M028 · Rope constraints and work

Pulley Network & Mechanical Advantage

Choose fixed, movable or four-strand compound reeving. Follow the actual rope length, individual sheave speeds and unequal strand tensions, then compare pull work, load energy and bearing loss.

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 $0\,\mathrm N$

PREDICT / SAMPLE / AUDIT

Where does the saved force go?

Physical forcesOne fixed force scale per trialComplete 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\,\%

Positive load motion is upward; positive input travel is downward. Count the supporting strands, not the sheaves. Mass includes the whole moving assembly. Inertia and bearing loss make the strand tensions unequal; a zero tension ends the taut-rope model.

The sensor acquires only through the selected cursor time. Noise is bounded and reproducible; editing creates a fresh trial. A quadratic position fit needs three samples and assumes constant acceleration. Rope weight and viscous drag can violate that assumption; inspect the fit residual and instantaneous acceleration. CSV exports the acquired positions and their fit, separately from model truth.

Three defined planar topologies with a massless, taut, inextensible rope and no rim slip. Moving assembly mass includes translating sheaves; every sheave has the selected inertia and viscous axle coefficient. Radius is fixed at 18 cm. The record ends before impact or at first zero tension; slack rope motion is omitted.

Start with the essentials

Focus question: Does adding more wheels always save more force?
One-sentence intuition: The supporting strand count sets ideal force and distance ratios. Actual wheels introduce different angular speeds, inertial loads and tension losses.

Core mathematical model

Rope length is the constraint

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

Load motion is upward positive and input motion is downward positive. Three predefined reeving geometries keep the drawn rope length constant.

Each sheave has a different speed

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

Number wheels from the anchored side toward the free end. In the compound rig, moving wheels have odd factors and fixed wheels have even factors, with opposite rotation senses.

Solve the complete assembly

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

Assembly mass includes translating sheaves. Each wheel has the same selected inertia and axle drag, independently of its schematic 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)

The sum of the supporting tensions accelerates the assembly. All tensions must stay nonnegative; first zero tension ends the taut-rope model before it becomes invalid.

Saved force is not free work

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

Kinetic energy includes translational and rotational motion. The ideal quasistatic force is weight divided by strand count; acceleration and loss change it.

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.

Dynamic force is not static weight

Typical misconceptionInput force times strand count always equals weight.

Better mental modelA moving assembly can accelerate and spin wheels; those energy and force terms matter.

A rope cannot push

Typical misconceptionThe same equation remains valid at negative tension.

Better mental modelThe model stops at the first slack boundary instead of applying a compressive rope force.

Run the experiment

01
Redirect a force
Use the fixed-wheel preset and compare load and input travel.
What to observe: The direction changes but the travel magnitudes agree.
02
Exchange force for distance
Compare the ideal two- and four-strand presets.
What to observe: They have the same load drive, while the four-strand input force is halved and travel doubled.
03
Audit each wheel
Choose the lossy compound rig and complete the record.
What to observe: Wheel speeds and strand tensions differ; bearing work closes the energy ledger.
04
Find a constraint boundary
Reduce the assembly mass and increase pull and axle drag.
What to observe: A travel limit or first zero tension ends the record; no invalid negative tension is simulated.