# The misery of rotating frames

• Leo Liu
In summary, the authors in "An Intro to Mech" discuss Newton's experiment with a rotating bucket of water and conclude that rotational motion is absolute. However, there is a paradox known as Mach's principle, where the fixed stars seem to determine an inertial system. The authors suggest that this may be due to the proper motion of the stars. This paradox has been widely discussed and further insight is requested.

#### Leo Liu

I have been reading Kleppner's An Intro to Mech recently and have found an interesting discussion on the nature of rotational motion in the book.

The authors wrote:
Newton described this puzzling question in terms of the following experiment: if a bucket contains water at rest, the surface of the water is flat. If the bucket is set spinning at a steady rate, the water at first lags behind, but gradually, as the water’s rotational speed increases, the surface takes on the form of the parabola of revolution discussed in Example 9.6. If the bucket is suddenly stopped, the concavity of the water’s surface persists for some time. It is evidently not motion relative to the bucket that is important in determining the shape of the liquid surface. So long as the water rotates, the surface is depressed. Newton concluded that rotational motion is absolute, since by observing the water’s surface it is possible to detect rotation without reference to outside objects.
And:
Nevertheless, there is an enigma. Both the rotating bucket and the Foucault pendulum maintain their motion relative to the fixed stars. How can the fixed stars determine an inertial system? What prevents the plane of the pendulum from rotating with respect to the fixed stars? Why is the surface of the water in the rotating bucket flat only when the bucket is at rest with respect to the fixed stars?

I am intrigued by this paradox (or property?), which is named Mach's principle, because I think it is bizarre that we can't know whether a frame is inertial in such cases. Would you mind sharing your insight into it?

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Delta2
Fixed stars determine non rotational frame. If the bucket is still and all the stars rotate around the bucket, the center water surface may be depressed but we have no way to check it, Mach's principle, in experiment.
And more in detail stars move from their "fixed" position. It is called as proper motion.

Leo Liu
Leo Liu said:
I am intrigued by this paradox (or property?), which is named Mach's principle, because I think it is bizarre that we can't know whether a frame is inertial in such cases. Would you mind sharing your insight into it?

Regarding this bit:
Leo Liu said:
How can the fixed stars determine an inertial system?
It doesn't have to be a causal influence by the current distal masses. But note that before the expansion all the mass in the universe in was closer together.

PS: Did you mean "mystery" in the thread title? Makes sense both ways though.

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Leo Liu

## What is meant by "The misery of rotating frames"?

The misery of rotating frames refers to the challenges and complexities that arise when trying to describe and understand the physical laws and phenomena in a frame of reference that is rotating or accelerating. This concept is a fundamental aspect of classical mechanics and is important in fields such as astrophysics and engineering.

## Why is it difficult to analyze rotating frames?

Analyzing rotating frames is difficult because the laws of physics, such as Newton's laws of motion, are based on observations in inertial frames of reference - frames that are not accelerating or rotating. When dealing with rotating frames, these laws must be modified or extended to account for the effects of rotation and acceleration.

## What are some common examples of rotating frames?

Some common examples of rotating frames include a merry-go-round, a spinning top, and a rotating planet like the Earth. These frames are constantly rotating or accelerating, making it challenging to analyze the physical laws and phenomena occurring within them.

## How does the Coriolis effect relate to rotating frames?

The Coriolis effect is a result of the rotation of the Earth and its effect on moving objects. In rotating frames, the Coriolis force must be taken into account as it can impact the motion of objects, causing them to deviate from their expected path. This is one example of the complexities of analyzing rotating frames.

## What are some practical applications of understanding rotating frames?

Understanding rotating frames is crucial in many practical applications, such as navigation systems, spacecraft dynamics, and weather forecasting. By accounting for the effects of rotation and acceleration, engineers and scientists can design and predict the behavior of systems and objects in rotating frames more accurately and effectively.