Elephants in Stilletos

I was intending to post this yesterday but my brief intoduction to Inktober ended up taking a lot longer than planned, so I postponed this one…

Yesterday’s prompt word for Inktober was “Pressure”. As usual, I thought about several possible avenues for interpreting this prompt, one of which was the scientific definition of pressure as force over area. This reminded me of a fun fact I learned while I was at school, namely that a woman in stiletto heels will do more damage to a wooden floor than an elephant, because although she weighs a lot less than the elephant (assuming average-sized women and elephants), all her weight is concentrated over a very small area compared to the elephant’s feet.

While thinking about this yesterday, it occurred to me that this was all based on the assumption (to be fair, probably a fairly safe one) that the elephant isn’t also wearing stiletto heels. This set me off on a rare foray into cartoon-style illustration:

Probably not one of my best sketches ever, but it was quite fun to make. I also took a bit of time to do my own calculations to verify the assertion that a woman in stilettos exerts more force than a barefoot elephant.

To do this, I looked up a few figures and estimated a few others.

According to an article appearing in the Independent in 2017, the average weight for a woman in the UK is 11 stone. That’s slightly higher than I expected, but I decided to go with that figure. Converting to metric units, that’s close enough to 70kg, so we can use that for our woman’s weight.

Except that it’s actually her mass, since weight is a force (gravity) acting on a massive object (i.e. an object that has mass, not necessarily a particularly large one) and is dependent on the strength of the gravitational field it’s in. We actually need the weight for our calculation (as pressure is force/area), but that’s easy enough to calculate from the mass. Newton’s 2nd law says that force is mass times acceleration (F=ma if you like equations, as I do) and in this case the acceleration is that due to gravity. That varies from place to place around the world but it’s roughly 9.81 metres per second squared. For my rough calculations, I decided that a nice round figure of 10m/s2 would do fine. So our average woman weighs about 700 Newtons.

I didn’t make a note of where I found the figures for an elephant but apparently a female African Bush Elephant weighs on average around 3 tons. I’m not sure if that’s supposed to be long tons or short tons, but either way I decided that just calling it 3 metric tonnes (3000kg) would be close enough. Again, that’s actually the elephant’s mass (everyday language tends to be shockingly imprecise when it comes to such things), and her weight would be 30,000N using the same figure of 10m/s2 for the acceleration due to gravity. Incidentally, I decided that since our woman is (by definition) female, I’d go with a female elephant too (they tend to be a bit smaller than the males) and since I tend to think of African savannas before African forests or any part of India when thinking of elephants, I opted for an African Bush Elephant (a species that’s generally somewhat bigger than the the other two varieties).

That bit was relatively easy. Working out the areas was slightly more problematic, especially for the woman in stilettos. You will probably be relieved to hear that I don’t have any stilettos in my own shoe collection, and I was too lazy to go out and find a woman with high heels so I could measure the surface area of her heels and toes together or figure out how much of her weight would be concentrated on each part of her foot. For the initial calculation, at least, I wanted to work on the assumption that both the woman and the elephant would be standing with their weight evenly distributed across all their legs (that sounds a bit weird for the woman – obviously “all” is just “both” in her case!). A bit of online research revealed that stiletto heels usually have a diameter of no more than one centimetre, but I couldn’t find anything out about the area of the front part of the foot that would be in contact with the ground and presumably bear its share of the weight. I settled for a rough estimate of about 1cm2 for the surface area of each heel and 50cm2 for the surface area of the toe/ball of each foot. For convenience I tweaked the latter down to 49cm2, giving a total surface area of 100cm2 for both feet (heels and toes combined).

The elephant’s foot size was actually a bit easier to determine. Apparently a typical African Elephant has feet between 40 and 50cm in diameter. I decided to give the woman a bit of a helping hand by assuming our elephant had relatively small feet (hence providing less area to spread the weight) and therefore a 40cm diameter, or 20cm radius which, if we assume that the feet are circular, gives a surface area of about 1250cm2 per foot or 5000cm2 for all four feet.

