Exploring Mechanics / 5. Simple Machines
Mechanical advantage and the principle that ideal machines trade force for distance.
Make a prediction first: a machine lets you lift a 200 kg load using only 40 kg worth of push. Does that mean the machine created extra energy from nowhere?
Input work = output work, for an ideal (frictionless) machine.
No machine creates energy. A machine with mechanical advantage MA lets you push with 1/MA of the force the load actually needs — but only by having your effort move MA times farther than the load does. Multiply force by distance on both sides and the two products (the input and output work) always come out equal, in the ideal case. This is the "no free lunch" of mechanics: every simple machine in this section is really just a different way of packaging that same trade.
Starting from energy conservation with no losses: Findin = Foutdout. Rearranged, Fout/Fin = din/dout — mechanical advantage (the force ratio) and velocity ratio (the distance/speed ratio) are the exact same number, just measured two different ways. That's why, in every widget in this section, computing one automatically tells you the other.
Set a mechanical advantage and an effort force, then watch the required effort distance and the resulting load distance move in exactly opposite proportion, keeping the work bars equal.
Real machines lose some input work to friction and heat. To deliver the same useful output work, a less efficient machine demands more input than the ideal case — mechanical advantage doesn't disappear, but you stop getting it for free.
Mechanical advantage compares force in to force out; efficiency compares work out to work in. A machine can have a huge mechanical advantage and still be quite inefficient (a rusty pulley system), or a small mechanical advantage and be nearly perfectly efficient (a smooth, direct lever). The two numbers describe different things and neither one determines the other.
Make a prediction first: a crowbar, a wheelbarrow, and your own forearm are all levers, but they don't feel similar to use at all — a wheelbarrow multiplies your lifting force, while your forearm curling a dumbbell clearly does not. Are these really the same kind of machine?
τ = F·r (torque = force × lever arm) · Balance: τeffort = τload
All three levers in the prediction above obey the exact same balance rule; what differs is only where the fulcrum sits relative to the effort and the load. That single arrangement choice is what decides whether a lever multiplies force (like the wheelbarrow) or sacrifices force for speed and reach (like your forearm).
Switch between the three lever classes and watch the fulcrum jump to a different spot on the beam — that's the entire difference between them.
A third-class lever always has reffort < rload, so mechanical advantage reffort/rload is always less than 1 — it takes more force than the load itself to operate. What you get in exchange is exactly what the "no free lunch" rule promises: the load end moves through a much larger distance (and speed) than the effort end. Your forearm muscle contracts only a little, but your hand sweeps through a wide, fast arc — precisely why arms and legs are built as third-class levers, trading force for speed and reach.
Drag either person along the seesaw. It balances only when both torques exactly match — not when both people have the same weight, and not when they sit at the same distance.
A stuck bolt needs a fixed amount of torque to turn, no matter what tool you use. A longer handle lets you supply that same torque with dramatically less force.
Make a prediction first: a pulley bolted to the ceiling lets you pull a rope down to lift a load up. Does that single pulley make the load any easier to lift, or does it only change which direction you pull?
MA = number of rope segments directly supporting the load · Effort force = load / MA · Rope pulled = MA × load's rise
The prediction above resolves right here: a single fixed pulley changes direction only. Attach that pulley to the load instead — a movable pulley — and two rope segments now share the load, cutting the needed force in half (at the cost of pulling twice the rope).
Add more rope segments between a fixed block and a movable block and the mechanical advantage climbs directly with the segment count — this is exactly how cranes and sailing-ship rigging lift enormous loads with modest hand force.
Every rope segment supporting the movable block shortens by the same amount when you pull — that's the "rope-length constraint," and it's the real reason pulling in n times the rope only raises the load 1/n as far: the total rope is fixed, so it has to be divided among all n supporting segments equally. The two-hanging-masses version of a pulley system (no mechanical advantage, just redirected tension) was already built as the Atwood Machine in Newton's Laws → Tension and Connected Objects — the version below revisits it here with the pulley itself front and center.
Two masses hang from opposite ends of a rope over a single fixed pulley — release them and watch the heavier side fall, dragging the lighter one up at exactly the same acceleration.
Pull the rope a full meter and the load only rises half a meter — trace input work against output work as you drag the slider and watch them stay exactly equal.
