Simple Machines

The six simple machines

Machine	Principle	MA formula	Examples
Lever	Torque balance	MA = d_effort / d_load	Crowbar, see-saw, scissors
Pulley	Distribution of tension	MA = number of supporting segments	Block-and-tackle, flagpole
Wheel and axle	Concentric levers	MA = R_wheel / R_axle	Steering wheel, doorknob, bicycle
Inclined plane	Force decomposition	MA = length / height	Ramp, road switchbacks
Wedge	Force redirection	MA = length / thickness	Knife, axe, splitting maul
Screw	Inclined plane wrapped around cylinder	MA = circumference / pitch	Screws, jacks, presses

Mechanical advantage

Mechanical advantage (MA) is the ratio of output force to input force:

MA = F_load / F_effort

For an ideal machine (no friction), conservation of energy implies:

F_effort × d_effort = F_load × d_load

So MA also equals the ratio of distances:

MA = d_effort / d_load

For real machines, efficiency η = (work out) / (work in) < 100%. Actual MA = ideal MA × η.

Lever

A lever has a fulcrum (pivot), and force applied at distances d_effort and d_load from the fulcrum.

Class	Fulcrum position	MA range	Example
1st class	Between effort and load	Variable, can be >1 or <1	See-saw, crowbar, scissors
2nd class	At one end, load between fulcrum and effort	Always ≥ 1	Wheelbarrow, bottle opener
3rd class	At one end, effort between fulcrum and load	Always < 1	Tweezers, fishing rod, broom

Class 3 levers have MA < 1 — they require MORE effort but multiply DISTANCE/SPEED. Used when precision or speed matters more than force.

Pulley systems

System	Supporting ropes	MA (ideal)
Single fixed pulley	1 (rope just changes direction)	1
Single movable pulley	2	2
Block and tackle (2-pulley pair)	4	4
Block and tackle (3-pulley pair)	6	6
Compound block and tackle (4-pair)	8	8

Screws and inclined planes

A screw is an inclined plane wrapped around a cylinder. For a screw with pitch p (distance between threads) and radius r:

MA = 2π·r / p

A typical 6mm wood screw with 3mm pitch — MA = 2π × 3 / 3 = 6.3. With a 10cm screwdriver, mechanical advantage is 100 / 6 × 6.3 = 105 — applying 1 N force on the handle gives 105 N at the screw tip.

JavaScript — simple machine calculations

// Lever
function leverEffort(loadForce, loadDistance, effortDistance) {
  return loadForce * loadDistance / effortDistance;
}

// Lift 1000 N with crowbar: load arm 0.5 m, effort arm 2.0 m
console.log(`Effort needed: ${leverEffort(1000, 0.5, 2).toFixed(0)} N`); // 250

// Pulley system: ideal MA = number of supporting segments
function pulleyEffort(load, supportingSegments) {
  return load / supportingSegments;
}

console.log(`200 kg load, 4-segment pulley: ${pulleyEffort(2000, 4).toFixed(0)} N`); // 500

// Inclined plane
function inclineEffort(load, height, length) {
  // Frictionless: F_effort = load × sin(θ) = load × (height/length)
  return load * height / length;
}

console.log(`Lift 1000 N, ramp 1:5: ${inclineEffort(1000, 1, 5).toFixed(0)} N`); // 200

// Screw
function screwEffort(load, pitch, radius) {
  return load * pitch / (2 * Math.PI * radius);
}

console.log(`Push 1000 N with 3mm pitch screw, 5mm radius: ${screwEffort(1000, 0.003, 0.005).toFixed(2)} N`); // ~95

// Wheel and axle
function wheelAxleEffort(load, axleRadius, wheelRadius) {
  return load * axleRadius / wheelRadius;
}

console.log(`Lift 500 N with capstan, axle 5cm, wheel 50cm: ${wheelAxleEffort(500, 0.05, 0.50).toFixed(0)} N`); // 50

// Mechanical advantage
function mechanicalAdvantage(loadForce, effortForce) {
  return loadForce / effortForce;
}

// With friction efficiency
function actualMA(idealMA, efficiency) {
  return idealMA * efficiency;
}

console.log(`Old pulley, ideal MA=4, efficiency 60%: actual MA = ${actualMA(4, 0.6).toFixed(2)}`); // 2.4

// Total mechanical advantage of compound machines
function compoundMA(...mas) {
  return mas.reduce((p, m) => p * m, 1);
}

// E.g., a screw turned by a wrench: screw MA × lever MA
console.log(`Compound: screw MA=10 × lever MA=5 = ${compoundMA(10, 5)}`);

Where simple machines show up

• Hand tools. Hammers (lever), screwdrivers (lever + screw), wrenches (lever), pliers (lever), can openers (wheel-axle + lever).
• Construction. Cranes (compound pulley), levers, ramps, screws (jackhammers, presses).
• Vehicle parts. Steering wheel (wheel-axle), gear systems (compound levers), brakes (lever + hydraulic).
• Engineering. Most complex machinery is a combination of simple machines. Engines, presses, turbines all use simple-machine principles.
• Sports and human body. Joints function like levers — bicep flexes elbow (3rd class lever, MA < 1, fast motion). Foot levers (2nd class) for walking.
• Education. Foundation of mechanics teaching; intuitive way to understand work, energy, force, and conservation.
• Daily life. Bottle openers, doorknobs, scissors, staircase ramps, ramps for accessibility, vises, presses.

Common mistakes

• Thinking simple machines reduce work. They reduce FORCE, not work. Total work (input ≈ output, neglecting friction) is the same.
• Forgetting the trade-off. Higher MA means more distance (or more time) for the same load lift. There's no "free lunch."
• Confusing classes of levers. Different positions of fulcrum/load/effort. 1st class can be either MA > 1 or < 1. 2nd class always > 1. 3rd class always < 1.
• Counting pulley supporting ropes wrong. Count rope segments that support the load (not all visible rope). For block-and-tackle with 2 pulley pairs, 4 segments support load → MA = 4.
• Ignoring friction. Real machines have efficiency < 100%. Don't assume ideal MA in real applications; check empirically or use efficiency factor.
• Treating the screw or wedge as different from inclined plane. Both are wrapping/redirecting an inclined plane. Same physics.