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Calculate the resulting tensile force

This exercise is about the following questions:

• How to calculate a component of a force?
• How to calculate a resulting force?

Two ropes are attached to an eyelet. Calculate the resulting tensile force on the eyelet. Does it act perpendicular?

given:
F1 = 8,2 kN
F2 = 9,7 kN
α = 50°
β = 60°

Solution

The following video is in german language.

The x-components of the two forces result to

$\tag{1} \sum F_x= F_1\cos(\alpha)+F_2\cos(\beta)$

$\tag{2} \sum F_x= 10.12 kN$

Assuming the transverse direction of the x-axis as the y-axis, a perpendicularly force would have to have a transverse component of zero. This is checked by

$\tag{3} \sum F_y= -F_1\sin(\alpha)+F_2\sin(\beta)$

$\tag{4} \sum F_y= 2.12 kN$

The transverse force is not zero, so the tensile force on the eyelet does not act perpendicular.

The resulting force F ist calculated from the x- and y-components as follows:

$\tag{5} F= \sqrt{F_x^2+F_y^2}$

$\tag{6} F = 10.34 kN$