M030 · Cable shape and friction

Capstan & Belt Friction

With a rope wrapped around a drum, how little force can hold a heavy load? Very little: the holding force can fall exponentially with friction coefficient times wrap angle, so a few turns can hold a heavy load. That minimum applies only when the rope is about to slip. A rope at rest can sit anywhere between the two slip limits.

Pull on one end of a rope wrapped around a fixed drum and find the range of pulls that keeps the load still. Probe the tension along the contact with noisy readings and see that a resting rope can use less friction than the material coefficient allows.

Probe coordinate $0$

Physical probe tension $0\,\mathrm N$

Contact arc length $0\,\mathrm m$

Model boundary $\text{Static}$

Acquired tension stations $0$

Fitted mobilized logarithmic slope $\text{Need 3 stations}$

Tension fit RMS residual $\text{Need 3 stations}$

Maximum larger / smaller tension ratio $1$

Minimum holding effort $0\,\mathrm N$

Maximum effort before reverse slip $0\,\mathrm N$

Required signed logarithmic slope $0$

Admissible-profile normal force per arc length $0\,\mathrm{N/m}$

Admissible-profile friction per arc length $0\,\mathrm{N/m}$

Required holding torque on the drum $0\,\mathrm{N\,m}$

Physical static power $0\,\mathrm W$

Measure the tension along a cable

With a rope wrapped around a drum, how little force can hold a heavy load?

Tension vectorsFixed scale per trialFixed drum radius

R

0.35\,\mathrm m

Contact wrap angle

\beta

180.00\,{}^\circ

Static friction coefficient

\mu_s

0.30\,

Effort end tension

F

30.00\,\mathrm N

Load end tension

W

60.00\,\mathrm N

Probe fraction along contact

\xi

0.50\,

Bounded relative tension noise

\varepsilon

0.00\,\%

Below the slip threshold, static friction does not fix a unique tension distribution. The teal curve is one smooth distribution that friction allows, with a constant logarithmic slope. The two dashed curves mark the limits. Outside them, the red curve shows only the distribution the settings would require, and no physical forces, measurements or static power are given. Slip speed and kinetic friction are not simulated.

The fitted logarithmic slope measures how much friction this chosen distribution actually uses, not the static friction coefficient of the material. The equal and opposite arrows at the cut are a diagnostic. Changing the radius changes the contact force per unit length and the holding torque, but the ideal limit on the tension ratio stays the same. The drum is fixed, and the rope is massless and flexible and lies flat in one plane.

The rope is massless, flexible and flat, and it rests on a fixed cylindrical drum. The Coulomb friction coefficient is the same everywhere, the rope wraps part way around the drum in one plane, and there is no V-groove to add grip. Pull stations set both end tensions. The tension profile shown is one possible static profile; static friction alone does not fix a unique one. Past either limit the model reports slip but does not simulate the motion that follows.

Start with the essentials

Focus question: Does a rope at rest always use the same amount of friction?
One-sentence intuition: Static friction has a maximum, and the rope uses all of it only when it is about to slip.

Core mathematical model

The friction limit on each short arc

\left|\frac{\mathrm d\ln T}{\mathrm d\alpha}\right|\leq\mu_s

The rope presses on each short arc of the drum, and that normal force sets the most friction the arc can supply. The angle is measured from the effort end toward the load.

The range of pulls that holds the load

W e^{-\mu_s\beta}\leq F\leq W e^{\mu_s\beta}

With too little effort the load side slips; with too much the rope would have to slip the other way. At either equality the rope is about to slip.

One possible static tension profile

T(\alpha)=F e^{k\alpha},\qquad k=\frac{\ln(W/F)}{\beta}

The Lab picks a profile whose logarithm has a constant slope. It is allowed whenever that slope stays within the friction limit, but static contact alone does not pick one profile over the others.

Radius changes the force per metre and the torque

n=\frac TR,\qquad |f|=\frac{|k|T}{R},\qquad \tau_{\mathrm{hold}}=R(W-F)

The normal and friction forces are spread along the contact, so they are measured in force per metre. A positive holding torque opposes the positive torque that the rope applies to the fixed drum. Nothing moves, so the power is zero.

A fit measures friction in use, not the coefficient

\ln T_i^{\mathrm{obs}}\approx b+\widehat k\alpha_i

The fitted slope approaches the static coefficient only when you know from elsewhere that the rope is about to slip. Below that point, the fit measures the friction actually in use in the chosen profile.

Common difficulties

Static friction is a limit, not a fixed value

Typical misconceptionA rope that holds must be using all the friction it can get.

Better mental modelThe limit only caps the friction. Many different tension profiles stay below it, and each of them holds the rope still.

Slip has two directions

Typical misconceptionFor a fixed load, pulling harder is always safer.

Better mental modelPast the upper limit the effort wins, and the rope slips the other way.

Friction in use is not the material coefficient

Typical misconceptionFitting the tension of a held rope measures the material's friction coefficient.

Better mental modelBelow the slip limit, the fit only gives the slope of the chosen static profile.

Run the experiment

01
Find the holding range
Choose Static hold and compare the effort with both limits.
What to observe: The chosen tension profile lies between the two dashed limit curves.
02
Cross each limit
Try the Insufficient effort and Reverse slip demand presets.
What to observe: The red curve is the balance you asked for, which friction cannot supply, so the Lab takes no readings.
03
Change only the radius
Keep the wrap angle, friction coefficient and end tensions fixed, and change the drum radius.
What to observe: The holding range stays the same, but the torque and the force per metre change.
04
Fit the tension slope
Go back to a held trial, add noise and scan the contact.
What to observe: The fitted slope comes close to the slope of the chosen profile, not to the material coefficient.