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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.

Diagram of a simple fixed pulleyA 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

Diagram - basic climbing setup, shows carabiner acts as pulleySubstitute 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.

Diagram - shows forces on top anchor and belayer in a 3 kN climber fallWe 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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