Accelerometers - and build one yourself

This has been on my list for quite some time. Really, it must be since i posted about measuring acceleration in free fall with an iphone. So, this post will be all about accelerometers.

How does an accelerometer work? Really, an accelerometer measures force some way on a known mass. Let me show an extremely simple accelerometer - a mass on a spring.

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(image from Science Buddies where they have instructions on building such a device)

Suppose I put this accelerometer in a stationary and non-accelerating elevator. Let me draw a free body diagram for the mass on the end.

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No magic here, right? The total force must be the zero vector since the mass is not accelerating. Now, here is the trick. The force the spring exerts is related to the amount the spring is stretched. This is called Hooke's law and can be written as:

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The vector delta s is the stretch of the spring (or compression). The negative sign says that the spring force is in the opposite direction as the stretch of the spring. k is the spring constant, or a measure of how stiff the spring is. So, now suppose the elevator is accelerating up. The free body diagram would now be:

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Besides the forces, I have also included a vector for the displacement of the mass (stretching) and a vector for the acceleration. For this type of accelerometer, you would measure the amount of the stretch. From that (and the mass and the spring constant, which don't change) you could determine the acceleration. In the vertical direction, the force equation would be:

i-5596250f33eb94346024ce328b7a440a-la_te_xi_t_1_11.jpg

Here delta s is negative (since it stretches down). This makes the spring force in the positive direction. So everything is good. There you have your accelerometer. Here is an example of such an accelerometer in use from TeaWithBuzz.

Notice that if you held the spring-accelerometer horizontal instead of vertical, it would give a different reading. The accelerometer doesn't know the difference between acceleration and the gravitational force. (Einstein's Equivalence Principle says that the effects of gravity are the same as an accelerating reference frame).

What other types of accelerometers are there? There are many variations of the above idea. If you know a mass and can measure a force on that mass, you have an accelerometer. Maybe it is a mass on something that bends. If you know a relationship between how that thing bends and the force, you can make it an accelerometer (assuming you can also measure how much it bends).

Another way to measure the force is with a piezoelectric sensor. This is very similar to a spring, but when it compresses, it produces a change in electric potential. So, for this as an accelerometer, you still need a known mass, but you don't need to read a displacement, you can read a voltage. So, really this is still the same as the spring.

Wikipedia lists more types of accelerometers than you could shake a stick at. Actually, you could also use a pendulum as an accelerometer. Since the period of oscillation depends on the gravitational field, if this were in an elevator accelerating upwards the period would change. Of course, that would not be practical in something like an iphone.

Build your own accelerometer

I met this teacher at the winter AAPT meeting in Chicago (sorry, but I can't remember your name). She gave me materials to build my own accelerometer. Unfortunately, I could not fit these in my already stuffed bag, so I had to give it away. I was able to reproduce the general idea. Let me just show you a picture:

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If my pictures are not clear, this is an upside down jelly jar filled with water. Inside the jar is a cork attached to a string that is anchored in the lid. In the design from the AAPT person, it used a plastic peanut butter jar with a fishing floaty thing (bob?). To use this, look at it from the top. The cork will move in the direction of the acceleration. This is same as the famous floating balloon in an accelerating car problem.

For some real fun, try holding this (or a better version) while moving in a circle.

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The cork will move in the direction of the acceleration. In Uncle Al's universe the buoyant cork moves in opposition to gravitational acceleration - or it would sink not float. Inertial case by the Equivalence Principle. Centripetal case is transverse Doppler. Untether the submerged bob and view its local and external observer path in all three cases.

Mirror the bob's flat top as one arm of a laser pointer interferometer and wait for the moon or a passing fattie. Inertial footsetps and gravitational mass are both detected.

A spun-up ping pong ball on the spinning turntable. Place three orthogonal circles of "corks in water-filled jars" in free fall orbit about the Earth. A local observer otherwise hermetically isolated from the universe can plot his free fall orbit by watching. All the fun is in the footnotes.