To ensure our final units are correctly expressed as Pascals, or Newtons per square metre, it’s handy at this stage to convert those areas into square metres rather than square centimetres. The woman, standing with both feet firmly on the floor is putting all her 700N of weight through 0.01m2 of the floor, while the elephant’s 30,000N is being spread across 0.5m2 with the net result that the woman is exerting 70,000N/m2 or 70kPa of pressure on the floor, while the elephant is exerting only 60,000N/m2 or 60kPa. So our average woman is indeed liable to do a bit more damage to our delicate wooden floor than our average elephant, though the figures are actually quite close.

The difference gets more pronounced if they both put all their weight on a smaller area. I’m not sure how practical it would be to rest all your weight on one heel while wearing stilettos (mind you, I’m not convinced it’s very practical to wear stilettos in the first place) but suppose she’s able to do so, our woman is now channeling 700N through an area of just 1cm2 or 0.0001m2 which makes for 7MPa of pressure (that’s 7 Megapascals, 7 million Pascals or 7×106Pa if you’re not afraid of scientific notation – it’s definitely much more convenient than long trails of zeroes at either end of your numbers). Assuming that it’s enough of a challenge for our elephant to stand on just one foot, without going up on her toes or heels, she would be putting 30kN through 0.125m2, which amounts to 2.4×105Pa, which is 240kPa or 0.24MPa – significantly less than the woman on one heel.

Since my cartoon was based on the idea that an elephant wearing stilettos would do more damage to the floor than a woman in stilettos, I couldn’t leave this set of calculations without considering the pressure exerted by our elephant if she were to don a set of stiletto heels. Presumably these would have to be custom made and I’ve no idea how big they would be, nor whether she’d wear them on all four feet or just two, so let’s assume that the heels themselves culminate in points the same size as the woman’s ones, i.e. 1cm2 each and the elephant has somehow managed to contrive to stand with all her 30kN of weight bearing down on just one of these heels. That would make for a pressure of 3×108Pa, or 300MPa. As we would expect, our elephant in stilettos would do considerably more damage to any floor than our woman. It’s probably just as well that elephants are not, as far as I’m aware, in the habit of wearing stiletto heels.

I should probably add that it’s been a good few years since I last did this sort of calculation, so I hope I haven’t made any major mistakes with my units or figures, or any assumptions that are too crazy (apart from the basic premise itself, perhaps). Still, I’m fairly confident, at least that the claim made by my cartoon is fundamentally correct:

An average-sized woman in stiletto heels exerts more pressure on the floor than an average-sized elephant…

… unless, of course, the elephant is also wearing stilettos!

(Magnus Forrester-Barker, 2021-10-09)

Something to torque about

For quite a long time now, I’ve had at least a theoretical grasp of the importance of correct torque settings for bike mechanics. Recently, though, I had practical confirmation of this — and of why my recent purchase of a torque wrench was money well spent. In the process I also discovered a handy alternative to my previously limited repertoire of ways of dealing with a stripped screw head.

Torque is essentially a measure of turning force and in this context it amounts to how tightly you do up your nuts (stop sniggering at the back there!). If you want to know more about the gory details, Google can furnish you with many happy hours of reading material.

The reason that torque is important for bike mechanics (and I’m sure also for many other engineering-related disciplines but I’ll try to stay focused here) is that if you don’t tighten up nuts and bolts sufficiently, parts tend to drop off the bike at inopportune moments, while if you overtighten them you not only make them harder than necessary to remove when the time comes for maintenance but run the risk of doing damage to various components. Unfortunately, while it’s generally fairly obvious which bits need to be tightened a lot and which bits can and should be left looser (though there may be a few surprises in store), it is pretty difficult – even with quite a lot of experience – to correctly judge torque settings by feel alone. As a rule of thumb, which seems to be borne out by my own experience, the tendency is to overtighten parts which shouldn’t be too tight and to insufficiently tighten ones which should be good and tight.

