Airplane 3 Nov 2008

Peter Fisher

11/26/08

8.033

I screwed up explaining this in Recitation 1, so I thought I better write it up. The problem is this: you are in an airplane accelerating at 2 m/s2 horizontally down the runway. Of course, the airplane also feels the force of gravity pulling it down. Seated inside the airplane, you are holding a helium balloon. What direction does the balloon tilt?

The equivalence principle says that you can replace an acceleration with a gravitational field. If the airplane accelerates to the right, an occupant feels an acceleration like gravity pointing to the left. That is, the occupant who cannot see outside of the airplane cannot tell if the plane is accelerating to the right or if there is a gravitational field with a component point to the left. The vector addition is shown to the left. The total gravitational vector points down and to the left. We know the buoyant force of the balloon pushes the balloon up opposite to the direction of the gravitational vector, so the balloon points in to the right, in the direction of the acceleration of the airplane. The equivalence principle is easy to state, but has important consequences. I am sitting on my couch next to the cat. As usual, she is at rest. However, the equivalence principle says she is accelerating upwards at 10 m/s2. How could this be? Gravity wants her to be falling toward the center of the Earth, accelerating with 1 g relative to the surface of the Earth. But, she is being held up by the couch, so relative to what she should be doing, she is accelerating upwards. You can see this in the last lecture and problem 1 on PS8: in both cases, we use the equivalence principle to think of a box or rocket in a gravitational field as accelerating and then find Doppling shifts and angular deflections.