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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.

enter image description here

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?

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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.

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  • $\begingroup$ You describe a convention where using the same notation for pseudoforces, like "net force" or "centripetal force," and for physical interactions like gravitational force, makes something that's different from a "proper" free-body diagram. That convention is relatively new to the physics pedagogy, and books from the 20th century generally won't adhere to it. Reader beware. $\endgroup$ – rob Jan 25 at 23:28
  • $\begingroup$ @rob Could you help me clarify the answer, please? My intent was to talk about the non-rotating frame where a net force is needed. $\endgroup$ – Bob Jacobsen Jan 25 at 23:52
  • $\begingroup$ I thought your answer was perfectly clear. My point was just that many good books by smart people (like Hartle, here) will contain these sorts of "improper" free-body diagrams. They're not quite wrong; we just did things less clearly in the past. $\endgroup$ – rob Jan 26 at 0:20
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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.

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  • $\begingroup$ In other words, the tension vector plus the gravitational force vector must necessarily sum up to be the centripetal force vector (because for us to be rotating in the first place, the centripetal force vector must exist?) $\endgroup$ – Programmer Jan 25 at 23:22
  • $\begingroup$ @Programmer Yes, just so. $\endgroup$ – rob Jan 25 at 23:22

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