Having had a few bad experiences in both directions, I finally got round to getting myself a torque wrench about 3 years ago. I went for a traditional beam-type one since they are relatively cheap and robust. I already had a socket set (probably one of the single most useful tool purchases I’ve ever made, especially for non-specific bike tools) so this, along with a handy chart of torque settings found in the back of my bike repair handbook, enabled me to get at least into the right ballpark with my torque settings. The downside of the beam-type wrench is that it can be quite difficult to accurately read the scale as you need to be able to look at it head on to avoid parallax errors; this is sometimes virtually impossible to achieve while you’re actually using it (or maybe it’s just my poor technique?). Even if you do manage to get it at a good angle, this particular wrench doesn’t have a particularly finely grained scale so the results are necessarily fairly approximate.

I therefore decided a few weeks ago to invest a little bit more in a click-type wrench. This is basically a ratchet driver which is set up so you can set it to a desired torque setting and it will “click out” when you reach that level of torque. They are somewhat more delicate than the beam-type, and may need occasional recalibration, but as long as it’s well looked after a click-type wrench should work well for a good long time. It certainly makes it much easier to achieve a specific torque setting as you can feel rather than having to see when you get there.

The main disadvantage of click-type wrenches is that they tend to have a relatively limited operating range of possible torque settings, and it’s just about impossible to find a single wrench that will cover all the settings needed on a typical bike. I decided to start by getting one that would do the low-torque settings (mine goes up to 24Nm) since I’ve usually found I’m more likely to overtighten than undertighten things and my beam wrench also seems to work a bit better for giving it quite a bit of welly (to use the technical term). At some point I may also get a click-type wrench to cover the higher torque range, but it’s not a very high priority at the moment.

When my new torque wrench arrived the other day, I was enthusiastic to start straight away on checking that all the various fittings on my bike (or at least the low-torque ones) were tightened to appropriate levels. This went well until I got to one of the bolts holding the brake arms on to the frame. The bolts for these particular brakes (which I put on as replacements for the original front brakes a couple of years ago) appear to be made of aluminium and I discovered that last time I took them off for maintenance I overtightened one of them to the point that the recess had started to deform and the hex key would no longer grip sufficiently to be able to undo it. Looking on the bright side, it’s better to discover the problem now rather than waiting until the brake had seized up and needed imminent maintenance while I was stranded on a roadside miles away from home.

My usual trick when faced with a screw or bolt with a damaged head that will no longer take its usual driver is to use a hacksaw to cut a slot so that I can attempt to use a flathead screwdriver on it instead. Sometimes this works and often it doesn’t (not to mention being a non-starter when the entire bolt head is recessed and you can’t get in to cut it. On this occasion I tried my usual trick and it failed.

Since the brake is a pretty important component to be able to keep in good working order, I didn’t really have the option of giving up on it. I didn’t fancy trying to drill the bolt out in the hope that I could somehow cut a new thread and save the frame to be able attach the brake again. Nor was I quite ready to admit defeat and take the wretched thing to the bike shop, so I used the time-honoured research method known as Google to explore alternative ways of fixing the problem.

This turned up two possible solutions I’d previously been unaware of. One is a little gizmo known as a screw extractor — basically a drill bit that you use to drill into the stuck screw in such a way that it basically unscrews itself. The other is a little bottle of stuff which isn’t quite glue but seems to have the effect of providing a better grip so that your tool can get enough purchase on the damaged screw to be able to turn it. There appear to be several brands, pretty much all of them of American origin, which may or may not be the same stuff under the different labels (it seemed to be pretty vague about what’s actually in it); the cheapest I could find was one called Screw Medic (costing about £3 for a small bottle, but it’s supposed to have a pretty unlimited shelf-life and only require a couple of drops per application, so it should last a fair while). This latter option seems a bit more straightforward to use than the screw extractor but only work on screws with fairly light damage, while the extractor should get pretty much any screw out.

Undecided as to which option to go for, impatient to wait and try one before ordering the other if the first didn’t work, and figuring that knackered screwheads seem to crop up fairly often (perhaps due to my previous lack of care and attention to appropriate torque levels — or making sure I’m not using a Pozidriv screwdriver with Phillips screws or vice versa), I decided to order both. The Screw Medic was the first to arrive and was awaiting me when I got home from work this evening. I’m delighted to say that it worked wonderfully and I was easily able to unscrew the damaged bolt and replace it with the one that I’d saved from the old brake (evidently a steel bolt and much sturdier than the other, though I still took care to tighten it carefully to only the required 6Nm with my new torque wrench).

