Energy density of a spring (or a "mainspring")
My 2-year-old son loves wind-up toys, and that set me thinking... what is the energy density (the energy per unit mass) of a wound-up spring? After a bit of googling, I have come to the conclusion that this is one of the few questions to which the internet does not know the answer!
Quite a few people have already asked the question.
For example, on the xkcd forum " today I began thinking about mainsprings, the coiled springs typically used to power wind up clocks, watches, etc. While reading up on them I began to notice a trend where articles comment on how much energy they can contain (usually described as "a lot" rather than anything useful.) This led me to try to find a source for the potential energy of a mainspring, something that I've found rather difficult to find." The same question has been asked more than once on physicsforums.
And there must be plenty of experts who know the answer... for example these Birmingham researchers, and about 400 years of clock-making experts, and Trevor Bayliss who made the wind-up radio. I don't know why they are so secretive! :-)
Let's figure out a rough answer to the questionA coiled spring stores energy in the same way as a bent beam. You can read about the energy stored in a bent beam in my lecture on the musical note produced by the "beams" of a marimba or xylophone.
It's interesting stuff, but actually we don't need all that detail to get the answer. The key insights we need are
The energy per unit mass in a bit of the spring that is strained with a strain of ε is
0.5 Y ( ε2 ) / ρwhere Y is the Young's modulus, and ρ is the density.
The stress τ is (roughly) related to the strain by
τ = Y ε
- and the maximum stress you can cope with [in a spring that is to be reused many times] is called the Yield strength, which I'll denote by the symbol τmax.
|Material||Y||τmax||ρ||0.5 (τmax)2 / ( Y ρ )|
|Steel (structural ASTM A36 steel)||200||250||8000||0.005|
|Carbon fibre [.]||230||4000||1600||6.0?|
|Steel (Micro-Melt 10 Tough Treated Tool (AISI A11))||200||5000||7450||2.3|
We can compare these energy densities with those of other energy storage systems featured in my book by looking at page 199. Sadly, the wind-up spring doesn't get close to the energy density of even the worst rechargeable batteries (30 Wh/kg).
[Next steps: quality-assure the numbers in the table, and do a real-world check against the actual weight and actual energy stored in real clock mainsprings.]