How Does Anchor Angle Affect Load Distribution in Climbing?

[rambo5330]

So I am a climber and I also love physics. I only have first year physics under my belt, and I do not know the concept behind the following question I often think about while setting up a climbing anchor.

Attached is a diagram of two 2-point anchor systems. the red lines building the two sides of the anchor can be thought of as two lengths of rope, and 3rd length of rope attached to the vertex of the anchor goes to the climber (load). In situation (a) when the anchor is less than 120 degrees the load is shared between the two bolts, each seeing less than 1 x the load.. i.e if a 150 lb person is attached to the anchor each bolt will see maybe 75 pounds. therefore making it advantagous to use the 2-point system to reduce overall load on individual bolts.
In situation (b), However; when the angle is greater than 120 degrees the force is actually multiplied at each bolt. so if you have a 150 lb climber (load) attached to the anchor, each bolt will actually see around 200+ lb, makeing it extremely dangerious to use this system.

Why does the force multiply like that and decrease so much when the angle is decreased? with my first year physics all I know how to do is use the angles and the weight of the load to determine the tension on each leg of the anchor and hence the force on each bolt...the numbers work but i don't know WHY they are increaseing and decreaseing as the angle changes? is it acting as a sort of lever causing torque on the bolt.. can someone please explain this?

 

[Valerie Park]

The reason why the force multiplies as the angle increases is because of the trigonometric law of sines. This law states that the ratio of the sides of a triangle are related to the angles by the following equation: a/sin(A) = b/sin(B) = c/sin(C) In your example, if the angle of the anchor is greater than 120 degrees, then the ratio of the sides of the triangle (lengths of rope) increases, which in turn increases the tension in each side. This means that each bolt will see an increased load, up to double the initial load. On the other hand, when the angle of the anchor is less than 120 degrees, the ratio of the sides of the triangle decreases, which reduces the tension in each side. This means that each bolt will now see less than the original load. To summarize, the force on each bolt is directly related to the angles of the anchor and the lengths of the ropes. When the angle increases, the force on each bolt also increases, and when the angle decreases, the force on each bolt also decreases.