So when the screw extractor set (there are actually 4 or 5 different bits for different sizes of screw) arrives, hopefully in the next few days, it may be a while before I need to use it. And if I make good use of my torque wrench (and my Philips and Pozidriv screwdrivers) I may never need to use it. Which would be fine by me.

To clip or not toe clip?

Just over a week ago, my bike started making some alarming squeaking noises from the vicinity of its transmission system (i.e. chain, bottom bracket etc.).

At first I didn’t have time to properly investigate, so I just slapped on some WD-40 (in case it was a simple lubrication issue) and hoped for the best.  That seemed to clear it up for about a day, but the problem soon came back and in addition to the disturbing noise I was becoming more aware of something feeling decidedly out-of-kilter.

When I investigated further, I discovered that the problem was in the right-hand pedal, whose bearings seemed to be on their last legs.  This was both a good and a bad thing, but on balance mostly good.

The main downside is that the pedal is essentially a sealed unit so there is no way to get in and mend it and the only option is to get a new one (or ideally two, so you retain a matched pair).

The first positive thing is that it’s a lot easier and cheaper to replace a pair of pedals than the entire bottom bracket assembly, which I had feared was about to go the way of all flesh.

The second positive is that I didn’t actually have to buy new pedals as I was able to take the ones off my mountain bike (which is currently and probably permanently off the road due to a bottom bracket shell issue that I’ve previously mentioned) and put them on my road bike.  Fortunately, unlike many of the other fittings of this bike (which is a fairly old French one), the pedals seem to use the same standardised size of fittings as most other bikes.

The third, and possibly biggest, positive is that, as a result of putting my mountain bike pedals on the road bike I’ve finally got round to getting toeclips onto it, which I’ve been intending to do more-or-less since I started riding this bike (or at least since I restarted using it a couple of years ago – I hadn’t become a convert to the joys of toeclips when I first had the bike).

Toeclips are designed to enable both of your legs to exert force the whole time you are pedalling, rather than just the leg which is pushing down at any given time.  This, fairly obviously, increases the efficiency of pedalling and is an especially noticeable benefit when you’re cycling up steep hills (a more or less unavoidable feature of cycling in Wales).  As an extra benefit, they also ensure your feet remain in a more-or-less optimal position for pedalling (assuming you’ve got the bike set up correctly), with the balls of the feet making contact with the pedal.

In both these respects, pedals with clips are better than traditional pedals without clips (which, confusingly, are not the same thing as clipless pedals), while retaining the convenience of being able to use them with more or less any shoes.  The only real downside is that the clips are a bit bulky and can get caught up on passing obstacles when you’re wheeling the bike, but it’s not a great problem.

Clipless pedals are ones which come with some system of cleats (there are also quite a few mutually-incompatible clipless systems), which enable you to attach your feet securely to the pedals to get the same benefits as using clips but to an even greater degree.  I’ve never tried them myself but they are supposed to be better than clips both in taking up less space (in fact, they are usually quite a bit smaller than ordinary pedals without clips) and providing better energy transfer.  They potentially also make it harder for somebody to grab your bike and ride off with it since you can’t easily ride them without the proper shoes.  That, of course, is also the main downside since you need to get a special pair of shoes (in some cases, it’s an ordinary pair of shoes to which you add cleats) to use the bike and, I think, you’d probably need to carry another pair of shoes to change into when you got off the bike as it’s probably not very comfortable (or good for the cleats) to walk far on them.

In any case, I’m very happy with my clipped pedals, which I’ve had on my mountain bike for the past 8 or 10 years and show no immenent signs of wearing out.

