[paul cotter (OP)]

As a result of answering a query by our esteemed colleague, Hamdani, this question arose in my mind. Imagine one has a rocket propelled craft with a power plant capable of producing a proper acceleration of 1g. Firing up the engine will produce an acceleration of 9.81m/s and an accelerometer in the craft will register 1g.  Switch off the engine, bring the craft to a halt and now release the craft from a height of 1mile above the surface of the earth. Neglecting air resistance the craft will now accelerate at ~9.81m/s but in this case the accelerometer will register zero. In both cases the acceleration is the same(roughly) but the accelerometer readings are divergent. I have some ideas to solve this anomaly, though I am interested in any input.

[alancalverd]

Your accelerometer presumably consists of a lump of material on a spring, the other end of which is attached to the rocket. The engine pushes the rocket and the spring has to compress to push the test weight.

In free fall, gravity attracts the rocket and the test weight equally and simultaneously.

[Halc]

Why does gravity not cause proper acceleration?

Short answer: Because gravity (under relativity) is not a force, and it takes a force to produce proper acceleration, per F=ma.. Under Newtonian physics, gravity IS a force, and the accelerometer reads zero anyway for the reasons Alan points out.

Imagine one has a rocket propelled craft with a power plant capable of producing a proper acceleration of 1g.

Such a rocket will not leave the launch pad, but will just hover there. It won't get to a mile up, but perhaps it started there for some reason.

Switch off the engine, bring the craft to a halt and now release the craft from a height of 1mile above the surface of the earth. Neglecting air resistance the craft will now accelerate at ~9.81m/s but in this case the accelerometer will register zero.

The latter is coordinate acceleration relative to a non-inertial coordinate system, not proper acceleration.  Under relativity, the rocket (after engines off) goes straight (or stays put), no proper acceleration, and it is the ground that undergoes proper acceleration up to meet the rocket, propelled by the force exerted by the ground below it.

[paul cotter (OP)]

Thank you both. I was looking for an answer without GR and Alan's answer is what I was thinking about myself. In the engine powered acceleration the engine pushes on the craft and the measuring element in the accelerometer will be pushed back due to it's inertia. In the gravity fall every atom in the craft experiences the same "force" and hence there can be no differential in the accelerometer.

[Eternal Student]

Hi,

   I'm not even entirely sure what the question was and I'm not sure I understand what your ( @paul cotter ) accelerometer is supposed to be doing.   
   What I'm surprised about is that @alancalverd hasn't mentioned something that he almost always does mention:

   To the best of my knowledge, when an aeroplane (and presumably also a spacecraft) is just sat on the runway (or launchpad),  without any engines running, prior to any launch / take-off....   the accelerometer in the cabin reads 1g  and most certainly NOT  0 g.
     It won't say 0 g unless you have chosen to calibrate the accelerometer to display 0 while you were on the runway (or launchpad),  which would normally be a completely inappropriate thing to do.      In an aeroplane the accelerometer is important for knowing how much stress you can put parts of the aeroplane under in your aerobatics.    So it's important to know, for example, that the wings are already being pulled downward and away from their attachment to the fuselage with a force corresponding to 1g  even while the aeroplane is just sat on the runway.

   I don't know many more details... ask @alancalverd  - the aeroplane enthusiast.    For example, that 1g that is displayed in the cabin while on the runway is that described as an upward acceleration or a downward accelertaion?   I'll assume it's like a conventional accelerometer as described here:  https://en.wikipedia.org/wiki/Accelerometer      so that this will be displayed as an upward accelertion.

    Anyway, this is important because I am not clear about what it is that you ( @paul cotter ) think your accelerometer will be displaying.     

....Firing up the engine will produce an acceleration of 9.81m/s and an accelerometer in the craft will register 1g....

    If the engines can only produce 1g and the spacecraft was starting on the surface of the planet, then the accelerometer in the cabin doesn't actually move or change one little bit.   If I've understood the situation correctly, then it would still read 1g exactly as it did before the engines fired up.   The only difference is that the engines are now fighting against gravity rather than the ground that the spacecraft was resting on.  The accelerometer in the cabin just doesn't care who or what was countering the gravity and indeed anything left lose in the cabin, like a pen the pilot takes out of his pocket and releases in the air, still behaves the same as it did before the engines fired up,  the pen experiences a downward acceleration of just 1g.

        Wikipedia defines the term as follows:   
...Proper acceleration is the acceleration (the rate of change of velocity) of the object relative to an observer who is in free fall...

