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[Phys-l] accelerometers



1) By way of background, recall that an ordinary real-world accelerometer
reads "1" when sitting in the terrestrial lab frame:
http://www.airworlduk.com/2004pix/accel.jpg

Also note that such an instrument measures the acceleration /vector/, or
rather the projection of the acceleration vector in a particular direction.
There is implicitly an arrow attached to the case of the instrument,
pointing upwards when the instrument is oriented as shown in the photo.
The lab frame is accelerating at 1 Gee /upward/ relative to free fall.

If you turn the instrument upside down, the readout will indicate "-1",
which makes sense: if you ask about the /downward/ acceleration of the
lab frame, it is -1 Gee.

If you take the instrument into the Galileo Memorial Freely Falling Elevator,
and hold it at rest in that frame, it will read zero, and will be insensitive
to orientation. This also makes sense; it is reporting a vector of zero
magnitude and indeterminate direction.

Note that the instrument can be considered a self-contained black box: The
reading is based entirely on the physics going on inside the instrument.
It does not make use of information about anything going on outside the box.

The traditional design of such an instrument involves little more than a
mass on a spring, plus some damping and a readout mechanism.

========

2) Let's play a little game. Your mission, should you decide to accept it,
is to design a new type of accelerometer, one that would read zero in the lab
frame. Let's call this a "Type-II" accelerometer.
a) It must be a small, self-contained, black-box instrument. (Otherwise the
task would be too easy; anybody can use a rangefinder to observe the position
of some fixed point in the lab frame, and calculate the acceleration by
differentiating.)
b) It must read zero when moving freely in interstellar space.
c) It must read +1 when accelerated upward at 1 Gee, and read -1 when
accelerated downward at 1 Gee, relative to the lab frame. (Otherwise the
task would be too easy; you could just have a chunk of solid wood painted
to read 0 no matter what.)

Q1: Does your instrument also read zero when turned upside down in the lab
frame? If not, please explain the meaning of this reading.
Q2: What does your instrument read in the freely falling elevator frame,
in your preferred orientation?
Q3: What does your instrument read if you turn it upside down relative
to your preferred orientation, in the freely falling elevator frame?
Q4: Last but not least, what are the physics principles that your instrument
uses to ascertain the acceleration?

Note that it will not suffice to take an ordinary accelerometer and offset
the reading by -1 Gee so that it reads zero when sitting in the lab frame.
This would produce
-- an unacceptable reading (-1) in interstellar space,
-- an unacceptable reading (-2) when turned upside down in the lab frame, and
-- a hard-to-explain insensitivity to orientation in the freely falling frame.

==========================

Hint: Item (2) is a dare, or a rhetorical question, intended to make a point
about the meaning and significance of the equivalence principle. I is worth
/trying/ to design a Type-II accelerometer, so you can appreciate how hard it
is, and why it is hard.

You do not need to study general relativity in order to understand the equivalence
principle, or to attack this problem. A modest familiarity with Newtonian
gravitation suffices. Forsooth, the idea that things would be weightless relative
to a freely-falling observer was published by Galileo in 1638, decades before
there was any Newtonian theory of gravitation.