# Fall factor

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.

$f={\frac {h}{L}}$ ## Impact force

The impact force is defined as the maximum tension in the rope when a climber falls. Using the common rope model of an undamped harmonic oscillator (HO) the impact force Fmax in the rope is given by:

$F_{max}=mg+{\sqrt {(mg)^{2}+2mghk}}=mg+{\sqrt {(mg)^{2}+2mgEqf}}$ where mg is the climber's weight, h is the fall height and k is the spring constant of the rope. Using the elastic modulus E = k L/q which is a material constant, the impact force depends only on the fall factor f, i.e. on the ratio h/L, the cross section q of the rope, and the climber’s weight. The more rope is available, the softer the rope becomes which is just compensating the higher fall energy. The maximum force on the climber is Fmax reduced by the climber’s weight mg. The above formula can be easily obtained by the law of conservation of energy at the time of maximum tension resp. maximum elongation xmax of the rope:

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

$F_{max}=mg+{\sqrt {(mg)^{2}+F_{0}(F_{0}-2m_{0}g_{0}){\frac {m}{m_{0}}}{\frac {g}{g_{0}}}{\frac {f}{f_{0}}}}}$ 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. 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 behavior 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 leads to an effective rope length smaller than the available length L and thus increases the impact force. Dry friction is also responsible for the rope drag a climber has to overcome in order to move forward. It can be expressed by an effective mass of the rope that the climber has to pull which is always larger than the rope mass itself. It depends exponentially on the sum of the angles of the direction changes the climber has made.