[tex]\displaystyle\\
\text{Descompunem fiecare forta in doua componente, dupa axele Ox si Oy.}\\\\
\text{Vom tine cont ca fortele sunt inclinate la 45 de grade }\\
\text{fata de axa Oxsau fata de axa Oy cu exceptia fortei }F_1.\\
\text{Gasim cate o forta in fiecare cadran.}\\\\
\text{Vom tine cont ca: } \sin 45^o = \cos 45^o = \frac{ \sqrt{2} }{2}\\\\
\text{la care vom adauga unul din semnele + sau - in functie de cadran.}
[/tex]
[tex]\displaystyle\\
F_1 = 3\sqrt{2}~~~F_{1x}= 3\sqrt{2}~~~F_{1y}= 0\\\\
F_2=4~~~F_{2x}= \frac{4\sqrt{2}}{2}~~~F_{2y}=\frac{4\sqrt{2}}{2}\\\\
F_3=5~~~F_{3x}= -\frac{5\sqrt{2}}{2}~~~F_{3y}=\frac{5\sqrt{2}}{2}\\\\
F_4=7~~~F_{4x}= -\frac{7\sqrt{2}}{2}~~~F_{4y}=-\frac{7\sqrt{2}}{2}\\\\
F_5=2~~~F_{5x}= \frac{2\sqrt{2}}{2}~~~F_{5y}=-\frac{2\sqrt{2}}{2}\\\\
[/tex]
[tex]\displaystyle\\
F_{1x}+F_{2x}+F_{3x}+F_{4x}+F_{5x} =3\sqrt{2}+\frac{4\sqrt{2}}{2}-\frac{5\sqrt{2}}{2}-\frac{7\sqrt{2}}{2}+\frac{2\sqrt{2}}{2}=\\\\
=\frac{6\sqrt{2} +4\sqrt{2} -5\sqrt{2}-7\sqrt{2}+2\sqrt{2}}{2}=\frac{(6+4+2-5-7)\sqrt{2}}{2}=0\\\\
F_{1y}+F_{2y}+F_{3y}+F_{4y}+F_{5y} =0+\frac{4\sqrt{2}}{2}+\frac{5\sqrt{2}}{2}-\frac{7\sqrt{2}}{2}-\frac{2\sqrt{2}}{2}=\\\\
=\frac{0+4\sqrt{2}+5\sqrt{2}-7\sqrt{2}-2\sqrt{2}}{2}=\frac{(4+5-7-2)\sqrt{2}}{2} =0
[/tex]
[tex]\Longrightarrow~~\text{Componentele x si y ale rezultantei sunt nule.}\\\\
\Longrightarrow~~ \overrightarrow{R} = 0\\\\
\Longrightarrow~~\texttt{Punctul material O este in echilibru.}[/tex]
