Resultant force

Calculate mathematically and graphically the resultant of three forces with a common center if:

F1 = 50 kN α1 = 30°
F2 = 40 kN α2 = 45°
F3 = 40 kN α3 = 25°

Correct answer:

F =  113.1225 kN
f =  65.613 °

Step-by-step explanation:

$$\begin{gathered} F_1=50\text{kN} \\ {F}_2=40\text{kN} \\ {F}_3=40\text{kN} \\ {a}_1=30{}^{\circ } \\ {a}_2=a_1+45=30+45=75{}^{\circ } \\ {a}_3=a_2+25=75+25=100{}^{\circ } \\ x=F_1\cdot \cos a_1+F_2\cdot \cos a_2+F_3\cdot \cos a_3=F_1\cdot \cos 30°+F_2\cdot \cos 75°+F_3\cdot \cos 100°=50\cdot \cos 30°+40\cdot \cos 75°+40\cdot \cos 100°=50\cdot 0.866025+40\cdot 0.258819+40\cdot (-0.173648)=46.7081 \\ y=F_1\cdot \sin a_1+F_2\cdot \sin a_2+F_3\cdot \sin a_3=F_1\cdot \sin 30°+F_2\cdot \sin 75°+F_3\cdot \sin 100°=50\cdot \sin 30°+40\cdot \sin 75°+40\cdot \sin 100°=50\cdot 0.5+40\cdot 0.965926+40\cdot 0.984808=103.02934 \\ F=\sqrt{x^2+y^2}=\sqrt{46.7081^2+103.0293^2}=113.1225\text{kN} \end{gathered}$$

$f=\frac{180°}{\pi }\cdot \arctan (y/x)=\frac{180°}{\pi }\cdot \arctan (103.0293/46.7081)=65.613^{\circ }=65°36^{\prime }47"$

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