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The capstan equation [1] or belt friction equation, also known as Euler–Eytelwein formula [2] (after Leonhard Euler and Johann Albert Eytelwein), [3] relates the hold-force to the load-force if a flexible line is wound around a cylinder (a bollard, a winch or a capstan).
The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them: [11] Diagram of two masses attracting one another = where F is the force between the masses; G is the Newtonian constant of gravitation (6.674 × 10 −11 m 3 ⋅kg −1 ⋅s −2);
The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object. [ 3 ] A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances .
This does cause frictional shear stresses in the contact area. In the final situation the bollard exercises a friction force on the rope such that a static situation occurs. The tension distribution in the rope in this final situation is described by the capstan equation, with solution:
The speed values in these tables are produced from the formula using an "average" coefficient of friction (μ) of 0.7, and a perception-reaction time of 1.5 seconds. Speed values specific to a given circumstance can be obtained with the same formula using the appropriate reference constants specific to the circumstance.
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