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So, my guess (and I've been drinking) is that the limit is when the vertical component of the propulsion (rider weight on the pedals causes bike to move forwards and up a slope) is counter balanced by the gain in gravitational potential energy from moving up the slope.
Lower gear = less forward motion (and therefore a smaller vertical component).
ICBA with the maths right now, will see if being sober in the morning helps.
Greenbank-
My calculation was purely based on a series of gear ratios. Torque at cranks creates torque at chainrings creates tension in the chain creates torque at the cog creates torque on the ground through the back wheel. The upward component of the force on the ground from the back wheel has to offset the weight of rider and bike.
You could almost certainly do the calculation with energy though... Perhaps comparing the GPE of the rider with the cranks parallel to the ground to GPE of rider and bike with the cranks perpendicular to the ground but further up the slope? That would give you a vertical ascent. Then you can derive a maximum gear ratio based on the ground covered at a given gradient. Interesting idea! I'd be keen to see what you find! Might even do some more calculations myself tomorrow.
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Okay, I was too interested and had to do a quick calculation...
GPE down the slope:
Rider is k above the minimum position on the bike (k is one crank length - cranks are parallel with the ground)
Bike and rider at minimum position on the slope
GPE_start = m_rider * g * kGPE up the slope:
Rider at minimum position on the bike (cranks perpendicular to the ground, riders weight on lower pedal)
Bike and rider distance d up the slope
GPE_end = (m_rider + m_bike) * g * d sin(theta)d = k / [(1 + m_bike/m_rider) * sin(theta)]
d is (metric) gear inches / 4 - we only considered a quarter revolution of the cranks
For k = 0.175m, m_bike = 10kg, m_rider = 75kg, theta = 20deg, we get d = 0.451m or 17.8 inches
So that's 71.2 gear inches, which is bang on a 43/16 on a 25mm, 700c tyre, a fairly typical fixie gear ratio! How satisfying!
To cycle up a 45 degree slope with the same bike weight, cranks, etc., you only need a 34/26!
Of course I'm ignoring drivetrain losses, loss of traction, what the upper foot is doing, the use of your muscles, etc.
frankenbike
I'll dig out the code and post it tomorrow. Apologies in advance to anyone who tries to read it...
Either way it suggested that you could get up a 20% gradient with a 50/12 or something which I found a little hard to believe. I made it because I was wondering about the maximum gradient you could get up on a fixie/SS, given that the maximum torque you can apply is based on your body weight, and whether you could base your gear choice off that. There are a bunch of factors I didn't account for in my "model" though and I wonder whether anyone can devise anything better.