\( \DeclareMathOperator{\abs}{abs}
\newcommand{\ensuremath}[1]{\mbox{$#1$}}
\)
Schwerpunkt und Gewichtskraft
1 Winkelprofil
1.1 a) Massen
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(%i3)
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A_1: 200^2; A_2: 800·300; A_ges: A_1+A_2; |
\[\tag{A\_ 1}40000\]
\[\tag{A\_ 2}240000\]
\[\tag{A\_ ges}280000\]
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(%i5)
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m_1: 10·A_1/A_ges;m_2: 10·A_2/A_ges; |
\[\tag{m\_ 1}\frac{10}{7}\]
\[\tag{m\_ 2}\frac{60}{7}\]
1.2 b) Schwerpunktlage
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(%i9)
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S_x: 1/10·(m_1·100+m_2·400); float(%); S_y: 1/10·(m_1·400+m_2·150); float(%); |
\[\tag{S\_ x}\frac{2500}{7}\]
\[\tag{%o7} 357.1428571428572\]
\[\tag{S\_ y}\frac{1300}{7}\]
\[\tag{%o9} 185.7142857142857\]
1.3 c) Kräfte
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(%i15)
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F_G: 10·981/100; beta: 20/360·2·%pi; F_H: F_G·sin(beta); float(%); F_N: F_G·cos(beta); float(%); |
\[\tag{F\_ G}\frac{981}{10}\]
\[\tag{beta}\frac{\ensuremath{\pi} }{9}\]
\[\tag{F\_ H}\frac{981 \sin{\left( \frac{\ensuremath{\pi} }{9}\right) }}{10}\]
\[\tag{%o13} 33.5521760602481\]
\[\tag{F\_ N}\frac{981 \cos{\left( \frac{\ensuremath{\pi} }{9}\right) }}{10}\]
\[\tag{%o15} 92.18384609909761\]
2 Schwerpunkt gebohrter Quader
2.1 a) Massen
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(%i18)
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A_1: %pi·80^2; A_2: 400·250; A_ges: A_2−A_1; |
\[\tag{A\_ 1}6400 \ensuremath{\pi} \]
\[\tag{A\_ 2}100000\]
\[\tag{A\_ ges}100000-6400 \ensuremath{\pi} \]
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(%i20)
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m_1:10·A_1/(A_2−A_1); float(%); |
\[\tag{m\_ 1}\frac{64000 \ensuremath{\pi} }{100000-6400 \ensuremath{\pi} }\]
\[\tag{%o20} 2.51661470815755\]
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(%i22)
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m_2:10·A_2/(A_2−A_1); float(%); |
\[\tag{m\_ 2}\frac{1000000}{100000-6400 \ensuremath{\pi} }\]
\[\tag{%o22} 12.51661470815755\]
2.2 b) Schwerpunktlage
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(%i24)
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S_x: 1/10·(m_2·200−m_1·100); float(%); |
\[\tag{S\_ x}\frac{\frac{200000000}{100000-6400 \ensuremath{\pi} }-\frac{6400000 \ensuremath{\pi} }{100000-6400 \ensuremath{\pi} }}{10}\]
\[\tag{%o24} 225.1661470815755\]
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(%i26)
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S_y: 1/10·(m_2·125−m_1·150); float(%); |
\[\tag{S\_ y}\frac{\frac{125000000}{100000-6400 \ensuremath{\pi} }-\frac{9600000 \ensuremath{\pi} }{100000-6400 \ensuremath{\pi} }}{10}\]
\[\tag{%o26} 118.7084632296061\]
3 Schwerpunkt und Gewichtskraft gebohrter Zylinder
3.1 a) Massen
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(%i29)
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A_1: %pi·50^2; A_2: %pi·15^2; A_ges: A_1−A_2; |
\[\tag{A\_ 1}2500 \ensuremath{\pi} \]
\[\tag{A\_ 2}225 \ensuremath{\pi} \]
\[\tag{A\_ ges}2275 \ensuremath{\pi} \]
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(%i33)
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m_1: 3·A_1/(A_1−A_2); float(%); m_2: 3·A_2/(A_1−A_2); float(%); |
\[\tag{m\_ 1}\frac{300}{91}\]
\[\tag{%o31} 3.296703296703297\]
\[\tag{m\_ 2}\frac{27}{91}\]
\[\tag{%o33} 0.2967032967032967\]
3.2 b) Schwerpunktlage
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(%i35)
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S_x: 1/3·(m_1·50−m_2·(50−15)); float(%); |
\[\tag{S\_ x}\frac{4685}{91}\]
\[\tag{%o35} 51.48351648351648\]
