Pendulum mayhem

One of the main intentions of our 2nd year Experimental Physics paper at the University of Waikato is to have students learn how to put together a physics experiment that measures something, and to measure that thing in a systematic and robust manner. What that means in practice is dealing with uncertainties.  Whereas the average person might want everything to be absolute-without-any-doubt-certain, it is rare that we can be absolutely unequivocal about anything in science. But what we can say, and should say, is exactly how certain we are about something.

In a physics experiment we tend to do the opposite – and speak about how uncertain the final result might be, and what are the main drivers of that uncertainty. A good experiment will be one where thought has been given to where these uncertainties arise, and effort taken to minimize these.

With all this in mind, I had my students perform a small ‘test’ last week. I stuck them in a lab, and told them to make up a pendulum and use it to measure the acceleration due to gravity. On the face of it, this is easy, and if you’ve studied physics at all you might well have done it. The theory states that (for small amplitudes) the period of a pendulum is given by 2 times pi times the square root of (pendulum length divided by the acceleration due to gravity).  So if you know the pendulum length, and measure the time for a swing, you can work out the acceleration due to gravity.

But is it that simple? There are in fact a whole host of problems. What is the pendulum’s length? The mass at the end certainly won’t be a point mass. In fact, a systematic uncertainty in the measurement of length can be accounted for by plotting a graph of period squared against length – the gradient of this line won’t be affected; the systematic uncertainty will simply mean that it won’t go through the origin.

Another uncertainty is the timing of the period.  We could set up laser-gated timing, but stopwatches are easier. There is the issue of whether the pendulum is actually at rest when you release it and start timing – e.g. do you tend to push it along to start with – i.e. it’s not released from rest?  But you can get around that as well, by not starting the stopwatch when you release it; rather waiting for it to do a complete swing first. This also accounts for any systematic uncertainty in pushing the button on the watch – if you tend to be slightly late, you will press it slightly late at the start, AND slightly late at the finish, and the two effects will cancel. The time on the clock will be correct. Additionally, timing lots and lots of swings is much better than one; the uncertainty in time is reduced when we divide it by 100 swings.

What is a small angle? How secure is the pivot – i.e. does the effective length vary in the course of a swing. Moreover, does the pendulum stay the same length throughout? We used a mass on a string, and the string might easily stretch throughout the experiment.

These are just some of the things that students had to think of when doing their experiment, which is rather harder than it might appear.

After the ‘test’, I had a go myself, trying to take into account all these issues and more. I used the same equipment as the students – string, stopwatch, small masses, a retort stand – and tried to get the best measurement I could. It’s a tough one. 

My measurement, for the record, is not as certain as I would like. I’ve measured the acceleration due to gravity, in our lab, as 9.87 metres per second squared, with an uncertainty of 0.10 metres per second squared. I reckon, after working through it, it’s a tough ask to get that uncertainty much lower. But if you want to give it a shot, please do and let us know how you get on.


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