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,
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.
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 multipitch climbing, or in any climb that starts from a position such as an exposed ledge, 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 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.
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.
When modeling the rope as an undamped harmonic oscillator (HO) the impact force Fmax in the rope is given by:
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.
Conservation of energy at rope's maximum elongation xmax gives
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
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:
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.
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.
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:
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of a periodically applied force is equal or close to a natural frequency of the system on which it acts. When an oscillating force is applied at a resonant frequency of a dynamic system, the system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies.
Climbing protection is any of a variety of devices employed to reduce risk and protect others while climbing rock and ice. It includes such items as nylon webbing and metal nuts, cams, bolts, and pitons.
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.
This glossary of climbing terms is a list of definitions of terms and jargon related to rock climbing and mountaineering. The specific terms used can vary considerably between different English-speaking countries; many of the phrases described here are particular to the United States and the United Kingdom.
A wide range of equipment is used during rock or any other type of climbing that includes equipment commonly used to protect a climber against the consequences of a fall.
Abseiling, also known as rappelling, is a controlled descent off a vertical drop, such as a rock face, by descending a fixed rope.
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.
Lead climbing is a climbing style, predominantly used in rock climbing. In a roped party one climber has to take the lead while the other climbers follow. The lead climber wears a harness attached to a climbing rope, which in turn is connected to the other climbers below the lead climber. While ascending the route, the lead climber periodically connects the rope to protection equipment for safety in the event of a fall. This protection can consist of permanent bolts, to which the climber clips quickdraws, or removable protection such as nuts and cams. One of the climbers below the lead climber acts as a belayer. The belayer gives out rope while the lead climber ascends and also stops the rope when the lead climber falls or wants to rest.
Projectile motion is a form of motion experienced by an object or particle that is projected near the Earth's surface and moves along a curved path under the action of gravity only. This curved path was shown by Galileo to be a parabola, but may also be a line in the special case when it is thrown directly upwards. The study of such motions is called ballistics, and such a trajectory is a ballistic trajectory. The only force of mathematical significance that is actively exerted on the object is gravity, which acts downward, thus imparting to the object a downward acceleration towards the Earth’s center of mass. Because of the object's inertia, no external force is needed to maintain the horizontal velocity component of the object's motion. Taking other forces into account, such as aerodynamic drag or internal propulsion, requires additional analysis. A ballistic missile is a missile only guided during the relatively brief initial powered phase of flight, and whose remaining course is governed by the laws of classical mechanics.
Rock climbing is a sport in which participants climb up, down or across natural rock formations or artificial rock 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.
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.
In rock climbing, rope drag is the friction of the rope plus its weight that the climber feels when pulling a rope through a number of protection points, or over rock prominences. A large number of anchor placements, especially if they form a zig-zag rather than a straight line, can make the rope drag so bad that the climber can hardly move forward.
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 Eytelwein's formula, relates the hold-force to the load-force if a flexible line is wound around a cylinder.
Belt friction is a term describing the friction forces between a belt and a surface, such as a belt wrapped around a bollard. When a force applies a tension to one end of a belt or rope wrapped around a curved surface, the frictional force between the two surfaces increases with the amount of wrap about the curved surface, and only part of that force is transmitted to the other end of the belt or rope. Belt friction can be modeled by the Belt friction equation.
The classical probability density is the probability density function that represents the likelihood of finding a particle in the vicinity of a certain location subject to a potential energy in a classical mechanical system. These probability densities are helpful in gaining insight into the correspondence principle and making connections between the quantum system under study and the classical limit.