Quantum Spectacles

What happens if you tether a helium balloon to the bottom of a closed air-filled cabin and continuously accelerate the cabin to the right at a constant rate?

That’s easy you’d say. If you’re sitting in a car that lurches forward, you’d feel a force backward and be pressed against the seat. So if the cabin accelerates to the right, the balloon should feel a force to the left so that the string inclines backwards, right?

Wrong! Precisely the opposite happens. The string inclines forwards in the direction of acceleration. This little physics surprise is a nice demonstration of…. (drum roll) … THE EQUIVALENCE PRINCIPLE!!!

Well, the equivalence principle is the short answer to the problem, so let’s see it first. The principle states that a uniformly accelerating frame of reference is indistinguishable from an inertial frame in which a uniform gravitational field (opposite to the acceleration) is present. As an illustration, suppose you’re in a rocket which is accelerating upwards (whatever upwards means) far away from any gravitating bodies. Since you’d experience a pseudo-force downwards, you wouldn’t be able to tell (without looking outside) whether you were indeed accelerating upwards or whether you were stationary on the surface of some planet which is exerting a gravitational field on you.

Equivalence Principle

So how does that help with a balloon in a cabin? All you need to do is forget that the cabin is accelerating to the right and replace the acceleration with an extra “gravitational field” to the left. Combined with the original vertical gravitational field, you get a net gravitational field pointing downwards and left (diagonally). Since the balloon aligned opposite to the gravitational field that it “felt” when there was no acceleration, it should align opposite to the new effective gravitational field. Hence, the balloon and string incline towards the right.

You might rightly ask, aren’t things a little more complicated? Isn’t there air inside the cabin that experiences a force too? And whatever happened to the pseudo-force that pushes things backwards? To answer these questions, let’s look at things in some more detail.

Firstly, the air inside the cabin feels the pseudo-force too and behaves as if there is a gravitational field in the horizontal direction as well. How does a gas respond to a gravitational field? Roughly speaking, the air tends to pile at the bottom so you get a gradient of density (and pressure) in the direction of the field.(Precisely, it takes up an exponential distribution which can be approximated to be linear over small distances.) A linear gradient in pressure in a fluid means that an object immersed in it will experience a buoyant force and in the case of an object less dense than the surrounding fluid, the buoyant force dominates and the object floats. This is of course what made the helium balloon rise up in the first place. But now, since you have a pressure gradient horizontally, you get a horizontal buoyant force too. This force dominates over the pseudo-force to the right since the balloon is less dense than air, just as in the case of the vertical force. So the net force is up and to the right.

Interestingly, the relatively involved analysis in the previous paragraph is completely contained in the statement “a helium balloon aligns opposite a gravitational field” which combined with the Equivalence Principle provides a simple answer to this deceptive problem.

More on the equivalence principle soon…