3.3 c) Kräfte
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(%i38)
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F_G: 3·981/100; float(%); beta: 20/360·2·%pi; |
\[\tag{F\_ G}\frac{2943}{100}\]
\[\tag{%o37} 29.43\]
\[\tag{beta}\frac{\ensuremath{\pi} }{9}\]
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(%i40)
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F_N:F_G·cos(beta); float(%); |
\[\tag{F\_ N}\frac{2943 \cos{\left( \frac{\ensuremath{\pi} }{9}\right) }}{100}\]
\[\tag{%o40} 27.65515382972929\]
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(%i42)
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F_H:F_G·sin(beta); float(%); |
\[\tag{F\_ H}\frac{2943 \sin{\left( \frac{\ensuremath{\pi} }{9}\right) }}{100}\]
\[\tag{%o42} 10.06565281807443\]
4 Signalarm
4.1 a) Massen
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(%i46)
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R_a: 1/2; R_i: 3/10; m_S: 12; m_Z: 20; |
\[\tag{R\_ a}\frac{1}{2}\]
\[\tag{R\_ i}\frac{3}{10}\]
\[\tag{m\_ S}12\]
\[\tag{m\_ Z}20\]
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(%i49)
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A_1: %pi·R_a^2; A_2: %pi·R_i^2; A_ges: A_1−A_2; |
\[\tag{A\_ 1}\frac{\ensuremath{\pi} }{4}\]
\[\tag{A\_ 2}\frac{9 \ensuremath{\pi} }{100}\]
\[\tag{A\_ ges}\frac{4 \ensuremath{\pi} }{25}\]
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(%i53)
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m_1: m_Z·A_1/A_ges; float(%); m_2: m_Z·A_2/A_ges; float(%); |
\[\tag{m\_ 1}\frac{125}{4}\]
\[\tag{%o51} 31.25\]
\[\tag{m\_ 2}\frac{45}{4}\]
\[\tag{%o53} 11.25\]
4.2 b) Schwerpunkte
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(%i55)
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beta: 30/360·2·%pi; l_s: 8/10; |
\[\tag{beta}\frac{\ensuremath{\pi} }{6}\]
\[\tag{l\_ s}\frac{4}{5}\]
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(%i59)
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l_sx:l_s/2·cos(beta); float(%); l_sy:l_s/2·sin(beta); float(%); |
\[\tag{l\_ sx}\frac{\sqrt{3}}{5}\]
\[\tag{%o57} 0.3464101615137755\]
\[\tag{l\_ sy}\frac{1}{5}\]
\[\tag{%o59} 0.2\]
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(%i63)
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l_rx:(l_s+R_a)·cos(beta); float(%); l_ry:(l_s+R_a)·sin(beta); float(%); |
\[\tag{l\_ rx}\frac{13 \sqrt{3}}{20}\]
\[\tag{%o61} 1.12583302491977\]
\[\tag{l\_ ry}\frac{13}{20}\]
\[\tag{%o63} 0.65\]
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(%i65)
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S_x: 1/(m_S+m_Z)·(m_S·l_sx+m_Z·l_rx); float(%); |
\[\tag{S\_ x}\frac{\frac{4 {{3}^{\frac{3}{2}}}}{5}+13 \sqrt{3}}{32}\]
\[\tag{%o65} 0.8335494511425221\]
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(%i67)
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S_y: 1/(m_S+m_Z)·(m_S·l_sy+m_Z·l_ry); float(%); |
\[\tag{S\_ y}\frac{77}{160}\]
\[\tag{%o67} 0.48125\]
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(%i72)
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S_x2: 1/(m_S+m_Z)·(l_s/2·m_S+(l_s+R_a)·m_Z);
S_x: S_x2·cos(beta); float(%); S_y: S_x2·sin(beta); float(%); |
\[\tag{S\_ x2}\frac{77}{80}\]
\[\tag{S\_ x}\frac{77 \sqrt{3}}{160}\]
\[\tag{%o70} 0.8335494511425222\]
\[\tag{S\_ y}\frac{77}{160}\]
\[\tag{%o72} 0.48125\]
4.3 c) Kräfte
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(%i78)
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F_G: (m_S+m_Z)·981/100; float(%); F_N: F_G·cos(beta); float(%); F_H: F_G·sin(beta); float(%); |
\[\tag{F\_ G}\frac{7848}{25}\]
\[\tag{%o74} 313.92\]
\[\tag{F\_ N}\frac{436 {{3}^{\frac{5}{2}}}}{25}\]
\[\tag{%o76} 271.8626947560109\]
\[\tag{F\_ H}\frac{3924}{25}\]
\[\tag{%o78} 156.96\]
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