Pulley Systems

Definition

A pulley is a grooved wheel mounted on an axle, with a rope or cable running over the groove. A pulley system uses one or more pulleys, threaded by a continuous rope, to lift or move a load. Pulleys are one of the six classical simple machines.

Pulleys come in two basic types:

• Fixed pulley. Anchored to a support (a ceiling beam, a tree branch, a crane top). Only redirects the rope; mechanical advantage = 1.
• Movable pulley. Attached to the load itself. Moves with the load when the rope is pulled. Mechanical advantage = 2 (in the basic configuration).

Combine multiple pulleys — fixed and movable — into a block-and-tackle (or compound pulley) for greater mechanical advantage.

How it works — the tension argument

In an ideal pulley (massless, frictionless), the tension in the rope is the same on both sides of every pulley. Here's the proof: apply Newton's second law to the pulley itself, which is massless. The net torque on it must be zero (otherwise infinite angular acceleration). So T_left·r = T_right·r, hence T_left = T_right.

This single fact — equal tension everywhere in an ideal rope — is all you need to analyze any pulley system. Just count the number of rope segments directly supporting the load. The mechanical advantage equals that count.

Three configurations

Single fixed pulley

One pulley anchored to a beam. The load hangs from one end of the rope, you pull on the other. Each side has tension T = W (the load weight). You pull with full load force, but you can pull downward instead of lifting upward — your weight can help, your arms work in a more comfortable position. MA = 1.

Single movable pulley

The load is hung from a pulley, which itself hangs from a single loop of rope. One end of the loop is fixed (anchored to the ceiling); the other end is pulled by you. The load is held by two rope segments, each carrying tension T. So 2T = W, giving T = W/2. You pull with half the load's weight — but the load only rises by half as much rope as you pull (because shortening both segments by Δd raises the load by Δd, while you pull 2·Δd of rope). MA = 2.

Compound (block-and-tackle)

Combine fixed and movable pulleys. Thread one continuous rope through all of them. The mechanical advantage equals the number of rope segments holding up the load (or equivalently, 2× the number of movable pulleys when configured optimally). A double block-and-tackle (2 movable, 2 fixed) gives MA = 4. A triple gives MA = 6. A quad gives MA = 8.

Configuration	Movable pulleys n	Mechanical advantage	Force to lift 100 kg
Single fixed	0	1	100 kg-force (981 N)
Single movable	1	2	50 kg-force (490 N)
Double block-and-tackle	2	4	25 kg-force (245 N)
Triple block-and-tackle	3	6	≈ 17 kg-force (163 N)
Quadruple block-and-tackle	4	8	≈ 12.5 kg-force (123 N)
Heavy crane (10:1)	5	10	10 kg-force (98 N)

Worked example — n = 4 movable pulleys

You design a compound pulley with 4 movable pulleys (and 4 corresponding fixed pulleys for the upper block). The load is 800 kg. Find the pull force, rope length needed to lift the load 2 m, and total work.

Step 1 — Mechanical advantage.

MA = 2n = 2·4 = 8

So the load is supported by 8 rope segments. Tension in the rope:

8·T = W = m·g = 800·9.8 = 7840 N
T = 7840 / 8 = 980 N (about 100 kg-force)

Step 2 — Rope to pull. To raise the load 2 m, each of the 8 supporting segments must shorten by 2 m. The total rope you pull through the system is:

d_pull = MA · d_lift = 8 · 2 = 16 m

Step 3 — Work. Verify conservation of energy:

W_in  = F_pull · d_pull = 980 · 16 = 15,680 J
W_out = W·d_lift = 7840 · 2 = 15,680 J  ✓

Answer. With n = 4 movable pulleys, MA = 8. You pull with 980 N — manageable — but must pull 16 meters of rope to lift the load 2 meters. Real pulleys lose about 5–20% to friction, so add 5–20% to the pull force (and the time spent pulling).

The rope-counting rule

The fastest way to find MA is to count rope segments touching the load (or the movable-pulley block):

• Single fixed: 1 segment up to the load (above), 1 down to your hand. MA = 1.
• Single movable: 2 segments up to the support, 1 down to your hand. MA = 2.
• 4-pulley (2 fixed, 2 movable): 4 segments supporting the load. MA = 4.

If you pull down (rope ends going down to your hand), the segment to your hand also counts. If you pull up (around an extra fixed pulley to reverse direction), it doesn't. So the same set of pulleys can give MA = 2n or 2n+1 depending on the threading.

