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## Question

If you pull straight back on the lower pedal of your bicycle with a string going perpendicular to the ground, will the bike move forward or backward initially?

Why?

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I'll assume that you meant "parallel to the ground" rather than perpendicular, because you asked that we pull "back" rather than "up" or "down".

I'm impressed at how this puzzle seems so tricky to someone who has spent nearly his entire life riding a bike.

Here's my walk down the garden path. In all cases, I'm standing behind the bike, exerting force on the string towards me (ie, "pulling back")

Spoiler

The bike would NOT roll towards me, but would slide (if I pulled hard enough) towards me:
( 1 ) if the string were attached to the bottom of the rear tire, directly between the tire and the ground

But it would roll towards me if the string were attached:
( 2 ) to the rear axle
( 3 ) to a spoke on the rear wheel, regardless of whether it's above or below the axle
( 4 ) to the derailleur above or below the axle
( 5 ) to the chain just below and in front of the derailleur
( 6 ) to the chain below the front sprocket
( 7 ) to the lower pedal below the front sprocket

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Spoiler

I think it would depend on what gear you're in.

If you're in an extremely low gear like for hill climbing, the pedals would rotate faster than the tires. (Maybe, I haven't tested this out experimentally to see if that's really true for very low gears.) So if the bike were to roll forward then a pedal at the bottom of its arc would be moving backward (as observed by a stationary person away from the bike). In that case, pulling the pedal backward would roll the bike forward.

If you're in a high gear then a tire would rotate faster than the pedal would. So if the bike is moving forward then the pedal would also be moving forward (as observed by a stationary person away from the bike), even though a rider would feel like they're pushing the pedal backward from their frame of reference. That means that the only way the pedal could move backward is if the entire bike is rolling backward. So if you pull the pedal back while it's in high gear, the bike would roll backward.

If you're at a gear setting where the gear ratio is equal to the ratio of the pedal's radius to the tire's radius, then you're at the boundary between those regimes; a pedal at the bottom of its arc would have no net movement whether the bike is going forward or backward because the bike would be moving forward at the same speed that the pedal appears to be getting pushed backward from the frame of reference of a rider. In that case, I guess if you pulled hard enough then the bike would have to just skid backward without actually rolling.

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On 6/8/2016 at 6:54 AM, plasmid said:

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I think it would depend on what gear you're in.

If you're in an extremely low gear like for hill climbing, the pedals would rotate faster than the tires. (Maybe, I haven't tested this out experimentally to see if that's really true for very low gears.) So if the bike were to roll forward then a pedal at the bottom of its arc would be moving backward (as observed by a stationary person away from the bike). In that case, pulling the pedal backward would roll the bike forward.

If you're in a high gear then a tire would rotate faster than the pedal would. So if the bike is moving forward then the pedal would also be moving forward (as observed by a stationary person away from the bike), even though a rider would feel like they're pushing the pedal backward from their frame of reference. That means that the only way the pedal could move backward is if the entire bike is rolling backward. So if you pull the pedal back while it's in high gear, the bike would roll backward.

If you're at a gear setting where the gear ratio is equal to the ratio of the pedal's radius to the tire's radius, then you're at the boundary between those regimes; a pedal at the bottom of its arc would have no net movement whether the bike is going forward or backward because the bike would be moving forward at the same speed that the pedal appears to be getting pushed backward from the frame of reference of a rider. In that case, I guess if you pulled hard enough then the bike would have to just skid backward without actually rolling.

Spoiler

That answer applies if you're standing on the ground behind the bike. But the OP doesn't specify where you are while you're pulling the string. If you're sitting on the bike while you're pulling the string backward, then the bike would roll forward regardless of which gear it's in because the net effect would be the same as pedaling.

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