Physics of Mountaineering

Climbing is a fascinating sport with very complex requirements to psyche and constitution. Aside from practicing this sport, there are several topics which are interesting from a theoretical point of view. 

For instance, using methods of physics we can model climbing ropes or the force a climber feels in taking a fall. The very interesting topic of grade systems in free climbing comes from psychophysics. In connection with multi-pitch routes two articles might be helpful: the climbing speed of teams as a function of the climbing grade, and a physical model of walking uphill.

We hope that the topics here may be useful to climbers, and perhaps even interesting to other people at the periphery of mountaineering and climbing.



Physics of Climbing Falls - Popular

Two simple formulas for climbing fall forces for static and dynamic belays: A new impact force formula for movable belayers is presented and compared with the standard impact force formula.

Climbing Fall with a Movable Belayer

The mechanics of a climbing fall with a belayer who can be lifted: In sport climbing, a common method of belaying is to use a static rope brake attached to the belayer’s harness, but the belayer can move freely. This paper investigates the dynamics of a climbing fall with such a belayer. The dynamics are nontrivial because of the belayer’s constraint to be always at or above his initial position. An exact solution for a linear elastic rope is presented. Compared to a fix-point belay, one obtains a considerable force reduction on the belay-chain. However, there is a trade-off of a longer stopping distance of both climber and belayer. In order to calculate the stopping distance, friction between rope and the top carabiner has been taken into account. This article is published in JOURNAL OF SPORTS ENGINEERING AND TECHNOLOGY.

Swinging on a Long Rope

About the helplessness of a freely suspended climber or the inability to excite a pendulum oscillation on a long rope: The situation of a freely suspended climber is considered who tries to excite a pendulum oscillation on a rope from a rest position. It turns out that this is almost impossible for rope lengths larger than approx. 3 meters. The main reason is the low excitation frequency of longer ropes and the slow increase in the swing amplitude which strongly depends on the rope length.



Physics of Climbing Ropes - Popular

The omnipresent impact force formula for a climbing rope: This work demonstrates the omnipresence of the known impact force formula. Although originally derived only for the straight fall with a linear elastic rope, it applies almost unchanged for many other, more complex models and falls.

Physics of Climbing Ropes, Part 1

The physics of a climbing rope: In this theoretical paper, the force-elongation behavior of a climbing rope in a heavy fall is investigated and compared with experiments.  The experiments show that the state-of-the-art viscoelastic description of a climbing rope with time-independent friction is not able to explain the rope tension as a function of time. A proper description has to take into account time-delayed friction, i.e. a transition from a low-friction regime to strong friction near the force maximum which leads to a fast relaxation of the rope into its equilibrium position. Furthermore, a climbing rope has to be described by a nonlinear tension with increased stiffness for large elongations in order to agree with experiments with varying fall masses. Finally, observed second-mode force oscillations are explained by a continuum description of the rope taking into account its mass.
This work is published in a more detailed version in Journal of SPORTS ENGINEERING AND TECHNOLOGY. Volume 231 Issue 2, June 2017 (first published June 16, 2016).

Physics of Climbing Ropes, Part 2

Impact forces, fall factors and rope drag: For all kinds of climbing situations, the impact force and the maximum elongation of a climbing rope are calculated by taking into account dry friction between the rope and the protection points. The climbing situations can include an arbitrary number and placement of protection points. In addition, we calculate the rope drag that a climber has to overcome in order to move forward.

Physics of Climbing Ropes, Part 3

Viscous and dry friction combined, rope control and experiments: The full equations of motion for a fall in a climbing rope are set up and solved when both internal viscous friction and external dry friction between the rope and one anchor point are taken into account. An essential part of the work is to discuss how the belayer can control the fall by adjusting the rope slip in the belay device. The theory can fully explain measurements of the maximum impact force, the force on the belayer and on the anchor point with and without rope control.

Physics of Climbing Ropes, Part 4

The fracture of a climbing rope: a phenomenological approach: For the fracture process of a climbing rope, two mechanisms are responsible: plastic deformation and local damage of the contact zone between rope and anchor. These mechanisms are described by two analytical models represented by nonlinear difference equations. The plastic deformation equation can be linked to a catastrophe-theoretical model. From the equation describing local damage accumulation, the Palmgren-Miner rule can be derived.  The used energy-based approach allows the combination of these models and thus the calculation of the number of falls to failure as a function of the ratio of fall energy/energy storage capacity. The behaviour of climbing ropes tested by subsequent UIAA falls can be quantitatively explained by these models.
The article is published in the Journal of SPORTS ENGINEERING AND TECHNOLOGY (first published in November 29, 2018).



Grade Systems for Free Climbing

Difficult Difficulties in Free Climbing: About Objective, Subjective, Maximum, Mandatory and Overall Climbing Grades (still in German): What is the relationship between the free climbing grade system which is a subjective perception and the underlying, not directly observable, objective difficulty? Do we need the French global grade system (IFAS), if the maximum and the mandatory difficulties of a climbing route are known?


Climbing Speed of Teams

Climbing speed of teams on alpine-style, multi-pitch routes (in German): We give a rule of thumb for the average climbing speed of two climber teams as a function of the average difficulty of the climb.



Mountain Hiking

About walking uphill: time required, energy consumption and the zigzag transition: A physical model for walking uphill is introduced. It is based on simple principles like the conservation of energy and a force dependent efficiency coefficient. Excellent agreement with experimental data was achieved.