JavaScript — pulley system analyzer

// Ideal pulley system: given load weight and number of supporting segments
function pulleyForce(load_kg, supportingSegments, g = 9.8) {
  const W = load_kg * g;
  const T = W / supportingSegments;
  return { tensionN: T, mechanicalAdvantage: supportingSegments };
}

console.log(pulleyForce(100, 1));   // 981 N — fixed pulley
console.log(pulleyForce(100, 2));   // 490 N — single movable
console.log(pulleyForce(100, 4));   // 245 N — block-and-tackle 2:2
console.log(pulleyForce(100, 6));   // 163 N — block-and-tackle 3:3
console.log(pulleyForce(800, 8));   // 980 N — heavy load, big MA

// How much rope to pull
function ropePulled(liftHeight, MA) {
  return liftHeight * MA;
}

console.log(ropePulled(2, 8));   // 16 m of rope for 2 m lift, MA=8
console.log(ropePulled(3, 4));   // 12 m of rope for 3 m lift, MA=4

// Account for efficiency
function realPullForce(load_kg, MA, efficiency = 0.85, g = 9.8) {
  const idealForce = (load_kg * g) / MA;
  return idealForce / efficiency;  // need more force to overcome friction
}

console.log(realPullForce(100, 4));         // ~288 N (vs 245 N ideal)
console.log(realPullForce(800, 8, 0.75));   // ~1307 N (vs 980 N ideal)

Applications

• Construction cranes. Tower cranes use compound pulleys with MA up to 10 or more, letting a small motor lift multi-ton steel beams.
• Elevators. Modern elevators use a counterweight roped to the cabin through a fixed pulley (1:1) or compounded (2:1) — the motor only has to lift the difference between cabin and counterweight, not the full weight.
• Sailing. Halyards (raising sails), sheets (controlling sail position), and tackle blocks all use pulleys. The term "block-and-tackle" comes from sailors' rigging.
• Mountain rescue. The "Z-drag" system uses 3:1 mechanical advantage to haul an injured climber up a cliff or out of a crevasse. A 5:1 or 9:1 can be improvised by stacking Z-drags.
• Garage doors. A counterweight balances the door (so it weighs near-zero in use), connected via a small pulley system. You pull a 70 kg door with about 3 kg of effort.
• Exercise machines. Cable-and-pulley machines redirect force and let users work muscles in ways free weights can't (e.g., overhead pulldowns redirect vertical weight to horizontal pull).
• Theatre rigging. Backdrops and lighting fixtures hang from compound pulley systems so stage hands can raise/lower hundreds of kilograms by hand.
• Bicycle/motorcycle/automobile gearing. Functionally equivalent — a different gear ratio is the gear-and-shaft analog of a pulley's MA.

Real-world losses

An ideal pulley has no mass, no friction, and a perfectly inextensible rope. Real pulleys deviate:

• Bearing friction. The pulley axle resists rotation. Every pulley costs a small fraction of the tension. For 6 pulleys in series, even 5% loss per pulley compounds to ~26% total loss.
• Rope stiffness. Bending a rope around a pulley costs energy (the "rope bending" loss). Thick steel cables on tower cranes lose 1–2% per wrap.
• Pulley mass. A heavy pulley has rotational inertia — it stores some kinetic energy when spinning, which must come from the input work.
• Rope stretch. Real ropes (synthetic, manila) stretch under tension. Part of your pull goes into stretching the rope, not lifting the load. Once stretched, you lift the load with reduced effective MA.

Total efficiency η = useful work out / total work in. Steel-on-steel ball-bearing pulleys can reach 95%+. Cheap plastic pulleys on nylon rope might be 60–70%. The mechanical advantage you compute from rope count is the ideal MA; the real MA is η times smaller.

Common mistakes

• Counting the wrong segments. Only segments directly attached to the moving load (or the movable pulley block) count toward MA. Segments running between fixed pulleys don't.
• Treating the fixed pulley as advantageous. It isn't, mechanically — it just redirects force. Still useful for ergonomics (pulling down with body weight vs. lifting up against gravity).
• Forgetting rope length increases proportionally. If MA = 8, you pull 8 m of rope to lift 1 m. Don't expect to save effort and time.
• Assuming massless rope. For tall lifts, the rope itself has weight — and that weight adds to your effective pull. Long mining cables can weigh more than the load.
• Ignoring friction losses. A 4-pulley system with 90% per-pulley efficiency delivers 0.9⁴ = 66% — not 100%. Real MA is significantly less than ideal MA.
• Drawing the force diagram wrong. The tension is the same in every segment of the same rope. Different ropes (or knot junctions) can have different tensions. Trace each rope separately.

Force and energy analysis

For an ideal pulley system, three quantities are conserved:

Tension  T  is the same in every segment of the same rope
Energy   W = F·d is unchanged (just redistributed)
Power    P  is unchanged if speed adjusts (slower pull, smaller force)

And the three trade-off relations are:

F_pull       = W_load / MA               (force is divided)
d_pull       = MA · d_lift               (distance is multiplied)
v_pull       = MA · v_lift               (speed is multiplied)

This is why pulleys are useful: human muscles produce a limited peak force (a strong person bench-presses about 1500 N) but nearly unlimited stamina. A pulley converts the peak-force-limited problem ("lift 5000 N") into a stamina-limited one ("pull 1000 N of rope for 5× longer").