Eytelwein's formula

The capstan equation or belt friction equation, also known as Eytelwein's formula, relates the hold-force to the load-force if a flexible line is wound around a cylinder (a bollard, a winch or a capstan).

Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small holding force exerted on one side can carry a much larger loading force on the other side; this is the principle by which a capstan-type device operates.

A holding capstan is a ratchet device that can turn only in one direction; once a load is pulled into place in that direction, it can be held with a much smaller force. A powered capstan, also called a winch, rotates so that the applied tension is multiplied by the friction between rope and capstan. On a tall ship a holding capstan and a powered capstan are used in tandem so that a small force can be used to raise a heavy sail and then the rope can be easily removed from the powered capstan and tied off.

In rock climbing with so-called top-roping, a lighter person can hold (belay) a heavier person due to this effect.

The formula is

where ${\displaystyle T_{\text{load}}}$ is the applied tension on the line, ${\displaystyle T_{\text{hold}}}$ is the resulting force exerted at the other side of the capstan, ${\displaystyle \mu }$ is the coefficient of friction between the rope and capstan materials, and ${\displaystyle \phi }$ is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle ${\displaystyle \phi =2\pi \,}$).

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