# How to differentiate between rotating frame and linearly accelerating frame?

Two friends, $A$ and $B$ are part of an experiment. $A$ is placed in a closed box and made to accelerate in free space at an acceleration $g$. $B$ is also placed in a closed box, but is made to rotate in a circle at uniform speed, such that the radial acceleration is also $g$. Can $A$ and $B$ perform some experiment from their boxes to tell who is moving radially and who is moving linearly?

Yes, I can think of two ways to do this and there may be more.

In a rotating frame the acceleration is a function of distance from the pivot. If B has sufficiently precise instruments the variation of acceleration with position will be detectable. However if B is confined to a very small space, or doesn't have precise enough instruments the variation of acceleration won't be detectable.

In that case B can construct a Foucault pendulum. For every rotation about the pivot B rotates once, and this rotation will change the plane of the pendulum. However you could defeat this by mounting B inside a gimbal. In that case I don't think there is any way to tell the difference between rotation and linear acceleration.

• LOL, if your answer is independent, that's quite a degree of agreement, even in the way how we split the methods. – Luboš Motl Aug 11 '14 at 6:16
• @LubošMotl: naturally I waited until you answered then plagiarised your answer :-) Now I'm going to have to spend all morning trying to think of a third technique to distinguish my answer! – John Rennie Aug 11 '14 at 6:18
• Haha, just to be sure, I didn't mean to accuse you of anything. In the same way, it may have been me who used a time machine and plagiarized you. ;-) – Luboš Motl Aug 11 '14 at 7:07
• @LubošMotl: :-) There is no third way to tell is there? If B is in a sufficiently small gyroscope stabilised shell then I can't think of any other way to detect the rotation. – John Rennie Aug 11 '14 at 7:19
• I can only think about thinking outside the box, like sending a light ray to a distant mirror and seeing that it returns back from the same side only after the multiplicity of the periodicity, or see some bending or something. Of course, the very local difference is very small. – Luboš Motl Aug 11 '14 at 7:39

The friend rotating and experiencing the centrifugal force may observe several effects that his linearly accelerating friend doesn't:

• the acceleration at different points of the box is slightly different i.e. the apparent gravitational field is non-uniform
• there is the extra Coriolis force acting on objects that are moving relatively to the rotating frame

A simple pendulum would be a good experiment to detect both non-inertial frames - rotating and linearly accelerating:

If you know weight of the pendulum in inertial frame, in rotating frame, its weight would decrease because of radially outward centrifugal force acting on it. (If earth stops rotating, our weight would increase!) Think: what would happen if you are in a rotating hollow sphere?

In linearly accelerating frame, the pendulum cannot be perpendicular to the direction of motion, but would make some angle wrt the 'vertical line' because of pseudo force. The measure of angle is proportional to the magnitude of acceleration. (This is a principle behind measuring acceleration & hence speed of a spacecraft in the outer space, where there is no considerable gravity).