Gears and levers

A friendly companion to the engineering tools: instead of designing a part, it shows why machines work. Every lesson is a drawing you drag — the numbers and force arrows update as you go — with a short explanation, a Try this challenge, and a box with the rule to remember. Each section ends with a short multiple-choice quiz that explains every answer.

The idea that runs through all of it: a machine trades force for distance. It can make a push smaller, but you push further — the work (force × distance) never goes up, and friction always takes a little away.

How to use it

Pick a section along the top: Levers, Pulleys, Wheel, slope, screw, Gears or Energy, then a lesson in the row of buttons beneath it.
Drag the coloured handles in the drawing (on a phone, with your finger): the orange hand is the effort, blue boxes are loads, the grey triangle is a fulcrum. Gears turn when you drag them round.
Use the sliders to change sizes, masses, tooth counts and friction; Play buttons animate pulling, lifting and turning.
Read the cards beside the drawing — effort, mechanical advantage, distances, speeds, torques — and try the challenge; it ticks itself off when you solve it, and Next challenge gives another.
Finish with the quiz.

What’s inside

Levers — first, second and third class (seesaw, crowbar, wheelbarrow, nutcracker, tweezers and the forearm), with a draggable fulcrum, load and effort. The tool shows the arms, both torques, the effort needed, the mechanical advantage and the push of the fulcrum; Lift and lower tilts the lever and traces how far each end moves. The Seesaw puzzle has five levels: drag (or tap) weights from the tray onto the seats until the moments balance.

Pulleys — a fixed pulley, a movable pulley, and a block and tackle with 2 to 6 supporting ropes. Drag the rope end or press Pull: the load rises 1/n as far as the rope comes through. Tick Include sheave friction to see each rope part carry a different tension and the actual mechanical advantage fall behind the ideal.

Wheel, slope, screw — the wheel and axle (MA = R/r), the inclined plane with optional friction and the force triangle (weight, normal, push, friction), the wedge, and a screw jack with the thread unrolled into its inclined plane: thread lead against handle radius, efficiency, and whether the jack holds the load when you let go.

Gears — two meshing gears you size by their teeth and turn by hand, with speed, torque and direction; an idler (ratio unchanged, direction flipped); a two-stage compound train with a target-ratio challenge; rack and pinion; a worm and wheel that you can try to back-drive; bevel gears turning a corner; a planetary set where you hold one member and drive another, with all six combinations and the Willis equation worked with your numbers; and a bicycle with chainring, cassette, wheel size, gear inches, development, gain ratio and speed.

Energy — where the work goes: work in, useful work out and heat, for typical machines and their efficiencies, plus a cheat sheet of every rule on the page.

The physics

Torque τ = F × d, with d the perpendicular distance from the pivot. A lever balances when load × load arm = effort × effort arm; MA = load ÷ effort = effort arm ÷ load arm.
Pulleys: the ideal MA is the number n of rope parts supporting the moving block, and the rope travels n times the lift. With a sheave efficiency η (tension out ÷ tension in), pulling down through the top block needs F = W(1 − η) / (η(1 − ηⁿ)).
Wheel and axle: MA = R / r. Inclined plane: MA = L / h = 1 / sin θ; with friction the push is W(sin θ + μ cos θ), and a block stays put on its own when μ ≥ tan θ. Wedge: MA = length ÷ thickness.
Screw jack (square thread, Shigley’s power-screw formulas): ideal MA = 2πR / l for lead l; raising torque T = (W dₘ / 2)(l + π μ dₘ) / (π dₘ − μ l), efficiency W l / (2π T), and the screw is self-locking when μ ≥ tan λ with tan λ = l / (π dₘ).
Gears: ratio = driven teeth ÷ driver teeth; speed divides and torque multiplies by it (ideal); each external mesh reverses the direction, so idlers flip the direction but cancel from the ratio. Compound stages multiply. Rack travel per turn = π m z. Worm ratio = wheel teeth ÷ starts, with lead angle tan λ = starts × m ÷ d₁, efficiency tan λ / tan(λ + φ) and self-locking when λ ≤ φ (tan φ = μ, pressure angle ignored). Bevel pitch cones: tan δ₁ = sin Σ / (z₂/z₁ + cos Σ).
Planetary: the Willis equation (ω_s − ω_c) / (ω_r − ω_c) = −z_r / z_s, or z_s ω_s + z_r ω_r = (z_s + z_r) ω_c, with z_r = z_s + 2 z_p. Planets are spaced at multiples of 2π / (z_s + z_r) so that they all mesh — unevenly if the teeth do not allow equal spacing.
Bicycle: development = (chainring ÷ sprocket) × wheel circumference; gear inches = wheel diameter in inches × ratio; gain ratio = (wheel radius ÷ crank length) × ratio (Sheldon Brown).

Assumptions: rigid parts, forces at right angles to the arms, g = 9.80665 m/s², and frictionless unless a friction slider says otherwise. The gear drawings use simple trapezoid teeth (the speeds are exact, the tooth shapes are not involutes — use the gears tool to design real ones). Efficiencies on the Energy tab are typical published ranges (Shigley’s Mechanical Engineering Design; Machinery’s Handbook; Spicer et al., "Effects of frictional loss on bicycle chain drive efficiency", 2001) and vary with load, speed, lubrication and wear.

Sources: R. Willis, Principles of Mechanism (1841); Budynas and Nisbett, Shigley’s Mechanical Engineering Design, ch. 8 (power screws) and ch. 13 (gears); Sheldon Brown, "Gain ratios" (sheldonbrown.com/gain.html).

Works with

How to use it

What’s inside

The physics