# Fall factor

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In lead climbing using a dynamic rope, the fall factor (f) is the ratio of the height (h) a climber falls before the climber's rope begins to stretch and the rope length (L) available to absorb the energy of the fall,

## Contents

${\displaystyle f={\frac {h}{L}}.}$

It is the main factor determining the violence of the forces acting on the climber and the gear.

As a numerical example, consider a fall of 20 feet that 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). This fall produces far more force on the climber and the gear than if a similar 20 foot fall had occurred 100 feet above the belayer. In the latter case (a fall factor of 0.2), the rope acts like a bigger, longer rubber band, and its stretch more effectively cushions the fall.

## Sizes of fall factors

The smallest possible fall factor is zero. This occurs, for example, in top-rope a fall onto a rope with no slack. The rope stretches, so although h=0, there is a fall.

When climbing from the ground up, the maximum possible fall factor is 1, since any greater fall would mean that the climber hit the ground.

In multi-pitch climbing (and big wall climbing), or in any climb where a leader starts from a position on an exposed ledge well above the ground, a fall factor in lead climbing can be as high as 2. This 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 fall factor drops below 2.

In falls occurring on a via ferrata, fall factors can be much higher. This is possible because the length of rope between the harness and the carabiner is short and fixed, while the distance the climber can fall depends on the gaps between anchor points of the safety cable (i.e. the climber's lanyard will fall down the safety cable until it reaches an anchor point); to mitigate this, via ferrata climbers can use energy absorbers. [1]

## Derivation and impact force

The impact force is defined as the maximum tension in the rope when a climber falls. We first state an equation for this quantity and describe its interpretation, and then show its derivation and how it can be put into a more convenient form.

### Equation for the impact force and its interpretation

When modeling the rope as an undamped harmonic oscillator (HO) the impact force Fmax in the rope is given by:

${\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+2mghk}},}$

where mg is the climber's weight, h is the fall height and k is the spring constant of the portion of the rope that is in play.

We will see below that when varying the height of the fall while keeping the fall factor fixed, the quantity hk stays constant.

There are two factors of two involved in the interpretation of this equation. First, the maximum force on the top piece of protection is roughly 2Fmax, since the gear acts as a simple pulley. Second, it may seem strange that even when f=0, we have Fmax=2mg (so that the maximum force on the top piece is approximately 4mg). This is because a factor-zero fall is still a fall onto a slack rope. The average value of the tension over a full cycle of harmonic oscillation will be mg, so that the tension will cycle between 0 and 2mg.

### Derivation of the equation

Conservation of energy at rope's maximum elongation xmax gives

${\displaystyle mgh={\frac {1}{2}}kx_{max}^{2}-mgx_{max}\$ ;\ F_{max}=kx_{max}.}

The maximum force on the climber is Fmax-mg. It is convenient to express things in terms of the elastic modulus E = k L/q which is a property of the material that the rope is constructed from. Here L is the rope's length and q its cross-sectional area. Solution of the quadratic gives

${\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+2mgEqf}}.}$

Other than fixed properties of the system, this form of the equation shows that the impact force depends only on the fall factor.

Using the HO model to obtain the impact force of real climbing ropes as a function of fall height h and climber's weight mg, one must know the experimental value for E of a given rope. However, rope manufacturers give only the rope’s impact force F0 and its static and dynamic elongations that are measured under standard UIAA fall conditions: A fall height h0 of 2 × 2.3 m with an available rope length L0 = 2.6m leads to a fall factor f0 = h0/L0 = 1.77 and a fall velocity v0 = (2gh0)1/2 = 9.5 m/s at the end of falling the distance h0. The mass m0 used in the fall is 80 kg. Using these values to eliminate the unknown quantity E leads to an expression of the impact force as a function of arbitrary fall heights h, arbitrary fall factors f, and arbitrary gravity g of the form:

${\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+F_{0}(F_{0}-2m_{0}g_{0}){\frac {m}{m_{0}}}{\frac {g}{g_{0}}}{\frac {f}{f_{0}}}}}}$

Note that keeping g0 from the derivation of "Eq" based on UIAA test into the above Fmax formula assures that the transformation will continue to be valid for different gravity fields, as over a slope making less than 90 degrees with the horizontal. This simple undamped harmonic oscillator model of a rope, however, does not correctly describe the entire fall process of real ropes. Accurate measurements on the behaviour of a climbing rope during the entire fall can be explained if the undamped harmonic oscillator is complemented by a non-linear term up to the maximum impact force, and then, near the maximum force in the rope, internal friction in the rope is added that ensures the rapid relaxation of the rope to its rest position. [2]

### Effect of friction

When the rope is clipped into several carabiners between the climber and the belayer, an additional type of friction occurs, the so-called dry friction between the rope and particularly the last clipped carabiner. "Dry" friction (i.e., a frictional force that is velocity-independent) leads to an effective rope length smaller than the available length L and thus increases the impact force. [3]

## Related Research Articles

A carabiner or karabiner, often shortened to biner or to crab, colloquially known as (climbing) clip, is a specialized type of shackle, a metal loop with a spring-loaded gate used to quickly and reversibly connect components, most notably in safety-critical systems. The word is a shortened form of Karabinerhaken, a German phrase for a "carbine rifle hook" used by a carbine rifleman, or carabinier, to attach his carbine to a belt or bandolier.

