Galilean Accelerating Reference

[Hyperreality]

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?

 

[Olias]

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.

 

[Hyperreality]

This might be some help for your rotational experiment.

http://scienceworld.wolfram.com/physics/Acceleration.html

 

[Doc Al]

Newton's 3rd Law

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! 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.

 

[arildno]

This might be some help for your rotational experiment.

http://scienceworld.wolfram.com/physics/Acceleration.html

Fix your link, hyperreality; in your first bracket, write only URL

 

[jdavel]

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.

 

[Doc Al]

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!