Measuring g with an index card and a pin

[bcrowell]

Ca. 1996 a colleague bragged to me about an undergraduate lab he'd worked out where they did a high-precision measurement of g (I think he claimed 3 sig figs) using an index card, a pin, and a stopwatch. He didn't tell me any details. I've long since lost contact with him, but I was thinking today about how to do it. Here's the best method I was able to come up with. Lay a ruler along a diagonal of the card and measure the length of the diagonal d. Make a pinhole with the pin along the main diagonal, and find the pinhole's distance L from the center. Measure the period of the pendulum. Optionally measure its Q. Then g is given by:
$$g=\left(\frac{4\pi^2L}{T^2}\right)\left[1+\frac{1}{12}\left(\frac{d}{L}\right)^2\right]\left[1-1/(4Q^2)\right]^{-1}$$
I couldn't find an index card around the house, so I used a postcard, which had a mailing label on it that I couldn't remove cleanly. Its Q seemed pretty large, so I didn't bother with the correction factor. The result I got was g=9.40 +- .07 m/s2 (random error due to time) +- .1 m/s2 (random error due to L). Anyone want to give it a shot and see if they can get high-precision results?

As far as I can tell, there are two systematic errors that are going to be hard to get rid of. (1) The card may not oscillate perfectly in its own plane; it may twist a little. (2) The treatment above assumes a damping torque that is proportional to the angular frequency, $\tau \propto \omega$. But this is really kinetic friction, which is probably independent of $\omega$.

Possibly #2 could be gotten rid of by measuring periods at different amplitudes.

 

[Andy Resnick]

This is excellent! Presumably, the stopwatch has a resolution of 0.01 sec?

 

[bcrowell]

Well, the stopwatch has a resolution as good as my reflexes :-) I did 8 trials, and the standard deviation was .08 s. It might be possible to improve this significantly using a photogate timer. But I'm guessing that the main errors in my initial try were systematic: the mailing label and errors (1) and (2) described above.

 

[bcrowell]

I spent some time this morning working out an analysis of how the results would be affected by kinetic friction (friction that isn't proportional to angular velocity). The result is that the period should be propotional to $1+1/\pi n$, where n is the number of cycles remaining until it halts. I carried out some measurements with the index card, and I actually don't see any systematic variation of more than about 10% in the period as a function of n, even for n of about 1 or 2, where you should get a decent-sized effect. Maybe the dominant effect is actually the torque due to air friction (which has a much longer lever-arm).

 

[rcgldr]

You sure this wasn't a hoax from you friend, such as using the stop watch to measure some elasped time event and writing down the results on a index card using a pen?