We had our departmental Christmas lunch on Tuesday, outside in the campus grounds. We had some lovely sunshine, but the wind did rather spoil things. I've certainly got used now to living in a very wind-free place – a fresh breeze is something quite unsual here. We were hanging on to our paper plates, but didn't expect to have to hang on to glass drink bottles as well. One particular gust was strong enough to take a newly opened individual-serving-sized glass bottle of lemonade and blow it over.
So, being a physicist I had a go at estimating just how strong the gust of wind needed to be to push over a lemonade bottle. As the wind hits the bottle it has to change direction, and this causes a change in its momentum. To do that requires a force – the force being the rate of change of momentum of the air. That gives us an estimate of the pushing force in terms of the speed of the wind – specifically the density of air, times the cross section of the bottle to the wind, times the speed of the wind squared. This generates a turning moment about a point on the base – to get this you can multiply the force by the distance of the centre of the bottle from the table.
The bottle will tip if this force is enough to overcome the turning moment due to gravity the other way. That's simply the weight times the radius of the bottle. Doing the calculation, gave an estimate of about 15 m/s or so, or a bit above 50 km/h. Not particularly high. I was a bit disappointed by the result.
But then I got thinking about something more interesting. In this case, the bottle tipped. But what determines whether it will tip over or slide along the table? To think about this, we need to work out how strong the wind needs to be in order to slide the bottle. In this case we can equate the sideways force exerted by the wind to the maximum amount of frictional resistance that the bottle-table interface can provide. The latter is simply the bottle's weight multiplied by the coefficient of static friction between the bottle and the table. Doing the maths again, with an estimate of the coefficient of friction of around 0.5, I got something marginally larger – about 60 km/h.
Now, the curious thing is the ratio of the force needed to slide the bottle to the force needed to tip it. Although each individual force is quite complicated to write down (so I'm not going to), the ratio turns out to be something really quite simple and elegant. Assuming a cylindrical bottle (!) the ratio of the force-to-slide to the force-to-tip is just the square root of the product of aspect ratio (height over diameter) and coefficient of static friction. This means if the coefficient of static friction is larger than the diameter over the height, it will tip rather than slide. If the coefficient of static friction is smaller than the diameter over the height, it will slide rather than tip.
As an example, if the coefficient of friction is low (e.g. the bottle is on ice) the force to get it to slide is much less than the force to tip it. If the wind blows hard enough, the bottle will slide, not tip. Having a small height compared to diameter also favours sliding rather than tipping – a squat ginger-beer bottle is rather more likely to slide when pushed sideways rather than a slender wine bottle.
One could potentially use this as an amusing way of measuring the coefficient of static friction. Use differently proportioned cylinders and apply a sideways force to each until they move. The squatter ones will slide, the more slender ones will tip. Somewhere in the middle will be one that does both at once. The coefficient of static friction is then just the diameter divided by the height of this cylinder. There are simpler ways to do it, such as measuring by the angle of a slope that is just steep enough for an object to slide down. The coefficient of friction is just the tangent of this angle.
That's it for this year, methinks. Have a Happy Christmas and enjoy the New Year. I think it's time I headed off to Seddon Park to see Mitchell Santner destroy the second-half of the Sri Lankan batting line-up.