Make a prediction first: a long, gently sloped ramp and a short, steep one both get a heavy crate up to the same loading-dock height. Which ramp needs less pushing force — and does either one save you any actual work?
A ramp is a simple machine with MA = 1/sinθ (frictionless case) — the same "no free lunch" trade from Mechanical Advantage, with the ramp's length standing in for the effort distance. This tab covers ramps and wedges; screws and the wheel-and-axle — both built from the exact same wrapped or spinning inclined plane — get their own dedicated tab next.
Compare lifting a crate straight up against pushing it up a ramp to the same height. The friction-by-friction force breakdown for a single incline was already covered in Newton's Laws → Inclined Planes; here, the focus is purely the mechanical-advantage trade between the two methods.
Direct lift work: Wh. Ramp work (frictionless): (Wsinθ)×(h/sinθ) = Wh — identical. The gentler ramp needs less force but a proportionally longer push, exactly canceling out. Any force savings you actually measure on a real ramp beyond that trade is really friction working in your favor in an unusual way, or more often, friction working against you and costing extra — never a genuine reduction in total work.
A wedge is just two inclined planes back to back. Driving it straight down converts a modest downward force into a much larger sideways splitting force — the thinner the wedge angle, the bigger that multiplication.
As the half-angle φ shrinks toward zero, tanφ shrinks even faster (proportionally to φ itself, in radians, for small angles), so 1/tanφ grows without bound. That's why a thin axe blade splits wood so effectively compared to a blunt wedge — and also why an extremely thin blade requires an impractically long, precise taper to manufacture, a real engineering trade-off this idealized formula doesn't capture.
Make a prediction first: a car jack lets one person lift an entire car by turning a small handle over and over. Is the jack's screw doing something fundamentally different from the ramp in the last tab, or is it secretly the exact same trick in disguise?
A screw is an inclined plane wrapped around a cylinder; a wheel-and-axle is really a lever that spins in a full circle instead of swinging back and forth. Both still reduce to force × distance staying constant.
Turn the handle once and it sweeps out a full circle of circumference 2πr — while the screw itself only advances one pitch-length, p. "Unwrapped," that's exactly a ramp of length 2πr rising a height p, which is where the mechanical advantage formula below comes from directly.
Halving the pitch p without changing r doubles the mechanical advantage 2πr/p — the "ramp" the thread unwraps into gets twice as shallow for the same circumference. That's exactly why fine-threaded screws and bolts (many turns per inch) can clamp with enormous force from modest hand torque, at the cost of needing far more turns to travel any given distance.
A steering wheel, doorknob, and winch handle are all the same machine: a large wheel turning a much smaller axle multiplies the torque you apply into a much larger force (or, run the other way, turns a small effort at the rim into fast rotation at the axle).
Make a prediction first: you insert a third gear (an "idler") between two meshing gears, without changing either original gear's tooth count. Does the overall speed ratio between the first and last gear change?
ω₁N₁ = ω₂N₂ for any two meshing gears · Every mesh reverses rotation direction
Gear teeth must mesh at the same linear (tangential) speed where they touch, which forces ω₁N₁ = ω₂N₂ — a smaller gear spins proportionally faster than a larger one it's meshed with. Torque works the opposite way: the smaller, faster gear delivers proportionally less torque, same trade-off as every other simple machine in this section.
Toggle the idler gear in and out. The overall ratio between the first and last gear never changes — only the final direction of rotation does.
With the idler: ωB = ωANA/NB, then ωC = ωBNB/NC = (ωANA/NB)(NB/NC) = ωANA/NC — NB cancels algebraically no matter what value it has. But each mesh (A–B, then B–C) reverses direction once, for two reversals total — back to A's original direction. Remove the idler and mesh A directly with C: exactly one reversal, so C spins opposite to A, even though its speed is identical either way.
A bicycle's chainring-and-cog pair is a gear train connected by a chain instead of meshing teeth directly, but the same ratio math applies — and it's the whole reason low gears help on hills while high gears help on flats.
A small chainring-to-cog ratio multiplies your pedaling torque at the wheel (helpful for hills), but proportionally reduces wheel speed for the same cadence — you pedal the same effort, spin your legs the same speed, but travel slower per pedal stroke. As always, nothing here creates or destroys energy; it only reshapes the force/speed trade at the wheel.