Post on 30-Dec-2015
description
Rock Climbing and Differential Equations: The Fall-Factor
Dr. Dan Curtis
Central Washington University
Based on my article:
“Taking a Whipper : The Fall-FactorConcept in Rock-Climbing”
The College Mathematics Journal,v.36, no.2, March, 2005, pp. 135-140.
Climbers use ropes and protection devices placed in the rock in order to minimize the consequences of a fall.
• Intuition says:
The force exerted on the climber by the rope to stop a long fall would be greater than for a short fall.
• According to the lore of climbing, this need not be so.
belayer
climber
protection point
belayer
climber
protection point
belayer
climber
protection point
L = un-stretched length of rope between climber and belayer.
DF
DT
The Fall-Factor: DT / L
Climbing folklore says: The maximum force exerted by the rope on the climber is not a function of the distance fallen, but rather, depends on the fall-factor.
Fall-factor about 2/3
Fall-factor 2
belay point
0
x
DF
DT
position at start of fall
position at end of free-fall
position at end of fall
2
2
d xm mgdt
2
2
d x dvv
dt dx
dvv gdx
21
2v gx C
During free-fall
0v when 0x so 0C 2 2v gx
Fx DWhen 2F Fv gD
After the rope becomes taut, the differential equation changes, since the rope is now exerting a force.
( ) 2
F
F F
dv kv g x Ddx mL
v D gD
2 22 ( )Fk
v gx x DmL
The solution is
Maximum force felt by the climber occurs when
Tx D and 0v
20 2 ( )T T F
kgD D D
mL
max 2 TDF mgkL
2 2( )T F T TD D k mgLD mgkDk
L L k L
The maximum force is given by