Coriolis force caused by tangential velocity

[Dale]

Now the physicist interprets the equilibrium condition r′=r′0=r0 as the point where no force acts on the pearl, i.e., in his interpretation the elongation of the spring measures the centrifugal force.

Remove the centripetal force and the elongation disappears even though the centrifugal force remains. So much for that interpretation.

Maybe we can agree on that, though I think the discussion of inertial forces are usually so overcomplicated that it comes to unnecessary discussions like this.

It is not as complicated as you are making it. My point 7 is correct, but instead of simply learning something new you want to argue. You and I have been down this road before and I did not enjoy it last time and am not willing to do it again. You are free to learn something new here or not. I will leave with one last post explaining the concept to the best of my ability and then I am done.

Suppose that we have an accelerometer involving a mass sliding on a frictionless rod with a spring oriented along the rod. At the equilibrium point the rod is marked with a 0.

We know the spring constant and the mass so we also mark the rod with a pair of 1’s on either side of the 0 for 1 g and a pair of 2’s for 2 g and so on, each indicating where the mass is for a given magnitude of acceleration, but at this time we do not mark directions. So now we have a device that can measure the magnitude of the acceleration along the axis but does not indicate the direction.

Now, to set the directions we experiment with our device. We place it horizontally on the table with the axis along the left-right direction. We note that the mass is at the 0 mark. We accelerate it to the right at 1 g and note that the mass is at the left 1 mark, so we mark an arrow pointing right next to the left 1 mark. We accelerate it to the left at 1 g and note that the mass is at the right 1 mark, so we mark an arrow pointing left next to the right 1 mark. And repeat.

In the end, we note that all of the arrows point in towards the 0 on the rod. A rightward acceleration is indicated by a leftward deflection.

Now, we put the accelerometer in a centrifuge on our table, oriented with the rod along the radial direction. We turn it on to 1 g and note that the mass deflects outward to the 1 g mark. An outward deflection indicates an inward acceleration, and sure enough if we look at the arrow it points inward, towards the 0 mark. This is consistent with the real inward centripetal force.

Now, an ambitious student seeks to analyze it in the rotating frame. In the rotating frame the real centripetal force still exists and a centrifugal force is added. The addition of the centrifugal force does not change the reading, so it should be immediately apparent that the accelerometer does not detect the centrifugal force, but the student wants to consider all possibilities.

The student enumerates all four possibilities. A) the accelerometer detects both forces. B) the accelerometer detects neither force. C) the accelerometer detects only the real centripetal force. D) the accelerometer detects only the fictitious centrifugal force. This is an exhaustive and mutually exclusive set of possibilities.

A) if it detects both then the reading should be 0. The reading is 1 g inward so A) is inconsistent with observation.

B) if it detects neither then the reading should be 0. The reading is 1 g inward so B) is inconsistent with observation.

C) if it detects only the real centripetal force then the reading should be 1 g inward. The reading is 1 g inward so C) is consistent with observation.

D) if it detects only the fictitious centrifugal force then the reading should be 1 g outward. The reading is 1 g inward so D) is inconsistent with observation.

The only possibility consistent with observation is that the accelerometer detects only the real centripetal force and does not detect the inertial centrifugal force.

Another way that it should be obvious that accelerometers do not detect inertial forces is to consider other non-inertial frames. There is nothing physically privileged about the co rotating frame. In other non inertial frames the inertial force will be different. By judicious choice of frame we can make it have any size and direction desired. Regardless of the size or direction of the inertial force the reading of the accelerometer is the same and the magnitude and direction are both given by the real force. Therefore, the accelerometer can not detect any inertial force in any non-inertial frame. The observation is always consistent only with the inertial forces being undetected by the accelerometer.

That is it. I am done here with you. You are free to learn or not as you choose, but I lack the desire to fight you on this. My statement 7 is correct, whether you will accept it or not.

 

[Swamp Thing]

This is how Wikipedia defines an accelerometer:

An accelerometer is a device that measures proper acceleration. Proper acceleration, being the acceleration (or rate of change of velocity) of a body in its own instantaneous rest frame, is not the same as coordinate acceleration, being the acceleration in a fixed coordinate system. For example, an accelerometer at rest on the surface of the Earth will measure an acceleration due to Earth's gravity, straight upwards (by definition) of g ≈ 9.81 m/s2. By contrast, accelerometers in free fall (falling toward the center of the Earth at a rate of about 9.81 m/s2) will measure zero.

And this is how the article explains how they work:

Conceptually, an accelerometer behaves as a damped mass on a spring. When the accelerometer experiences an acceleration, the mass is displaced to the point that the spring is able to accelerate the mass at the same rate as the casing. The displacement is then measured to give the acceleration.

Maybe the disagreement about Dale's statement

7) Inertial forces cannot be detected by an accelerometer

has a lot to do with which acceleration one is talking about. I think Dale's Statement 7 is true about "coordinate acceleration" -- or, to coin an alternative term, "geometric acceleration". An accelerometer won't really sense inertial forces that are associated with coordinate acceleration.

When it comes to proper acceleration on the other hand... By definition an accelerometer measures proper acceleration, and the way it does this is to first measure some effect (e.g. deflection) of the inertial force arising from the proper acceleration, then scale the result based on the known elasticity and mass to get acceleration. So in this case we necessarily measure inertial force in order to measure acceleration.

 

[Dale]

So in this case we necessarily measure inertial force in order to measure acceleration.

An accelerometer never measures inertial forces as shown exhaustively above.

the inertial force arising from the proper acceleration

I get the feeling from your comment that you do not understand what an inertial force is. “Inertial force” is another name for “fictitious force”. They arise from analyzing a scenario in a non inertial frame; they do not arise from proper acceleration.

https://en.m.wikipedia.org/wiki/Fictitious_force

 

[Swamp Thing]

Ok, I think I now see what you mean. My sentence...

The way (the accelerometer works) is to first measure some effect (e.g. deflection) of the inertial force arising from the proper acceleration, then scale the result based on the known elasticity and mass to get acceleration.

... should be corrected to read : "We allow a spring to provide the required centripetal force, then calibrate the spring's deformation to read out the acceleration." In this picture, clearly, the centrifugal force is not (and can not) be measured.