I am reading through Hartle's "Introduction to Einstein's General Relativity" and it discusses the Eotvos Experiment in Chapter 6. The free-body diagram (shown below) has me a little puzzled.
This experiment was designed to see if there's any difference in inertial mass mI (think Newton's 2nd law) and gravitational mass mG (think Newton's law of gravity). The setup is this: Imagine two masses of equal weight, connected by a rigid rod. This rod is then suspended by a fiber and hangs freely. At first, you would (incorrectly) think there are only two forces acting on the masses; the force of gravity mGg pulling down straight down to the floor, and tension T pointing straight up along the fiber.
BUT, we are on Earth, and Earth is rotating; thus, there is also centripetal force mIa acting the masses.
So in the free body diagram below, we have three forces acting on the rod/masses: tension, gravity, and centripetal force.
The book/diagram claims that the fiber hangs at a small angle such that a small component of the gravitational force can balance the centripetal acceleration. From the figure though, I don't see how any component of gravity can cancel the centripetal acceleration. If you broke gravity up into X and Y components, one component is pointing WITH the centripetal acceleration, and another component is perpendicular to it - there's no component of gravity that can possibly cancel the centripetal force, is there? What am I missing here? How can gravity cancel the centripetal force?