Galilean Accelerating Reference

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Discussion Overview

The discussion revolves around the implications of Newton's Third Law in the context of Galilean accelerating reference frames. Participants explore the mathematical formulations related to acceleration and forces in such frames, as well as the experiential aspects of acceleration, particularly in vehicles. The scope includes theoretical considerations, conceptual clarifications, and practical implications of measuring acceleration in non-inertial frames.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants present the mathematical derivation of force in accelerating frames, noting that Newton's F=ma does not hold without additional terms, which they refer to as inertial forces.
  • There is a question regarding the symmetry of Newton's Third Law in accelerating frames, with some arguing that it applies only to "real" forces and not to inertial forces.
  • One participant describes a common experience of being pushed back in a seat during acceleration and questions how motion can be observed in such a frame.
  • Another participant introduces the concept of 'Newton's Bucket' and relates it to the discussion, suggesting a personal experiment to understand the forces involved.
  • Participants discuss methods to measure acceleration in an accelerating frame, with suggestions including the use of accelerometers and the behavior of hanging weights.
  • Concerns are raised about the challenges of determining the direction of "down" in non-inertial frames, particularly in varying terrain.
  • One participant proposes a method for constructing an accelerometer using springs to measure acceleration consistently across different environments.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Newton's Third Law in accelerating frames, with no consensus reached on whether it is symmetrical. The discussion remains unresolved regarding the implications of inertial forces and the measurement of acceleration in non-inertial frames.

Contextual Notes

Participants acknowledge limitations in determining the direction of "down" in non-inertial frames and the dependence on specific conditions, such as the presence of hills or varying terrain, which complicate the measurement of acceleration.

Hyperreality
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Two frame of reference A and A'. A' starts accelerating with respect to A.

The distance of separation of the two frame of reference is

s = 1/2 at^2

x' = x - s
= x - 1/2 at^2

Differentiating twice with respect to time we get

d^2x'/dt^2 = d^2x/dt^2 - a

d^2'x/dt^2 + a = d^2x/dt^2

Therefore

F' = m(a + d^2'x/dt^2) and F = ma.

Is Newton's Third law symmetrical in a Galilean accelearating reference frame?

The two formulas are different, but since acceleration is a vector quantity, which means is simply the resultant acceleration for A'. So is Newton's third law symmetrical in an accelerating Galilean frame of reference? And how can we measure the acceleration if we are inside the accelerating frame of reference.

It is a common experience that people tend to be pushed back to the seat when the car is accelerating, is it possible to observe the change of motion of a body in an accelerating frame of reference while you are being pushed back at the same time?
 
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Hyperreality said:
Two frame of reference A and A'. A' starts accelerating with respect to A.

The distance of separation of the two frame of reference is

s = 1/2 at^2

x' = x - s
= x - 1/2 at^2

Differentiating twice with respect to time we get

d^2x'/dt^2 = d^2x/dt^2 - a

d^2'x/dt^2 + a = d^2x/dt^2

Therefore

F' = m(a + d^2'x/dt^2) and F = ma.

Is Newton's Third law symmetrical in a Galilean accelearating reference frame?

The two formulas are different, but since acceleration is a vector quantity, which means is simply the resultant acceleration for A'. So is Newton's third law symmetrical in an accelerating Galilean frame of reference? And how can we measure the acceleration if we are inside the accelerating frame of reference.

It is a common experience that people tend to be pushed back to the seat when the car is accelerating, is it possible to observe the change of motion of a body in an accelerating frame of reference while you are being pushed back at the same time?

Recently I have been trying to find info on 'Newton's Bucket', which I believe has relevence to your (and my recent) inquiry?

Having found very little info, I done some experimenting myself, and consequently the 'person and seat' experience an attraction due to their proximity, ie any person close to a seat will experience a sense of Directional force towards a seat, measured by the observation as 'direction of motion', both seat and person are traveling in a direction 'opposite' to acceleration direction.
 
Newton's 3rd Law

Hyperreality said:
Therefore

F' = m(a + d^2'x/dt^2) and F = ma.
Right. In an accelerating (noninertial) frame, Newton's F=ma does not hold without adding extra terms. These extra terms are sometimes called "fictitious" forces, but a better term would be inertial forces.
Is Newton's Third law symmetrical in a Galilean accelearating reference frame?
An interesting question. As I understand it, Newton's 3rd law would only apply to "real" forces: forces with an agent, not inertial forces. Thus in my accelerating reference frame of a car rounding a turn, I would feel an inertial force pull me to the outside. This force would have no third law "reaction" force. However, to keep me in the car, the car seat needs to exert a "real" (agented) force against me--and I will exert an equal and opposite force against the car seat. Newton's 3rd law would appear to hold for those forces.
The two formulas are different, but since acceleration is a vector quantity, which means is simply the resultant acceleration for A'. So is Newton's third law symmetrical in an accelerating Galilean frame of reference?
See my comment above.
And how can we measure the acceleration if we are inside the accelerating frame of reference.
An accelerometer! :smile: Seriously, you can measure the acceleration in many ways. What you would measure would be the deviation from F = ma due to your own frame's acceleration. For example: a weight hanging from a string will hang straight down in an inertial frame; but will hang at an angle as you round that turn.
 
Doc Al said (#4), "...you can measure the acceleration in many ways...a weight hanging from a string will hang straight down in an inertial frame; but will hang at an angle as you round that turn."

This would work for a car that wasn't going up or down a hill. But in general, you couldn't really tell which way is "down". Could you?

I'd make my accelerometer by attatching, with 3 springs, a single mass to each of 3 mutually perpendicular walls of my laboratory. Any experiment done in my lab should give the same result when done in any other lab whose springs are stretched the same as mine.
 
jdavel said:
This would work for a car that wasn't going up or down a hill. But in general, you couldn't really tell which way is "down". Could you?
Good catch. Shame on me! :rolleyes:
 

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