Just fiddling about with a bit of math(s), and I came up with the crank
length vs. wheel size calculator. It came from the fact that I’m not sure
whether to go for a 24 or 26 when I buy a muni… and then what length
cranks to use.

I figured one benchmark is the point at which you hit a slope so steep you
have to start pulling up on the seat to avoid being thrown off.

Yeah my angle came in low, although of course this doesn’t account for any momentum, and is purely steady climb. Also, Muni tyres are much bigger than their stated size, my 24" Muni is very nearly the size of the 26" bikes next to it in the rack, so maybe something a bit more specific there would help things.

> What formula are you using? The angles are suspiciously shallow.
>
> Naively, I figure the angle where pedalling becomes impossible even
> with horizontal cranks is
>
> arctan (m_rider * l_crank / (m_total * l_wheel),

I have the same, but I made it arcsin. In the force diagram I drew, the
gravitational force is the hypotenuse, and the force in the direction of
travel is the ‘opposite’.

I’m still not convinced by arctan. That suggests that you would need
infinite-length cranks to stay on a vertical wall. (I know this is
horribly theoretical!) But my method says that if the cranks are the same
as the wheel radius, and the unicycle has no weight you can sit
(force-wise) on a vertical wall. This seems intuitively correct to me.

Bear in mind that angles of tracks ARE low. They nearly always look lower
than you expect. Consider a 1 in 5 (20%) hill. That’s considered pretty
steep. Its angle from horizontal is only 11 degrees.

Another test is to look up at 45 degrees. Now measure it. You’ll nearly
always be looking up at only 20-30 or so. This is a pain when trying to
spot satellites and other celestial bodies given their elevation.

Anyway… I digress…

If anyone can come up with an argument why my force diagrams are wrong,
I’d be happy to discuss, but be wary of those angles looking too low -
you’d be surprised.

If I put my details into that form it gives me a slope of 1:2.3!

I used 28" wheel diameter (my 26x3" wheel is near enough 28" outside diameter), 165mm cranks, 168lbs (12st) rider weight and 22lbs uni weight (26x3 nimbus 2). The only figure I guessed was the uni weight (10kg ish), but I think it’s fairly close, if anything it’s lighter.

I know it’s only a theoretical calculation, not allowing for energy used to balance etc, but I ride up a 1:5 ish hill on the way home from work and there’s no way I could get up it without pulling on the saddle. If I didn’t pull on the handle I’m sure I would just stand up and stop.

1:2.3 seems far too optimistic - or is my technique really that bad?

> If I put my details into that form it gives me a slope of 1:2.3!

Ok… so now we’ve had people complaining about the slopes being too
gentle, now too steep

I guess the theory only states that on the given hill, you will be able to
stillstand with cranks horizontal and all your weight on the pedal which
is pushing upwards.

I guess once you’re riding, all sorts of dynamics appear, including cranks
not being horizontal, momentum etc.

I didn’t intend to complain - just comparing the outcome of the theory with actual riding. The “being able to do a stillstand” explanation makes more sense to me - the steepest rideable hill would presumably be considerably shallower.

This thread has got me wondering what I got (or if I passed) my A Level maths all those years ago. I can’t remember but after this thread, I’m thinking that maybe I didn’t take it & those 2 years of my life are someone else’s memory.

My bad. Arcsin it is. Thanks for a nice counterexample.

The reason the angles seem small to me isn’t just a misperception when
looking at terrain. I actually ride up bumps that are steeper than
your calculator predicts. At the BMX track, I have managed to climb a
slope averaging about 34 degrees (using a makeshift clinometer) on my
KH24, which maxes out at 27 degrees according to the formula. The
last foot or two of the bump must be closer to 40 degrees. I’ll admit
I’ve wondered whether quantum tunnelling has contributed to my success
every time I roll over the top of this bump!

Anyway, there’s nothing wrong with your formula. It just makes some
limiting assumptions: No pulling up on the handle; no use of momentum
to carry you forward; no bouncy riding.

Thanks for the mental calisthenics, and a measure of when slopes
really get hard!