Belt Friction and the Capstan Equation Made Simple

Belt friction is the kind of friction produced from wrapping a belt-like object around a cylindrical object, such as a bollard. Knowledge of the forces at work in such situations are exploited in various fields from marine applications to rock climbing and mountaineering. The Capstan equation can be used to determine the maximum supportable load with the assistance of belt friction:

Capstan Equation

where Tload is the loaded side (the heavier side), Thold is the resistive side, e is the mathematical constant, μ is the coefficient of static friction, and φ is the total angle swept by all turns of a given rope in radians (one full turn is equal to 2π radians). In other words, φ represents the total contact length the rope makes with the cylinder in radians.

Capstan Equation DiagramThere are certain requirements that must be met in order for this equation to hold true:

  • Tload is the maximum load where if any further load is added, sliding will occur.
  • The rope being used must be perfectly flexible and not rigid in any way. Any rigidity will require energy to overcome, and will therefore render the formula inaccurate.
  • The rope must also be inelastic.

From the equation you can see that by either increasing the number of wraps or increasing the coefficient of friction between the two materials, the maximum supportable load will increase exponentially. It also shows that the diameter of the cylinder the rope is wrapped around has absolutely no effect on the supportable load.

However, it should be noted that not only can a cylinder with a larger diameter generally support greater load (provided it’s anchored sufficiently of course), but because many rope or cable materials are not actually perfectly flexible, having a larger diameter will increase the range of capstan equation-obeying belt materials due to them not having to bend as sharply.

It is often the case that the intention of utilizing this mechanical advantage is not to necessarily lock the load in a fixed position, but to have the ability to slowly release it in a controlled manner. Sail boats for example, utilize this principle heavily in the manual adjustment of sails due to the need for constant tension lest the wind overpower the sailor.

Rock climbing is another field that takes advantage of this phenomenon. A person at the top of a steep rock can easily hold the weight of even several people with the assistance of belt friction. As an example, with coefficient of friction μ being equal to 0.3, wrapping the rope around a given cylinder 3 times, you would be able to support 286 kilograms with 1 kilogram of weight-force.

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