# Detecting external 'forces' in an inertial frame

Disclaimer: Science noob here, I may not use all terms correctly.


I've read somewhere of the following situation: An observer is inside a room (elevator) with no means of referencing anything outside. There is an apparent force that keeps everything stuck to the floor. 1) We can suppose that the room is standing on the surface of the Earth and objects inside are under the influence of gravity. 2) The room may be under uniform linear acceleration in the direction of the 'roof'. 3) The room may be swung around on a rope, with the floor opposite the direction of the centripetal force.

The observer inside the room can supposedly not determine which of the three situations pertain.

Question: Is this correct, and why?

This is not true. The only situations that are indistinguishable are the case of a uniform gravitation field and uniform acceleration.

Case 1

If the room is on earth, then technically, you have a gravitational force that depends on height and it looks like $$g(r) = g(0)\left(\frac{R_e}{R_e + r}\right)^2,$$ where $$R_e$$ is the distance from the center of the earth to the floor of the elevator and $$r$$ is your height from the floor.

Case 2

A uniformly accelerating elevator has the same $$g$$ everywhere inside it.

Case 3

In the case where the elevator is swung around with a rope, the centrifugal force changes as a function of your distance from the center ($$F\sim \omega^2 r$$). This means that the $$g$$ in the elevator changes as below.

$$g(r) = g(0)\frac{R - r}{R},$$

where $$R$$ is the distance from the center of rotation to the floor and $$r$$ is the height measured from the floor of the room. Moreover, you will also be able to detect a Coriolis force on moving objects in this case.

• I take it the Earth's gravitation is not uniform? Does a uniform gravitational field exit in nature? – christo183 Feb 4 '19 at 5:02
• Yes, the earth's field is non uniform and a uniform field is pretty hard to realize. One possible way is a hollow sphere inside which g=0 but this sphere should be the only object in the universe :) – nr2618 Feb 4 '19 at 9:28
• So practically, it can always be determined what kind of pseudo force one is experiencing. Thanks! – christo183 Feb 4 '19 at 17:18