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

A climbing harness is a device which allows a climber access to the safety of a rope. It is used in rock and ice climbing, abseiling, and lowering; this is in contrast to other activities requiring ropes for access or safety such as industrial rope work, construction, and rescue and recovery, which use safety harnesses instead.

A block and tackle or only tackle is a system of two or more pulleys with a rope or cable threaded between them, usually used to lift heavy loads.

Tree climbing is a recreational or functional activity consisting of ascending and moving around in the crowns of trees.

Glossary of climbing terms relates to rock climbing, mountaineering, and to ice climbing.

Rock-climbing equipment varies with the type of climbing undertaken. Bouldering needs the least equipment outside of shoes and chalk and optional crash pads. Sport climbing adds ropes, harnesses, belay devices, and quickdraws to clip into pre-drilled bolts. Traditional climbing adds the need for carrying a "rack" of temporary passive and active protection devices. Multi-pitch climbing adds devices to assist in ascending and descending fixed ropes. And finally aid climbing uses unique equipment.

Abseiling, also known as rappelling, is the controlled descent of a steep slope, such as a rock face, by moving down a rope. When abseiling, the person descending controls their own movement down the rope, in contrast to lowering off, in which the rope attached to the person descending is paid out by their belayer.

Belaying is a variety of techniques climbers use to create friction within a climbing system, particularly on a climbing rope, so that a falling climber does not fall very far. A climbing partner typically applies tension at the other end of the rope whenever the climber is not moving, and removes the tension from the rope whenever the climber needs more rope to continue climbing.

Rock climbing is a sport in which participants climb up, across, or down natural rock formations or indoor climbing walls. The goal is to reach the summit of a formation or the endpoint of a usually pre-defined route without falling. Rock climbing is a physically and mentally demanding sport, one that often tests a climber's strength, endurance, agility and balance along with mental control. Knowledge of proper climbing techniques and the use of specialized climbing equipment is crucial for the safe completion of routes.

The Munter hitch, also known as the Italian hitch, mezzo barcaiolo or the crossing hitch, is a simple adjustable knot, commonly used by climbers, cavers, and rescuers to control friction in a life-lining or belay system. To climbers, this hitch is also known as HMS, the abbreviation for the German term Halbmastwurfsicherung, meaning half clove hitch belay. This technique can be used with a special "pear-shaped" HMS locking carabiner, or any locking carabiner wide enough to take two turns of the rope.

Multi-pitch climbing is a type of climbing that typically takes place on routes that are more than a single rope length in height, and thus where the lead climber cannot complete the climb as a single pitch. Where the number of pitches exceeds 6–10, it can become big wall climbing, or where the pitches are in a mixed rock and ice mountain environment, it can become alpine climbing. Multi-pitch rock climbs can come in traditional, sport, and aid formats. Climbers have also free soloed multi-pitch routes.

An ascender is a device used for directly ascending a rope, or for facilitating protection with a fixed rope when climbing on very steep mountain terrain.

Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes. Not to be confused with friction, which is a dissipative force acting on a system. Friction can cause or be a factor of damping.

In rock climbing, an anchor can be any device or method for attaching a climber, a rope, or a load above or onto a climbing surface—typically rock, ice, steep dirt, or a building—either permanently or temporarily. The intention of an anchor is case-specific but is usually for fall protection, primarily fall arrest and fall restraint. Climbing anchors are also used for hoisting, holding static loads, or redirecting a rope.

A climbing rope is a rope that is used in climbing. It is a critical part of an extensive chain of protective equipment used by climbers to help prevent potentially fatal fall-related accidents.

A dynamic rope is a specially constructed, somewhat elastic rope used primarily in rock climbing, ice climbing, and mountaineering. This elasticity, or stretch, is the property that makes the rope dynamic—in contrast to a static rope that has only slight elongation under load. Greater elasticity allows a dynamic rope to more slowly absorb the energy of a sudden load, such from arresting a climber's fall, by reducing the peak force on the rope and thus the probability of the rope's catastrophic failure. A kernmantle rope is the most common type of dynamic rope now used. Since 1945, nylon has, because of its superior durability and strength, replaced all natural materials in climbing rope.

A belay device is a mechanical piece of climbing equipment used to control a rope during belaying. It is designed to improve belay safety for the climber by allowing the belayer to manage their duties with minimal physical effort. With the right belay device, a small, weak climber can easily arrest the fall of a much heavier partner. Belay devices act as a friction brake, so that when a climber falls with any slack in the rope, the fall is brought to a stop.

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

The figure 8 belay device is a piece of metal in the shape of an 8 with one large end and one small end.

## References

1. Davies, Carey (July 16, 2017). "Get into via ferrata: the gear". www.thebmc.co.uk. Retrieved 2019-02-16.
2. Leuthäusser, Ulrich (June 17, 2016). "The physics of a climbing rope under a heavy dynamic load". Journal of SPORTS ENGINEERING AND TECHNOLOGY. doi:10.1177/1754337116651184 . Retrieved 2016-06-29.
3. Leuthäusser, Ulrich (2011): "Physics of climbing ropes: impact forces, fall factors and rope drag" (PDF). Retrieved 2011-01-15.