How fast can you spin your arm?
This is a slightly odder one, but again (like the Pope and pizza question) it stems from a conversation with my friends. This one started from me providing tips to warm up cold hands, an ever-present problem in poorly-insulated student houses. As someone who's quite keen on mountaineering, especially in winter, cold hands are something I think I know about and the engineer in me can't help but try to solve a problem when one is presented.
A classic mountaineer's trick to get some blood flowing in the fingers is to windmill your arm vigourously, both generating some heat through movement and creating some extra artificial blood pressure. This bonus pressure forces some blood into the digits, which can be enough to give you a little more dexterity for a few minutes, handy for undoing zips, putting on crampons etc. While I was keen on the idea, one of my friends put forward the concern that maybe just forcing ciculation through constricted capillaries was a less-than-great idea: couldn't you do some damage?
But what kind of damage? I'm sure that those with more medical training would be able to intelligently discuss the response of interior tissues to excess pressure, but I'm not desperately interested in the exact answer to that question. Instead, let's treat this as a good exercise in applying some engineering analysis to a new problem, and coming up with a ballpark answer. The version of the question I saw as easiest to tackle is "How fast can I spin my arm before the skin of my fingertip bursts?"
The first step is to start to think of some sort of model for the system. In this case the simplest model is probably treating the finger as a bag of fluid covered in a layer of skin. To simplify even further, let's say that each of those finger bags is a cylindrical tube, with a hemispherical end. That finger has some radius, and there's an internal pressure, generated by our spinning arm. While extra pressures exist (atmospheric externally and normal blood pressure internally) my instinct at this phase is that these probably roughly cancel out, so we can ignore them 1.
This model is a good start - we've reduced one set of complexities (the geometry of a finger) down to something simple we can model with a straightforward equation:
This equation describes the stress (σ) in a thin-walled hemispherical pressure vessel in terms of the pressure P, the radius r, and the thickness t. Is the end of a finger a neat hemisphere? Absolutely not, and the skin is fairly thick there, so I suspect we're outside the range where the thin-wall assumption (that the stress is constant throughout the thickness) can be safely made. On the other hand, we have some good reasons to not try harder:
- How important is the output of this calculation? Not very - I'm certainly not presenting the answer as medical advice. If I were doing this to design an actual product, I'd spend more time and effort on the details, but here we're just doing a big-picture estimation, so it's not worth the work.
- Is this the largest source of error in the calculation? Probably not, we've already made some pretty big geometric and structural assumptions, so they're likely to contribute more in terms of errors.
- What's the logical next step up in terms of accuracy and precision? Closely linked to the last question, if we wanted to get a better result, we should focus on the areas that are most wrong now. here the best place to look would probably be at structure and blood vessel information, not the pressure assumption. It's entirely possible (I'd say likely in this example) that something we learned about a more accurate structural model would push us to use a different category of equation here, not just a more precise version of this one.
While this equation is nice and simple, right now it's hiding something. Whilst I've written P as a simple single term here, it's actually going to be a little trickier to work out, because we're looking at the pressure created by a column of fluid in a rotating tube. Standard fluid pressure due to gravity is simply P = ρgh, and we can try generalising this: Pressure = density * acceleration * depth. This only applies while our acceleration is constant throughout the depth. If we think about centripetal acceleration, we know that it's proportional to the velocity squared, divided by the radius. This means our acceleration changes along the radius of the arm, so we're going to need to go back to first principles.
Let's consider a small element of fluid somewhere in the middle of the arm 2.
Here we have some fluid, starting from some radius r, with a length dr, On the left-side of the diagram (nearest the shoulder of our hypothetical test subject), there's some force P acting on the element. By the time we've moved to r+dr, that force has changed by dP, giving P+dP. Next we need to figure out what dP is, and here we can use the simple pressure equation we mentioned earlier, as we consider dr to be small enough for the acceleration to be constant across it. Substituting in gives
where ac is the centripetal acceleration of this element, found from the equation
With a bit of rearranging and substituting in, we find
Now we're going to move dr to the left-hand-side of the equation. Technically, I understand this makes pure mathematicians sad, because we're treating calculus operators like ordinary variables, but I'm an engineer, and this works, so I'm not going to scrutinise it too hard. This allows us to write
Now we can integrate both sides
Elementary integration gives
Because we used the indefinite integral, we now have a + c term, but earlier we said we were discounting internal blood pressure, which means P = 0 at r = 0 (the shoulder joint). This means the c term is 0, and so we don't need to show it from here on.
This is great - we've derived a neat equation for the pressure in an incompressible column of fluid being spun about some point. This could certainly have some applications, like designing parts for a centrifuge, or looking at internal pressures in centrifugal pumps. For our purposes, there's something still missing, which is a limiting value for pressure. To find this, we need to know what pressure the fingertip can sustain, based on our earlier equation. A quick search gives the ultimate tensile strength of skin as 27.2 MPa [1], and a thickness of 0.57 mm [2]. For radius, I measured my middle finger and came to a value of about 8 mm. Rearranging for pressure, we get
Substituing in our values gives a failure pressure of 1.9 MPa, which is apparently roughly equivalent to the pressure inside a steam locomotive boiler. Trying to forget the image of a skin-based train, we can now subsititute this into the left-hand-side of the pressure-radius equation we had earlier. Rearranged to put angular velocity on the left, that looks like
The only values we're still missing are ρ, which can be found from the literature to be essentially the same as water, 1 g/cm3 [3], and arm length, which is easily sorted with a tape measure. My arm is pretty close to 0.7 m long from shoulder to middle finger tip, which is probably never going to be useful information outside of this calculation. Putting in all over values gives our final answer
So now we have an answer - if you span your arm faster than 88 rad/s, you'd be at risk of bursting your fingertip with the blood pressure. Let's do our best not to picture that too clearly, and think about that speed. It corresponds to a tip velocity of 62 m/s, or 222 km/h. That's a pretty speedy arm, and I don't think anybody's shoulder could generate the forces needed to produce it. Our conclusion (again, not actual medical advice) is that you're free to windmill your arm as fast as you like in the name of warming your hands, safe in the knowledge that your fingertips won't burst in the process.
Or you could buy a pair of gloves.
1 A simple demonstration of this is that while you do bleed if you prick a finger, it's more of a trickle than a jet, so the pressure differential between blood and the atmosphere can't be that large. If this were an artery, we would need to start taking blood pressure into consideration.
2 If your educational background is anything like mine, this phrase might be enough to trigger flashbacks to lots of mechanics lectures...
References
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