    So, just from that definition, if the only acceleration a body was experiencing was due to the gravitational field in the region, then that body has precisely 0  proper acceleration.   So, well... I'm not sure how to phrase this exactly....   It is apparent why gravity doesn't count as something causing proper acceleration:    The 0 point for proper acceleration is deliberately chosen to be precisely that situation where the body experiences what will appear under Newtonian mechanics and Newtonian gravity as only being due to gravity.
     There is some confusion about "proper acceleration" but that only exists when and if you interpret it as being an acceleration relative to an inertial frame.    If you're using Newtonian mechanics, just don't do this.   The definition of proper acceleration stipulates and therefore demands that acceleration was measured relative to an observer in free fall  - and as we all know in Newtonian mechanics, an observer in free-fall on or near planet earth is NOT something we would usually consider to have an inertial rest frame.   Under Newtonian mechanics and Newtonian gravity we would usually consider an observer in free-fall to be accelerated and therfore to have an accelerated rest frame.   To be clear, under Newtonian mechanics (including Newtonian gravity) you simply have NO RIGHT or ability to talk about proper acceleration as if it is an acceleration realtive to an inertial frame   (unless you're in the extremely privileged position where the local gravitational field was 0 and an object in free-fall doesn't really "fall" or get accelerated anywhere).
     However, you can legitimately talk about proper acceleration as an acceleration relative to an inertial frame when you use GR.   This is because GR defines inertial frames slightly differently to Newton's approach.   

Newtons inertial frames:
Under Newtonian mechanics, an inertial frame is one where you expect Newtons laws to hold including the law that states an object will move in a straight line unless acted on by a force.   

GR inertial frames:
In GR, an inertial frame is defined as a frame that is in free-fall.    You throw away some of the things that Newton may have expected.  For example, you don't expect or require objects to move in straight lines in an inertial frame when no force acts on them.   (As it happens, the main rule governing the motion of such objects is that they will move along geodesic paths,  these geodesics will not be straight lines unless the local spacetime curvature is 0).

[Image from Tenor.com    Seems ok to share when I'm not making a profit. ]

I'm hopefull that this may put some things in some order and perspective.    The Wikipedia article about "accelerometers" is, in my opinion, a bit sloppy because it uses the phrase "inertial frame" where only Newtonian mechanics has been used in the rest of the section / paragraph  and the reader would reasonably assume that Newton's notion of inertial frame was being used.   This is simply misleading, you must use the GR definition of inertial frame if you wish to consider an accelerometer as showing an acceleration relative to an inertial frame.   If you wish to use only Newtonian stuff, then a proper acceleration is simply one relative to an observer in free-fall (so that a 0 proper acceleration is almost always some non-zero acceleration in a Newtonian inertial reference frame).

Best Wishes.

[paul cotter (OP)]

Hi ES, Halc has already complained of insufficient info and my only excuse is that I'm an old fogey!. I will restate the question: in DEEP SPACE a rocket engine capable of producing a proper acceleration of 1g will propel our test vehicle to the same speed as a gravity drop to the surface of the earth but with conflicting accelerometer data. By the time I had formulated the question I had figured out the fundamental difference, I let the question stand to see what other inputs might be. I was looking for a simplistic answer without GR and i'm happy this has been achieved.

[vhfpmr]

when an aeroplane (and presumably also a spacecraft) is just sat on the runway (or launchpad),  without any engines running, prior to any launch / take-off....   the accelerometer in the cabin reads 1g  and most certainly NOT  0 g.
     It won't say 0 g unless you have chosen to calibrate the accelerometer to display 0 while you were on the runway (or launchpad),  which would normally be a completely inappropriate thing to do.

What's the frequency response of accelerometers in a plane?

My only experience of accelerometers is in vibration testing, and they read zero when the table is stationary, but being piezo devices, I wouldn't expect their frequency response to include zero. (The minimum frequency we were testing was 10Hz.)

[alancalverd]

There's lots of vibration in a small plane, so the g meter is as heavily damped as the others (rev counter, rate of climb indicator.....) with a time constant of at least 0.1 s.

 It is usually set to read 1g when stationary on the ground (because we are primarily interested in the bending load on the wing root and the effect of g on stall speed), unlike an accelerometer in a racing car which will read 0 when stationary (because we are interested in the horizontal forces on the driver and tyres)

[vhfpmr]

There's lots of vibration in a small plane, so the g meter is as heavily damped as the others (rev counter, rate of climb indicator.....) with a time constant of at least 0.1 s.

It was the LF response I was querying, I realise that pilots aren't particularly interested in measuring vibration.

[set fair]

  Switch off the engine, bring the craft to a halt and now release the craft from a height of 1mile above the surface of the earth.

Suppose instead you park the craft on top of a tower 1 mile high. Now the accelerometer will show that the craft is accelerating at ~1g (principle of equivalence) away from the earth. Then tip the craft over the edge of the tower...