A lever, a rope over a pulley and a pair of gears all make the same trade. You push with less force over a longer distance, or with more force over a shorter one, and the work, force times distance, comes out the same. Each lab below computes that trade from the geometry.
A lever is a stiff bar that turns about a pivot called the fulcrum. The load is the weight you want to move and the effort is the push or pull you supply. What decides the outcome is torque: a force multiplied by its distance from the fulcrum. When the torques on the two sides match, the bar balances, so a small force far from the fulcrum holds a big weight close to it. Mechanical advantage is how many times the lever multiplies your force. Drag the fulcrum, the load or the hand along the bar, or pick an example.
Try this: pick Pry bar, press Lift, and compare the two dotted lines: the hand travels far while the load barely moves.
F is a force in newtons (N), d a distance from the fulcrum in meters, τ a torque in newton-meters (N·m). The load's force is its weight, mass × g with g = 9.81 m/s². When the bar turns by an angle θ (in radians), each point moves along an arc of length d·θ, so the far point moves farther by exactly the factor it saves in force.
The boast credited to Archimedes is that with a place to stand and a long enough lever he could move the Earth. Put a person on one end and Earth's 5.97 × 10²⁴ kg from the fulcrum, and pretend both sit in the same uniform gravity, as on a giant seesaw.
Balance needs d = MEarth × dEarth / mperson ≈ — on the person's side. And the trade still holds: to raise Earth by , the person would have to push their end through —.
A pulley is a wheel, called a sheave, that guides a rope. A fixed pulley hangs from the ceiling and only changes the direction of your pull. A movable pulley rides on the load. A block and tackle combines several of each. In an ideal rope the tension is the same everywhere, so if n strands hold up the moving block, each carries 1/n of the weight and so does your hand. The price is rope: to raise the load by some height, every one of those n strands has to shorten by that height, so you pull n times as much rope through your hands.
How to read this: the dim pulleys are fixed to the ceiling; the green ones move up with the load. Count the gold-numbered strands holding the green block: your pull is the weight divided by that number. The amber marks ride on the rope, so watch how much faster they move on your strand.
Try this: set Strands to 1 and press Lift, then set it to 6 and Lift again: same rise, six times the rope.
W is the load's weight in newtons and h is how high it rises. Only strands that pull up on the moving block count. The strand in your hand comes down from the top block, so it adds nothing. In the friction model each sheave passes on only a fraction e of the tension, so the strands nearest your hand carry more than the far ones, and the n sheaves together cost you extra pull.
A gear is a wheel with teeth. Gears that mesh have teeth of the same size, so a gear's radius is proportional to its number of teeth, and the two rims move at the same speed where they touch. The gear ratio is the driven gear's teeth divided by the driver's. A small gear driving a big one turns it slower by that ratio and with that many times more torque (turning force). A compound train puts two gears on one shaft, so the ratios of the stages multiply.
Try this: watch the line on the gold gear crawl while the amber one spins, then swap the Driver and Driven sizes to make it race.
T is a tooth count, ω a turning speed (rpm, or rad/s inside P), τ a torque in N·m and P the power in watts. Where two gears touch, the rims share one speed, ω₁r₁ = ω₂r₂, and push on each other with one force, so torque grows with radius as fast as speed shrinks. Each mesh reverses the direction of turning. Real gear meshes lose a percent or two each; this lab is ideal.
A bicycle uses a chain instead of meshing gears, so the chainring and cog turn the same way. The chainring at the pedals and the cog on the rear wheel move the same number of chain links, so the cog turns (chainring teeth ÷ cog teeth) times per pedal turn and the wheel turns with it. Cadence is how many times a minute you turn the pedals. Development is how far the bike rolls per pedal turn, and gear inches is the older way to say the same thing: the diameter of a direct-drive high-wheeler wheel that would go as far per pedal turn. Pick a combination or click a cell in the table. For why the bike stays upright at those speeds, see E01 · Why bikes stay up.
Cadence is in pedal turns per minute, N is a tooth count, C is the rolling circumference of the rear wheel in meters (v comes out in m/s), and D = C/π is the wheel's diameter. The wheel sizes here use typical rolling circumferences from cycle-computer tables; tire brand and pressure shift them by a percent or two.