Attacked by an umbrella

We have a spring-loaded umbrella at home. The idea is that you press a button, and it automatically springs into shape – its shaft springs out and the canopy unfolds. I've often wondered about the wisdom of such a mechanism and thought what would happen if it went off in an inconvenient confined space, such as a shop packed with expensive china ornaments.To reset, after use, you press the button again to fold up the canopy, but then need to reset the spring mechanism by pushing half the shaft inside the other half. It's that resetting bit that got me yesterday morning. On coming back inside the house after tending to the chickens I tried to collapse and reset the umbrella, but didn't push the shaft hard enough for the mechanism to engage. Instead, when I let it go, the shaft sprung back and hit me in the face – hard enough to draw blood. 

Fortunately my injury was entirely superficial (but it hurt!) and I won't be appearing in the 2015 ACC funny list for the craziest effectors of injury*. 

So how hard did it hit me? Let's do a quick estimate. First, how much energy is in the mechanism when enabled? This depends on the force required to push the spring into place and the distance one needs to push it. The latter is the easy one – about 20 cm. The force is harder to estimate. Imagine putting the umbrella on its end and gradually applying an increasing weight to its shaft, until the spring is fully compressed. How much needs to be applied? About 3 kilograms of mass, maybe. That's 30 newtons of force. So the energy stored in the spring is the force times the distance compressed, divided by two. The factor of half comes from the fact that as the spring compresses the force required increases linearly with the compression. So that gives us about 30 newtons, times 0.2 metres, divided by 2, which equals 3 joules of energy. 

That energy stored in the spring got transferred to the kinetic (movement) energy of the shaft when I let it go. So, how hard did it hit me? One can do the calculation in reverse. About 3 joules has to be dissipated over a small distance as it hit me. Maybe I moved a couple of centimetres. So what force would dissipate 3 joules when acting over 2 cm. Divide 3 J by 0.02 m and we get 150 Newtons. 

That's the weight of 15 kg of stuff, applied through the end of the shaft, to my face. Imagine a suitcase perched on top of the umbrella with the other end supported by my lower lip. Ouch.  The force is substantially larger than that I need to compress the spring in the first place, because it is dissipated over a much smaller distance. However, the suitcase comparison shouldn't be taken too far because the umbrella force was applied only for a very brief period. 

I have now learned to be a bit more careful when resetting the umbrella. 

*But let's not forget the 1978 umbrella-assassination of Bulgarian dissident Georgi Markov on Waterloo Bridge in London



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