That Amazing Devil Gravity

Here we have a clip from the excellent movie adaptation of Tom Stoppard's play Rosencrantz and Guildenstern are Dead. In addition to engaging and nuanced performances by Gary Oldman, Tim Roth, Richard Dreyfus, and Iain Glen, the script is full of thought-provoking metaphysical introspection, and some delightful physics introspection as well. It's well worth renting.

A classic conceptual difficulty for first-year physics students -- and the public in general -- is the idea that all objects (in the absence of significant air resistance) accelerate toward the ground at the same rate. This is not intuitively obvious; in fact, for over 1,000 years, between the time of Aristotle and Galileo, the general consensus was that heavier things fall faster. Much of the confusion was, and is, the result of the masking effect of air resistance.

In the film scene, Rosencrantz (Gary Oldman) is fascinated when he watches an object constructed of feathers fall as fast as a heavy ball. He expects the feathers to take longer because he is used to seeing the effect of air resistance on light objects. But when he tries to demonstrate his initial counterintuitive result to Guildenstern (Tim Roth) using only a single feather, the feather does fall more slowly. Why? Because in the first case the feathers are fixed to a denser (although still light) base so that they are streamlined and relatively unaffected by air friction. With the single feather, the effect of air resistance is large because the feather is both light and has a lot of surface area exposed as it falls. Get rid of the air, and the feather falls as fast as the ball, as David Scott demonstrated in a famous experiment performed on the moon when he dropped a hammer and a feather off the steps of the lunar module.

But constant acceleration due to gravity is not necessarily conceptually obvious to physicists either! Here's why. Although we normally think of mass as, well, simply mass, we can actually separately define two distinct types of mass, which we call inertial mass and gravitational mass. Inertial mass is defined by the resistance of an object to a change in motion. For example, an object with more inertial mass will experience a smaller acceleration for a given amount of force compared to an object with less inertial mass. This property is encapsulated in Newton's Second Law, F = ma, where the mass refers to an object's resistance to acceleration.

On the other hand, we know that objects attract each other via the force of gravity. The more gravitational mass that the objects have, the greater that force. According to Newton's Universal Law of Gravitation, the force of gravity between two objects is given by F = Gm₁m₂/r², where G is a universal constant of nature, the ms are the gravitational masses of each object and r is the distance between the centers of the objects. Now, according to physicists, there is theoretically no requirement that gravitational mass should have the same value as inertial mass -- but it does! This is why all objects accelerate at the same rate. To see why this is true, let's look at the acceleration of a ball of mass m near the surface of the Earth, which has mass M. We get: F = GMm/r² = ma

The m on the left is the gravitational mass of the ball, while the m on the right is its inertial mass. Since the two are equal, they cancel on both sides of the equation and we get a = GM/r² for any mass of object. This acceleration due to gravity depends only on the gravitational constant, the mass of the Earth, and the distance to the center. On another planet this value of acceleration would be different than on Earth but it would still be equal for every object when on that planet.

Esoteric? Maybe. But not if you're a physicist, and, I suspect, not if you're Rosencrantz or Tom Stoppard!

Adam Weiner is the author of Don't Try This at Home! The Physics of Hollywood Movies.