Galilean Accelerating Reference

In summary, The distance of separation of the two frame of reference is 1/2 at^2. Acceleration between the two frame of reference is different. Newton's Third law does not hold in an accelerating Galilean frame of reference. To measure the acceleration, you would need to use an accelerometer.
  • #1
Hyperreality
202
0
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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  • #2
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.
 
  • #4
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.
 
  • #6
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.
 
  • #7
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:
 

Related to Galilean Accelerating Reference

1. What is Galilean Accelerating Reference?

Galilean Accelerating Reference (GAR) is a scientific concept that describes the motion of an object in a non-inertial reference frame, such as a frame that is accelerating or rotating. It is named after the famous Italian scientist Galileo Galilei, who first proposed this idea in the 16th century.

2. How does GAR differ from an inertial reference frame?

In an inertial reference frame, an object will remain at rest or in a state of constant velocity unless acted upon by an external force. In contrast, in a Galilean accelerating reference frame, an object will appear to experience fictitious forces, such as centrifugal force or Coriolis force, that are not actually present. This is due to the fact that the frame itself is undergoing acceleration or rotation.

3. What are some real-world examples of GAR?

One common example of Galilean Accelerating Reference is the experience of feeling pushed to the side when a car makes a sharp turn. This is due to the car's acceleration causing a fictitious force in the opposite direction. Another example is the apparent deflection of objects on a rotating merry-go-round, which is caused by the Coriolis force.

4. How is GAR used in scientific research?

GAR is an important concept in many areas of scientific research, particularly in the fields of mechanics and physics. It is often used to analyze the motion of objects in non-inertial reference frames, such as in space exploration or in the study of fluid dynamics. Understanding GAR is also crucial in designing and operating systems that experience acceleration or rotation, such as airplanes and satellites.

5. Are there any limitations to the use of GAR?

While Galilean Accelerating Reference is a useful concept in many situations, it does have its limitations. It is only applicable to systems that are experiencing constant acceleration or rotation, and does not account for relativistic effects at high speeds. In these cases, a more advanced framework, such as Einstein's theory of general relativity, must be used.

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