Simple Machines Reference
All six classical simple machines, each with its mechanical-advantage formula and one worked example at realistic, everyday values — computed live from the same functions behind the calculators on this site, not a separately typed-up table.
Formula, worked example, and mechanical advantage
| Machine | Formula | Mechanical advantage | Input → result |
|---|---|---|---|
| Lever | MA = effort arm length ÷ load arm length | 4 | Effort force needed: 100 N Load lifted: 400 N |
| Pulley | MA = number of rope segments supporting the load | 4 | Ideal effort force: 200 N Load lifted: 800 N |
| Inclined Plane | MA = ramp length ÷ ramp height (= 1 ÷ sinθ) | 5 | Ideal force needed: 200 N Load moved: 1000 N |
| Wheel & Axle | MA = wheel radius ÷ axle radius | 5 | Effort force applied: 20 N Load force at shaft: 100 N |
| Gear Pair | Ratio = driven teeth ÷ driver teeth | 5 | Output speed: 300 rpm Output torque: 10 N·m |
| Screw | MA = (2π × lever arm) ÷ pitch | 274.89 | Actual effort force: 109.1348 N Load lifted: 9000 N |
Every row above comes from calling that machine’s real calculator function against one fixed example scenario — change the underlying calculator’s math and this table updates with it automatically.
Each machine, worked out in full
Lever
Formula: MA = effort arm length ÷ load arm length
Example: Wheelbarrow: 1.2m effort arm, 0.3m load arm, 400N load (~41kg of soil) — mechanical advantage 4.
What’s traded: Your hands travel 4× farther through the air than the load rises.
Pulley
Formula: MA = number of rope segments supporting the load
Example: 4-part block and tackle lifting an 800N load, ideal (100%) efficiency — mechanical advantage 4.
What’s traded: Pull 4m of rope through your hands for every 1m the load rises.
Inclined Plane
Formula: MA = ramp length ÷ ramp height (= 1 ÷ sinθ)
Example: Loading-dock ramp: 5m long, 1m rise, 1000N load, frictionless ideal — mechanical advantage 5.
What’s traded: Push the load 5m along the slope (at 11.54°) to raise it 1m — real friction always raises this force above the ideal shown here.
Wheel & Axle
Formula: MA = wheel radius ÷ axle radius
Example: Screwdriver: 15mm handle radius, 3mm shaft radius, 20N grip — mechanical advantage 5.
What’s traded: The handle sweeps 5× farther per turn than the shaft's own surface.
Gear Pair
Formula: Ratio = driven teeth ÷ driver teeth
Example: Garage-door-opener pair: 12-tooth driver, 60-tooth driven gear, 1500rpm / 2N·m in — mechanical advantage 5.
What’s traded: A 5:1 ratio divides speed by 5 while multiplying torque by 5 — traded, not multiplied for free.
Screw
Formula: MA = (2π × lever arm) ÷ pitch
Example: Car jack: 0.35m handle, 8mm pitch, 9000N load, 30% real-world efficiency — mechanical advantage 274.89.
What’s traded: The ideal effort is only 32.7404N — friction more than triples it, which is also what makes the jack self-locking.
Why the numbers look so different across machines
Every simple machine on this site runs on the same underlying idea: force traded for distance (or, for a gear pair, rotational speed traded for torque), with nothing multiplied or created for free. What changes from machine to machine is only the geometry that sets the exchange rate — an arm-length ratio for a lever, a rope-segment count for a pulley, a length-over-height ratio for a ramp, a radius ratio for a wheel and axle, a tooth-count ratio for a gear pair, and a circular-sweep-over-pitch ratio for a screw. That's why a screw’s mechanical advantage of a few hundred and a lever’s mechanical advantage of 4 aren’t really different in kind — they're the same formula shape, applied to inputs that happen to sit in very different numeric ranges.
The screw row also shows something the other rows mostly don’t need to: a large gap between the frictionless ideal effort and the realistic actual effort. That's not an inconsistency — it reflects how much more friction a continuously wedged screw thread experiences compared to a lever's brief pivot or a pulley's rolling wheel, a difference covered in more depth in the mechanical-advantage-versus-efficiency article linked below.
Frequently Asked Questions
Are these numbers computed the same way as the calculators above?
Yes — every number on this page comes from calling the exact same lever, pulley, inclined-plane, wheel-and-axle, gear-ratio, and screw functions the six calculators on this site use, run against one representative example for each machine. Nothing here is a separately maintained or hand-typed table.
Why does the screw's mechanical advantage look so much larger than the others?
A screw's ideal mechanical advantage is 2π times its lever-arm length divided by its pitch, and a small pitch divided into even a modest circular sweep produces a large number almost automatically — mechanical advantages in the hundreds are normal for an ordinary screw. It isn't a different kind of physics, just a formula whose typical inputs land in a different range.
Why do the lever, pulley, and wheel-and-axle rows show an "effort force" while the gear row shows a speed and a screw row shows an actual vs. ideal force?
Each machine trades force for a different kind of distance: a lever, pulley, and wheel-and-axle trade force for linear or rotational distance moved, so their headline numbers are forces. A gear pair trades rotational speed for torque, so its headline numbers are speed and torque. A screw's friction losses are large enough that showing both the frictionless ideal and the realistic actual effort side by side is the more honest comparison — see the tradeoff column for exactly how much friction costs on the jack example.
Can I use these example numbers as a rule of thumb for my own build?
Treat them as a demonstration of each formula's shape, not a substitute for your own measurements. Every one of these examples uses specific dimensions and loads chosen to be realistic and round; your own lever arm, rope count, ramp angle, gear teeth, or screw pitch will very likely differ, so run your own numbers through the matching calculator rather than assuming these figures transfer directly.
Educational estimates only, based on ideal (or, for the screw example, a realistic but still illustrative) mechanics. Verify load-bearing or safety-critical figures with a qualified engineer.
Use it with
- Lever Calculator, Pulley Calculator, Inclined Plane Calculator, Wheel & Axle Calculator, Gear Ratio Calculator, and Screw Calculator— run your own dimensions through whichever machine you're working with.
- Mechanical Advantage vs. Efficiency — why every real machine needs more effort than its ideal formula predicts, and why the gap is so much bigger for a screw than a lever.
- Designing a Simple Machine to Hit a Target Mechanical Advantage — working backward from a target mechanical advantage to real, buildable dimensions for each machine.