Notes on a cup

Ritzenhoff Mugs

Experimental apparatus

An opportunity for an experiment with a cup of coffee. Sadly though, for the experiment itself, it would probably help if the mug were empty, so there are two choices: Either grab a coffee and drink it so that you have the empty cup next to you, or get an empty cup and wait for your coffee until later. There is though, perhaps a third choice, get two cups, one with coffee in it, one empty, that sounds a much better idea.

Now, get a pen or pencil and start to tap the rim of the cup, make note of the sound that the cup makes as you tap at a point next to the handle, moving around to 45º from the handle, 90º from the handle etc. Perhaps compare the sound of different mugs but, on going around any particular cup, what do you hear? The note that you will hear when you tap the mug just next to the handle, or at 90º intervals from the handle should be lower than the note that you hear at 45º angles to the handle. Why is that?

wobbly bridge, Millennium Bridge

“Couple at St Pauls”, photograph © Artemisworks Photography. The ‘wobbly bridge’ is in the background.

Before answering that question, and to give you some time to think about it, it may be time to consider a (related) anecdote. Back at the turn of the millennium, a new ‘shard of light’ was built across the Thames. The Millennium Bridge takes pedestrians from the Tate Modern on the South bank towards St Paul’s on the North bank (or vice versa). It opened on 10th June 2000 and then closed, two days later, owing to problems that left it labelled the ‘wobbly bridge’. Along with many people, I had been taken in by the newspaper headlines of the time saying that we had built a terrible and wobbly bridge. It wasn’t until I was researching St Katherine’s Docks for the White Mulberries cafe-physics review that I found David Blockley’s book, ‘Bridges, the science and art of the world’s most inspiring structures’ and learned the true story. It turns out that the reason the bridge wobbled was because of a previously unknown phenomenon. Dubbed ‘synchronous lateral excitation’, it is a human crowd response to a platform swaying under their feet. Apparently in response to a swaying platform, people will widen their gait slightly to compensate for the wobble, only this acts to increase the sideways force on the platform itself and so can amplify the wobble. This bit had been known, what had not been appreciated was how the ‘wobble’ would grow if a crowd were present. The reason that the wobbly bridge surprised everyone was that never before had the critical mass of pedestrians been walking on a susceptible bridge. According to Blockley, 156 people walking along a particular section of the (original) Millennium Bridge did not cause a problem, but 166 walking in a group along the bridge caused the wobble to quickly become very appreciable.

hitting Zorro

Poor Zorro being experimented upon.

The solution, of course, was to damp the structure, to add shock absorbers and weights to the bridge so that the oscillation decreased. The cup is behaving similarly. Each time you tap the cup, you are exciting a standing wave around the rim of the mug, this is what is exciting the sound. This vibration has four points of maximum oscillation (called anti-nodes) and four stationary points (nodes) around the mug spaced at equal intervals. If the cup is hit so that the handle (which adds a relative weight to one side of the cup) is at a point of maximum oscillation, the mass that is being moved is greater than if there is a node at the handle so it does not have to move. This change of mass shifts the frequency of the oscillation and so the note is lower than when the handle is at a point of zero movement. For more information on the standing waves in your cup click here.

So it’s not just science in your coffee cup, a world of engineering is mirrored in your brew too.

Bridges – the science and art of the world’s most inspiring structures, by David Blockley was published by Oxford University Press in 2010, it is well worth a read as it is a very accessible and informative guide to bridges as well as being entertaining.

If you notice any engineering in your coffee cup, why not let me know via the comments section below or by contacting me via email.

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