Science Puzzle
Which Gear Gets You Up the Hill?
A cyclist reaches the bottom of a long, steep hill and has to pick a gear. In gear A the chain runs from a 50 tooth chainring at the pedals to an 11 tooth sprocket on the back wheel. In gear B it runs from a 34 tooth chainring to a 34 tooth sprocket.
Which gear makes the climb easier on the cyclist's legs, and what do they give up in exchange?
The Answer
Gear B makes every push of the pedals far easier. The price is speed: the cyclist has to turn the pedals many more times to cover the same distance.
Count the teeth. The chain moves the same number of teeth on both wheels, so one turn of a 50 tooth chainring drags 50 teeth past the back sprocket. With only 11 teeth on the back, the wheel goes round 50 ÷ 11, about 4.5 times. In gear B, 34 teeth on each wheel means one turn of the pedals turns the wheel exactly once. A typical road wheel is about 2.1 metres round, so gear A moves the bike about 9.5 metres per pedal turn and gear B only about 2.1 metres.
Lifting a rider and bike up the hill takes the same amount of work whichever gear is used: their weight times the height gained. Work is force multiplied by distance, so a gear can't reduce the work; it can only change how it's shared out. Gear B spreads the job over about 4.5 times as many pedal turns, so each push needs less than a quarter of the force. On a steep slope that's the difference between spinning steadily upwards and grinding to a halt.
On the flat it's the other way round. There's little climbing to do, so the big gear lets the rider turn a comfortable number of pedal strokes and still go fast. Every lever, pulley and gearbox makes the same trade.
The principle: Mechanical advantage. Gears trade force for distance: a low gear needs less force on each pedal stroke but more strokes to cover the same ground, and the total work done stays the same, apart from a little lost to friction.