Editing Fall factor (section)

== 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 ''F<sub>max</sub>'' in the rope is given by:

:<math>F_{max} = mg + \sqrt{(mg)^2 + 2mghk},</math>

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 2 involved in the interpretation of this equation. First, the maximum force on the top piece of protection is roughly 2''F<sub>max</sub>'', since the two sides of the rope around that piece both pull downward with force ''F<sub>max</sub>'' each.
Second, it may seem strange that even when ''h=0'', we have  ''F<sub>max</sub>''=2''mg'' (so that the maximum force on the top piece is approximately 4''mg''). This is because a factor-zero fall is the sudden weighting of a slack rope with weight ''mg''. The climber falls, picking up speed as the earth pulls them downward and, simultaneously, the rope stretches and pulls upward on them. The climber starts slowing down when the upward force generated by the stretching rope equals the gravitational force ''mg''. They then keep moving downward because of their momentum but now they are slowing down, not speeding up. Eventually they come to a stop and at that instant the rope is at maximum tension pulling upward on the climber by a force of ''2mg''. Because this upward force by the rope is more than the weight ''mg'' downward, the climber then yo-yo's up. The yo-yo'ing will eventually stop when the fall energy has all dissipated by frictional forces between (and within) the rope, the protection pieces and the harness.

===Derivation of the equation===

Conservation of energy at rope's maximum elongation ''x<sub>max</sub>'' gives

:<math> mgh = \frac{1}{2}kx_{max}^2 - mgx_{max}\ ;  \      F_{max} = k x_{max}. </math>

The maximum force on the climber is ''F<sub>max</sub>-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

:<math>F_{max} = mg + \sqrt{(mg)^2 + 2mgEqf}.</math>

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 ''F<sub>0</sub>'' and its static and dynamic elongations that are measured under standard [[UIAA]] fall conditions: A fall height ''h<sub>0</sub>'' of 2 × 2.3&nbsp;m with an available rope length ''L<sub>0</sub>'' = 2.6m leads to a fall factor ''f<sub>0</sub>'' = ''h<sub>0</sub>/L<sub>0</sub>'' = 1.77 and a fall velocity ''v<sub>0</sub>'' = (''2gh<sub>0</sub>'')<sup>1/2</sup> = 9.5&nbsp;m/s at the end of falling the distance ''h<sub>0</sub>''. The mass ''m<sub>0</sub>'' used in the fall is 80&nbsp;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:

:<math>F_{max} = mg + \sqrt{(mg)^2 + F_0(F_0-2m_0g_0)\frac{m}{m_0}\frac{g}{g_0}\frac{f}{f_0}} </math>

Note that keeping ''g''<sub>0</sub> from the derivation of "''Eq''" based on UIAA test into the above ''F<sub>max</sub>'' 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.<ref name=leuthaeusser>{{cite journal|url=http://www.sigmadewe.com/bergsportphysik.html?&L=1|title=The physics of a climbing rope under a heavy dynamic load|date= June 17, 2016|accessdate =2016-06-29|author=Leuthäusser, Ulrich|journal=Journal of SPORTS ENGINEERING AND TECHNOLOGY|volume=231 |issue=2 |pages=125–135 |doi=10.1177/1754337116651184|url-access=subscription}}</ref>

===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.<ref name=uleuthaeusser>Leuthäusser, Ulrich (2011):{{cite web|url=http://www.sigmadewe.com/fileadmin/user_upload/pdf-Dateien/Physics_of_climbing_ropes_Part_2.pdf?&L=1|title=Physics of climbing ropes: impact forces, fall factors and rope drag|accessdate =2011-01-15}}</ref>