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Calculate the permissible angle

This exercise shows how to calculate the permissible rope angle for a given load.

Task

A load of G = 10 kN has to be attached to two suspension points with ropes. The maximum permissible tensile force of the rope is 20 kN. What value must the angle α have so the rope doesn’t break?

weight load hold by two ropes
Weight load hold by two ropes

Solution

The following video is in german language.

Zulässigen Winkel berechnen - Technische Mechanik 1, Übung 4

Due to the symmetrical arrangement of the suspension, the look on one side is sufficient. It is a system of concurrent forces in a plane.

Rope forces for exercise 4 (only one side)
Rope forces for exercise 4 (only one side)

The vertical component of the rope force equals to the half of the load G:

\[\tag{1} {F_{\mathit{1y}}}=\frac{G}{2}\]

and the relationship to the rope force is

\[\tag{2} {F_1} \sin{\left( \alpha \right) }={F_{\mathit{1y}}}\]

\[\tag{3} F_1 = \frac{F_1y}{\sin\left( \alpha \right) }\]

The rope force shall not be greater than 20 kN, so the following applies

\[\tag{4} 20 kN \geq \frac{F_1y}{\sin\left( \alpha \right) }\]

\[\tag{5} \alpha \geq \arcsin \left( \frac{5 kN}{20 kN } \right) \]

\[\tag{6} \alpha \geq 14.48° \]

So the angle α must stay or be equal to 14.48°, or the rope will break. This was the way to calculate it for a symmetrical setup. We have another exercise, where an asymetrical layout is investigated, just take a look into the category Engineering Mechanics I in the menu at the side.

This was the exercise about how to calculate the permissible rope angle if you have a given load and maximum rope force. If you enjoyed it, please share it!