous drag goes down (meaning that viscous drag becomes more important), and the ratio of gravitational force to drag force also goes down (meaning that drag becomes more important relative to gravity).



"If your approach to mathematics is mechanical not mystical, you're not going to go anywhere." -- Nassim Nicholas Taleb