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Determine missing force and angle with a given resultant force

This exercise is about how to determine a missing force and adresses the following questions:

  • How to calculate a missing force by a given resultant force?
  • How to determine the angle of a force?
  • How to change equations and solve unknown values (or how to solve a simple equation system)?


A ship is being pulled by three tugs. Tug 1 pulls with F1 = 7 kN, tug 2 with F2 = 6 kN. The force for tug 3 and its angle has to be calculated for a resultant force of 10 kN in x-direction.

tugs with pulling forces
Tugs with pulling forces


The following video is in german language.

Technische Mechanik 1, Übung 2 - Kraft und Winkel bei einer gegebenen Resultierenden ermitteln

The boundary conditions for the solution are: the sum of all forces in the x direction is 10 kN, the sum of all forces in the y direction is zero.

\[\tag{1} \sum F_x= 10 kN = F_1\cos(60°)+F_2\cos(45°)+F_3\cos(\alpha)\]

\[\tag{2} \sum F_y= 0 = F_1\sin(60°)-F_2\sin(45°)+F_3\sin(\alpha)\]

Equation 1 is solved for F3 and inserted into equation 2.

\[\tag{3} F_3= \frac{10 kN -- F_1\cos(60°) -- F_2\cos(45°)}{\cos(\alpha)}\]

\[\tag{4} 0 = F_1\sin(60°)-F_2\sin(45°)+\frac{10 kN -- F_1\cos(60°) -- F_2\cos(45°)}{\cos(\alpha)}\sin(\alpha) \]

Equation 4 is solved to

\[\tag{5} \alpha = \arctan\left(\frac{F_2\sin(45°)-F_1\sin(60°)}{10 kN -- F_1\cos(60°) -- F_2\cos(45°)}\right) \]

\[\tag{6} \alpha = -38.9° \]

and the missing force is

\[\tag{7} F_3 = 2.9 kN \]

Here we have some more exercises about resulting forces.