A climbers pendulum fall

We analyse a lead climber pendulum fall, a fall that can occur after the climber traverses horizontally away from their highest anchor. The model is based on a spring pendulum with a nonlinear component and incorporates air resistance. The resulting set of equations can be solved numerically using an Excel spreadsheet that is easily implemented by a student. The maximum tension and speed during the fall were calculated for a real dynamic climbing rope. The inclusion of rope elasticity can, under the right conditions, lead to a speed at the bottom of the swing that is less than that found for a stiff rope. The path of the climber is not closed as they swing back and forth, and we analyse radial oscillations to gain further insight. For multiple swings, the air resistance is important for removing energy from the system.