M028 · Rope constraints and work

Pulley Network & Mechanical Advantage

When a pulley system lets you pull with less force, where does the saved force go? Into distance. With more supporting strands you pull with less force, but you must pull proportionally more rope, so ideally the work stays the same. Real pulleys also have mass and bearing friction, so adding wheels can stop paying off.

Rig the rope over a fixed pulley, a movable pulley or a four-strand compound system. Follow the rope length, the speed of each pulley and the unequal tensions in the strands, then compare the work you put in with the energy the load gains and the energy lost in the bearings.

Load rise from release $0\,\mathrm m$

Strand 1 tension from anchor $0\,\mathrm N$

Strand 2 tension from anchor $0\,\mathrm N$

Strand 3 tension from anchor $0\,\mathrm N$

Strand 4 tension from anchor $0\,\mathrm N$

Strand 5 tension from anchor $0\,\mathrm N$

Recorded time $0\,\mathrm s$

Load coordinate, upward positive $0\,\mathrm m$

Signed load velocity $0\,\mathrm{m/s}$

Instantaneous load acceleration $0\,\mathrm{m/s^2}$

Generalized inertial mass $0\,\mathrm{kg}$

Translational kinetic energy $0\,\mathrm J$

Sheave rotation energy $0\,\mathrm J$

Rope kinetic energy $0\,\mathrm J$

Gravitational energy change $0\,\mathrm J$

Signed input travel $0\,\mathrm m$

Work at the free end $0\,\mathrm J$

Viscous bearing work $0\,\mathrm J$

Integrated energy residual $0\,\mathrm J$

Acquired positions $1$

Whole-record quadratic acceleration $\text{Need 3 samples}$

Quadratic position fit RMS $\text{Need 3 samples}$

Record end time $0\,\mathrm s$

First record boundary $\text{Travel boundary}$

Ideal support count $2$

Actual support force / input force $2$

Minimum strand tension $0\,\mathrm N$

Rope, motion and work

When a pulley system lets you pull with less force, where does the saved force go?

Physical forcesForce arrows keep one fixed scale for each trial.Complete moving assembly mass

m

2.00\,\mathrm{kg}

Downward pull at free end

F

12.00\,\mathrm N

Moment of inertia of each sheave

I

0.00\,\mathrm{kg\,m^2}

Viscous coefficient of each axle

B

0.00\,\mathrm{N\,m\,s}

Initial load coordinate

q_0

0.00\,\mathrm m

Position sensor sample period

\Delta t

0.10\,\mathrm s

Bounded position noise amplitude

\varepsilon

0.00\,\mathrm{mm}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

The load moves positive upward, and the input end travels positive downward. Count the strands that hold up the load, not the pulleys. The mass includes the whole moving assembly. Inertia and bearing losses make the strand tensions unequal; once any tension reaches zero, the taut-rope model stops applying.

The sensor records only up to the cursor time. The noise is bounded and reproducible, and editing any setting starts a fresh trial. The quadratic position fit needs three samples and assumes constant acceleration. Rope weight and viscous drag can break that assumption, so look at the fit residual and the instantaneous acceleration. The CSV holds the recorded positions and their fit, kept apart from the model's true values.

The Lab offers three fixed rope arrangements in a plane. The rope is massless, taut and cannot stretch, and it does not slip on the pulley rims. The mass of the moving assembly includes the pulleys that move with it, and every pulley has the chosen moment of inertia and viscous axle coefficient. The pulley radius is fixed at 18 cm. A frictionless guide keeps the moving carriage from pitching. The record ends before anything collides or when a strand tension first reaches zero, so a slack rope is not simulated.

Start with the essentials

Focus question: Does adding more wheels always save more force?
One-sentence intuition: The number of supporting strands sets the ideal force and distance ratios. Real pulleys spin at different speeds, take force to speed up and lower the rope tension a little at each wheel.

Core mathematical model

Fixed rope length links the motions

\Delta s_{\mathrm{in}}=n\Delta q,\qquad n\in\{1,2,4\}

Upward load motion and downward motion of the free end both count as positive. In all three rope arrangements, the rope length stays constant.

Each pulley turns at a different speed

|\omega_j|=\frac{j|\dot q|}{R},\qquad S=\sum_{j=1}^{n}j^2

Number the pulleys from the anchored end toward the free end. In the compound rig, the moving pulleys get odd numbers and the fixed pulleys even numbers, and the two sets turn in opposite directions.

Equation of motion for the whole assembly

\left(m+\frac{IS}{R^2}\right)\ddot q+\frac{BS}{R^2}\dot q=nF-mg

The assembly mass includes the pulleys that move with the load. Every pulley has the same chosen moment of inertia and axle drag, whatever size it looks in the drawing.

Trace tension back through the rope

T_n=F,\qquad T_{j-1}=T_j-\frac j{R^2}(I\ddot q+B\dot q)

Together, the supporting tensions accelerate the assembly. A rope can only pull, so every tension must stay at or above zero. The taut-rope model stops when the first tension reaches zero, before it would give wrong answers.

Less force does not mean less work

\Delta K+mg\Delta q=F\Delta s_{\mathrm{in}}-\int_0^t\frac{BS}{R^2}\dot q^2\,\mathrm dt

The kinetic energy includes both the moving load and the spinning pulleys. For a very slow pull with no losses, the force equals the weight divided by the number of strands; acceleration and losses change that.

Common difficulties

Count strands, not wheels

Typical misconceptionEvery added wheel doubles the advantage.

Better mental modelOnly strands pulling upward on the moving assembly contribute to its support.

Motion changes the force balance

Typical misconceptionInput force times strand count always equals weight.

Better mental modelWhen the assembly accelerates, part of the force goes into speeding it up and spinning the pulleys, so those terms cannot be ignored.

A rope cannot push

Typical misconceptionThe equations still work if a tension turns negative.

Better mental modelA negative tension would mean the rope pushes. The model stops the moment any strand goes slack instead.

Run the experiment

01
Redirect a force
Choose the Change direction preset and compare how far the load and the free end move.
What to observe: The direction of the force changes, but the load and the free end move the same distance.
02
Trade force for distance
Compare the Half the pull and Quarter the pull presets.
What to observe: Both rigs drive the load the same way, but with four strands you pull with half the force over twice the distance.
03
Look at each pulley
Choose Bearings cost work. Open Position sensor and more readings and press Complete record.
What to observe: The pulleys spin at different speeds and the strand tensions differ. The work you put in equals the energy the load and pulleys gain plus the energy lost in the bearings.
04
Find where the model stops
Lower the moving assembly mass, then raise the downward pull and the axle coefficient.
What to observe: The record ends when the load reaches its travel limit or a tension first hits zero. The Lab never simulates a negative tension.