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In climbing, (specifically in lead climbing) using a dynamic rope, the fall factor is the ratio of the length a climber falls before his rope begins to stretch and the amount of rope avaiable to absorb the energy of the fall.

R={\frac {l}{r}}

where R is the fall factor, l is the length of the fall before the rope begins to stretch, and r is the amount of rope available to absorb the energy of the fall.

The significance of the fall factor is that, when a fall is arrested by a belayer, if the tension in the rope is assumed to develop according to Hooke's Law, then the maximum impact force of the fall depends only the climber's mass, the elasticity of the rope, and the fall factor. In particular, the maximum impact force does not depend on the length of the fall.

Lead Climbing

A fall factor of 2 is the maximum that should be possible in a lead climbing fall, since the length of an arrested fall cannnot exceed two times the length of the rope. Normally, a factor-2 fall can occur only when a lead climber who has placed no protection falls past the belayer (two times the distance of the rope length between them), or the anchor if the climber is solo climbing the route using a self-belay. As soon as the climber clips the rope into protection above the belay, the distance of the potential fall as a function of rope length is lessened, and the fall factor drops below 2.

A fall of 20 feet is much more severe (exerts more force on the climber and climbing equipment) if it occurs with 10 feet of rope out (i.e. the climber has placed no protection and falls from 10 feet above the belayer to 10 feet below--a factor 2 fall) than if it occurs 100 feet above the belayer (a fall factor of 0.2), in which case the stretch of the rope more effectively cushions the fall.

Via Ferrata

In falls occurring on a via ferrata, fall factors can be much higher. This is possible because the length of rope between harness and carabiner is short and fixed, while the distance the climber can fall depends on the gaps between anchor points of the safety cable.

Maximum Impact Force

When a fall is arrested by a static belay, if the tension in the rope is assumed to develop according to Hooke's Law, then the maximum impact force on the climber is

F_{1}=mg+{\sqrt {(mg)^{2}+2kmgR}}

where $F_{1}$ is the maximum impact force on the climber, m is the mass of the climber, g is gravitational acceleration, k is the rope's modulus of elasticity, and R is the fall factor.^[1]

If the rope's modulus of elasticity is unknown, it can be calculated as

k={\frac {U(U-1.568)}{2.791}}

where U is the UIAA impact force rating of the rope.^[1]

If the rope has been clipped into an anchor between the climber and the belayer, the maximum impact force on the belayer will be less than that on the climber because of friction between the rope and the anchor. Various experiments have shown that this frictional force is approximately equal to ${\tfrac {1}{3}}F_{1}$ . Therefore, the maximum impact force $F_{2}$ on the belayer will be

F_{2}={\frac {2}{3}}F_{1}

.

Thus the maximum impact force on the anchor $F_{3}$ will be

{\tfrac {}{}}F_{3}=F_{1}+F_{2}={\frac {5}{3}}F_{1}

.^[1]

References

^ ^a ^b ^c Goldstone, R. (2006, December 27). The Standard Equation for Impact Force. Retrieved April 17, 2009, from http://www.rockclimbing.com/cgi-bin/forum/gforum.cgi?do=post_attachment;postatt_id=746;

Busch, Wayne. "Climbing Physics - Understanding Fall Factors". Retrieved 2008-06-14. {{cite web}}: Cite has empty unknown parameter: |coauthors= (help)

[goldstone-1] Goldstone, R. (2006, December 27). The Standard Equation for Impact Force. Retrieved April 17, 2009, from http://www.rockclimbing.com/cgi-bin/forum/gforum.cgi?do=post_attachment;postatt_id=746;

[1]

@@ Line 38: / Line 38: @@
 :<math>\tfrac{}{}F_3 = F_1 + F_2 = \frac{5}{3}F_1</math>.<ref name=goldstone />
-== An Approximate Relationship Between Maximum Impact Force and Fall Factor ==
-The severity of a fall (the force generated in the system) is approximately proportional to the square root of the fall factor, so that a factor 2 fall is considerably more serious than a factor 1 fall.  This can be seen by noting that the maximal force can be estimated by
-:<math>E=mgl=\int_0^{\Delta} dr  F \approx \Delta  F_{\rm max} </math>
-where <math>\Delta</math> is the distance over which the fall is stopped. The distance <math>\Delta</math> can be estimated by stating that the relative expansion of the rope is proportional to the force <math>F_{\rm max}</math>
-:<math>\frac{\Delta}{r}\propto F_{\rm max}</math>.
-Solving this equation for <math>\Delta</math> and inserting it into the above expression one arrives at
-:<math>F_{\rm max}\propto \sqrt{l/r} = \sqrt{FF}</math>
 == References ==