 

A cool tool

One of the blogs I keep an eye on is Gizmag, which I’m sure used to describe itself as “the emerging technology magazine”, although it now seems to have dropped that slogan.  According to its “About” page, Gizmag is over 10 years old and is “a celebration of human endeavor”; as you can see from the spelling it’s US based. The Gizmag owners say “We aim to inspire, not ridicule. We cover technology, not the politics or the money behind it.”  All in all, that seems to be a pretty laudable goal and it’s certainly a good blog to follow if you’re interested in technology.

One of the gadgets that recently featured was of particular interest to me as a cyclist.  This was the Nutter, a very cunning looking bike multi-tool.

Multi-tools are very popular amongst cyclists, as they enable you to carry several different screwdrivers and allen keys (and sometimes other things) in one small, handy package.  That’s great for taking with you on a bike for doing roadside repairs and could be used by more generally by cyclists on a tight budget, although I generally find it much easier to use dedicated tools when I’m working in the relative comfort of my own garage.

The small size of a multi-tool, which is one of its main strengths, is also one of its biggest weaknesses, since having very short handles they don’t provide much leverage and therefore make it harder to loosen very tight screws and allen bolts (or, conversely, to tighten them very hard — though for most purposes you can get things sufficiently tight with a good multitool).

The Nutter is designed to overcome this problem, by the simple expedient of making a longer tool.  It does this by combining it with a tyre lever, which is another tool that most sensible cyclists would carry as a matter of course.   From the photos, it looks a bit bigger than the tyre levers I usually use, but again the extra size would give more leverage and make it easier to get tyres on and off.  In any case, the increased size would be offset by the fact you wouldn’t need to carry separate tyre levers as well as the multitool.

In addition to the tyre lever and a fairly standard set of allen keys (aka hex keys) and screwdrivers – which are supplied as removable bits, rather than on separate shafts – the Nutter comes equipped with a 15mm spanner and a spoke spanner, as well as a handy carrying case.

Apparently the Nutter hasn’t yet gone on to the general market but if they become available at a price within my budget I’ll certainly consider getting one, as it looks like a very useful tool to have and a well-designed (and made) bit of kit.

155.891 smoots (or maybe two)

When I started my occasional series of posts about length measurements just over a year ago, I mentioned that there were two reasons why I had chosen to use the Menai Suspension Bridge as the reference object for all the different units (to be measured via the Google Maps DMT wherever possible).  One reason was that I regularly traverse this landmark.  The other, as I said, was a bridge-related connection to one of the units which was to be related in due course.  Now is the time!

The smoot is a unit that originated in October 1958 when a bunch of engineering students at MIT used one of their number (Oliver Smoot, later to be Chairman of the American National Standards Institute (ANSI) and President of the International Organization for Standardization (ISO)) as a measuring stick to measure the length of the Harvard Bridge.  One smoot is equal to Oliver Smoot’s height (at the time of the measurement), which was 5’7″ (i.e. 67″ or 1.7018m). The bridge’s length was measured to be 364.4 smoots plus or minus one ear, with the “plus or minus” intended to express uncertainty of measurement.  That’s about 620m in the rather more boring but somewhat more common metric system.

The Menai Suspension Bridge, according to my measurement on Google Maps, is 155.891 smoots long, so it’s a bit less than half the size of the Harvard Bridge.  Incidentally, the Menai Suspension Bridge looks very similar to the Széchenyi Lánchíd (Széchenyi Chain Bridge) in Budapest:

Széchenyi Chain Bridge, Budapest

The Budapest bridge was actually built about 15 years later by a different engineer (and modelled on a bridge over the Thames).  Wikipedia lists its length as 375m, which Google Calculator tells me is about 220 smoots.

In case you’re wondering about the possibility of two smoots indicated by the title of this post, it’s actually a reference to two Smoots since Oliver has a cousin, George Smoot, a physicist who won the Nobel Prize (for physics, unsurprisingly) in 2006 and has appeared as a guest star on The Big Bang Theory (which is quite appropriate since much of his physics work has been on the big bang).   Arguably, George is more famous than his cousin although I’m not aware that he has any units named after him.  I decided to go ahead and write this post after I discovered (from Wikipedia, where else?!) that today is George Smoot’s birthday.  So, happy birthday George (in case you should ever happen to read my blog, which is admittedly fairly unlikely)!