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Sven Riwoldt
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Aufgaben/Loesung002.tex Normal file
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\section{Aufgabe 2}
Welche Gegenkraft $\vec F$ hebt die folgenden vier Einzelkr<6B>fte, die an einem Massepunkt angreifen, in der Wirkung
auf?\\
\hspace{10mm}$ \vec F_1 = \left( {200N,110N} \right)$\hspace{10mm}$\vec F_2 = \left( { - 10N,30N} \right)$
\hspace{10mm}$ \vec F_3 = \left( {40N,85N} \right)$\hspace{14mm}$\vec F_4 = \left( { - 30N, - 50N} \right)$\\ \\
Von welchem Betrag ist $\vec{F}$? Unter welchem Winkel greifen $\vec{F_1}$ und $\vec{F_2}$ den Massepunkt an?
\subsection{L<EFBFBD>sung}
$ \overrightarrow F = - \left( {\overrightarrow {F_1 } + \overrightarrow {F_2 } + \overrightarrow {F_3 } + \overrightarrow {F_4 } } \right)\\
= - \left( {200N - 10N + 40N - 30N,110N + 30N + 85N - 50N} \right)\\
= - \left( {200N,175N} \right) \\
= \underline{\underline {\left( { - 200N, - 175N} \right)}}$\\
$\underline{\underline {\overrightarrow F }} = \sqrt { - \left( {200} \right)^2 + \left( { - 175} \right)^2 } N =\sqrt {\left( {8 \cdot 25} \right)^2 + \left( {7 \cdot 25} \right)^2 } N =25\cdot \sqrt{113}N \entspricht \underline{\underline{265.7536453 N}} $\\
\\
Der gesuchte Winkel sei $\alpha \text{, } 0\leq\alpha\leq \pi$.
\\
Dann gilt: $\cos\left(\alpha \right)=\frac{{\left\langle {\overrightarrow {F_1 } ,\overrightarrow {F_2 } } \right\rangle }}{{\left| {\overrightarrow {F_1 } } \right| \cdot \left| {\overrightarrow {F_2 } } \right|}}$. ${\Huge \otimes}$
$ \left\langle {\vec F_1 ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over F} _2 } \right\rangle = \left\langle {\left( {200N,110N} \right),\left( { - 10N,30N} \right)} \right\rangle = - 2000N^2 + 3300N^2 = 1300N^2 $
$\left| {\vec F_1 } \right| = \sqrt {\left( {200} \right)^2 + \left( {110} \right)^2 } N \cong 228.2542442N$
$\left| {\vec F_2 } \right| = \sqrt {\left( { - 10} \right)^2 + \left( {30} \right)^2 } N \cong 31.6227766N$\\
Das Einsetzen in ${\Huge \otimes}$ ergibt:
$\cos \left( \alpha \right) \cong 0.1801044696 \Rightarrow \arccos \left( \alpha \right) \cong \underline{\underline{1.38970367}} \left( {{\rm 79}{\rm .62415508}^ \circ } \right) $