FAQ: Climbing Physics  Climbing Forces  The Pulley Effect
Basics
Whether top roping or lead climbing, all climbers should be familiar
with the force multiplying effects of the simple pulley. In the
climbing system, the rope running over a carabiner creates a situation
which acts as a pulley. For our examples, we are going to make some
convenient assumptions  that the pulley has no friction as it turns,
nor does it have any mass (it doesn't weigh anything). We'll also
assume the rope has no mass. This allows us to focus on the basic
forces involved.
A
Simple Fixed Pulley
A pulley is used to either change the direction of a force or gain
mechanical advantage. A single fixed pulley used to lift a weight
only changes the direction of the lifting force. You pull down to
lift the weight instead of pulling it upward. It takes 10 lbs. of
force pulling down to lift the 10 lb. weight on the other end of
the rope. If we pull down 5 feet of rope, the weight rises 5 feet.
Using the pulley doesn't reduce the amount of force needed to lift
the weight, though it's easier to pull down on a rope than to pull
it up as you can use your body weight as a counterbalance. There
is no mechanical advantage to be gained in this situation.
We don't think about the force on the anchor to which the pulley
is attached, but it is the focus of this exercise. The load on the
pulley is equal to the sum of the loads on the ropes on either side
of it. This remains true so long as the two ropes are essentially
parallel (See anchor forces
to learn what happens as the ropes are separated). If you do the
simple math, there is a 10 lb. weight pulling down on one side of
the pulley, and 10 lbs. of force on the other end of the rope to
hold the weight off the ground. There is a total of 20 lbs. of downward
force on the pulley and whatever the pulley is anchored to.
A Simple Moveable Pulley
Rearranging
the system may help make things more clear. In this case, we will
lift the 10 lb. weight by attaching the pulley to it. One end of
the rope will be anchored above the weight /pulley. We'll pull up
on the other end of the rope. As we do, the weight /pulley will
be lifted. In this case though, we will only have to exert 5 pounds
of force to lift the weight/pulley. As a trade off, we'll have to
pull in twice as much rope. If we pull up 5 feet of rope, the weight
will rise only 2 ½ feet. In this example it is easier to see
how the force on the pulley (10 lbs.) is equal o the sum of the
forces on the ropes on either side of it ( 5 lbs. + 5 lbs.).
In this situation we have not changed the direction of the force
used to lift the 10 lb. weight. We pull the rope up to make the
weight go up. We have gained a mechanical advantage of 2:1. This
system makes us feel twice as strong. In lifting the 10 lb. weight,
we only exert 5 lbs. of force. Of course, we do have to pull up
twice as much rope to get the weight to the top.
A Climbing Situation
Substitute
a climber for the 10 lb. weight, a carabiner for the pulley, and
a belayer on the rope on the opposite side of the pulley, and you
have a basic climbing situation. We want to know the forces on the
climber, anchor, and belayer when the climber falls to see how the
pulley effect works in a climbing fall.
In reality, the carabiner does not act as our perfect frictionless
pulley does. There is a good bit of friction when the rope makes
a tight bend over the metal carabiner. According to data from Petzl,
this carabiner  rope friction force constant is about 66% or 1.66.
The impact force on the falling climber end of the rope will be
greater than the force on the belayer end of the rope. This is because
as the rope is pulled and stretched towards the falling climber,
it must cross the carabiner. To do so it must overcome the friction.
We can calculate the impact force of the falling climber mathematically
( See Climbing Forces  Some
Hypothetical Instances ). Since we know how much force is generated
through friction, we can also calculate the force on the top carabiner
/ anchor:
Climber Impact Force x 1.66 = force on carabiner / anchor.
We
now know the force on the climber, and the force on the top anchor.
We can now figure out the force on the belayer. The load on a pulley
is equal to the sum of the loads on the ropes on either side of
it. To find out the load on the belayer, subtract the load on the
climber from the load on the top anchor carabiner.
Force on top carabiner / anchor  climber impact force = force
on belayer
Looking at a fall which generates 3 kN of force on the climber,
we can calculate the force on the top anchor carabiner (3 kN x 1.66)
as 4.98 kN. Knowing this, we can determine the amount of force the
belayer experiences as 1.98 kN (4.98 kN  3 kN).
It is important to recognize how the pulley affect works when we
build our climbing anchors. The force on the anchors is the sum
of the forces experienced by the climber AND the belayer. Even in
a low impact situation such as top roping, surprising large loads
can occur with even small amounts of slack. Beware when larger amounts
are slack needed for instance such as climbing out from under a
roof, or when the belayer is inattentive. Much larger forces can
occur in lead climbing situations.
A dynamic climbing rope must be designed to absorb enough energy
so that the maximum force on an 80 kg. (180 lb.) climber in a factor
2 fall is not more than 12 kn. (2698 lb.). The maximum force on
a top anchor could approach 19.9 kN or 4475 lbs.
To see what kinds of forces are generated in a variety of climbing
situations, go to Climbing
Forces  Some Hypothetical Instances.
Links
Fall Factor
and Climbing: Impact force calculator
Climbing
Forces in Leader Falls (.PDF file)
Forces
on the falling climber depending on different belaying techniques
Planet
Climbing Training  Advanced Belay Techniques
Climbing
Ropes
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