Help Understanding this Free Body Diagram (Eotvos Experiment)

Question

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 m_I (think Newton's 2nd law) and gravitational mass m_G (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 m_Gg 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 m_Ia 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?

Answer 1

This isn’t a proper free body diagram, because the $ma$ line doesn’t represent a force. Only $T$ and $mg$ are forces. Only forces should be in a free body diagram.

Their resulting net force has to equal that $ma$ line because the Earth is rotating.

Now that you see that, note that $T$, the hanging direction, isn’t aligned to $mg$. That (small) difference is what’s being discussed.

Answer 2

Wanting the centripetal force to be "cancelled" is a common mistake among new students of physics, because too many early examples and problems in textbooks involve zero-acceleration systems.

If an object is moving in a circle (as you are, fellow earthling, unless you are a polar explorer), then its velocity is not constant: circular motion requires the direction of the velocity vector to change. The only way to have a changing velocity is to have a nonzero net force. That's Newton's second law.

Remember that "centripetal" is a direction, not an interaction. The leftover force that keeps Eotvos's dangling plumb bob from flying off into space arises because the gravitational attraction towards Earth's center and the tension from the plumb line are not quite antiparallel. That's the point of the figure.