- Definition: Raumwinkel = Fläche der Einheitskugel = Kalottenfläche / Radius^2
- Gesamtraumwinkel:
![{\displaystyle 4\pi {\frac {r^{2}}{r^{2}}}=4\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f3298e5e1beb2860b60ac582baedfb58a0c63a)
- Öffungswinkel:
![{\displaystyle \Omega =2\pi \left[1-\cos {\frac {\phi }{2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a039bc7b1ecd53df337f239e7b143bba3c1726c)
- Bedingt durch Apparatur, Eichung, etc.
- Größe unbekannt
- Messwert
schwankt um Mittelwert
![{\displaystyle S=\sum _{i}{\left({\overline {x}}-x_{i}\right)^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54db242bf403549d21f0e8d7dae3223a025552e3)
- Ziel:
![{\displaystyle \lim _{S\to 0}{\frac {\mathrm {d} S}{\mathrm {d} {\overline {x}}}}=2\,\sum _{i}{\left({\overline {x}}-x_{i}\right)^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5548899157bad3bcbb7bcd02c4a4c2dbd5c111ad)
![{\displaystyle {\overline {x}}={\frac {1}{n}}\,\sum _{n}{x_{i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/240e55b3ea9fbf4722dcc39b326c43fe745ac6f1)
![{\displaystyle x_{tat}=\lim _{n\to \infty }{\frac {1}{n}}\,\sum _{n}{x_{i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e12df5cc472f2c3dd7c825b8f1ade84c950e0fcb)
Fehler der Einzelmessung
![{\displaystyle \sigma ={\sqrt {\frac {\sum _{n}{\left({\overline {x}}-x_{i}\right)^{2}}}{n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce2750cf39bd51c866082d0f72e386493a9ac9a)
Fehler des arithmetischen Mittels
![{\displaystyle \sigma ={\sqrt {\frac {\sum _{n}{\left({\overline {x}}-x_{i}\right)^{2}}}{n\,\left(n-1\right)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf64de030f291c7566602b5b894c7de4579fcb4)
Unterscheidung: Rechts- und Linkssystem
![{\displaystyle (x,y,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4)
![{\displaystyle (r,\theta ,\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e918fbb45350470ed9142d8ba0016fed78e130)
- Linienelement
![{\displaystyle \mathrm {d} {\vec {r}}=\mathrm {d} r\cdot {\hat {e}}_{r}+r\cdot \mathrm {d} \theta \cdot {\hat {e}}_{\theta }+r\cdot \sin {\theta }\cdot \mathrm {d} \phi \cdot {\hat {e}}_{\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db3b7b051d9c9f2606061ea9458ad5e5cf572a3c)
- Volumenelement
![{\displaystyle \mathrm {d} V=r^{2}\cdot \sin {\theta }\cdot \mathrm {d} r\cdot \mathrm {d} \theta \cdot \mathrm {d} \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc65bff2112c1f22cd9732db2fe742df9e5f4bdc)
- Raumwinkel
![{\displaystyle \mathrm {d} \Omega =\sin {\theta }\cdot \mathrm {d} \theta \cdot \mathrm {d} \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d13fc61f5ffe980c96f37fbde511a41a88ab457)
- Transformation
![{\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}r\cdot \sin {\theta }\cdot \cos {\phi }\\r\cdot \sin {\theta }\cdot \sin {\phi }\\r\cdot \cos {\phi }\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5824f68a2140be53d64942464464a5d0b484131)
![{\displaystyle (\rho ,\phi ,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5959e4609bcc3d932ed3d957e52302c760716855)
- Linienelement
![{\displaystyle \mathrm {d} {\vec {r}}=\mathrm {d} \rho \cdot {\hat {e}}_{\rho }+\rho \cdot \mathrm {d} \phi \cdot {\hat {e}}_{\phi }+\mathrm {d} z\cdot {\hat {e}}_{\phi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f159dde5ae24409bdec7de75003246b9e6556a6c)
- Volumenelement
![{\displaystyle \mathrm {d} V=\rho \cdot \mathrm {d} \rho \cdot \mathrm {d} \phi \cdot \mathrm {d} z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/621071d308cfdf741e00087484b71b4e1aabde21)
![{\displaystyle \mathbf {r} =\omega \,\mathbf {r} _{0}=\left(1,{\dot {\theta }},{\dot {\phi }},{\dot {\psi }}\right)\,\left(0,x,y,z\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/612f6a770be86e0b98c80e8d0bb0c5c587a4bc89)
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Kreisfrequenz
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Winkelgeschwindigkeit
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Umlauffrequenz
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Periodendauer
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Rotationsgeschwindigkeit und -Beschleunigung
Bearbeiten
![{\displaystyle \mathbf {a} _{rot}={\dot {\mathbf {v} }}_{rot}={\ddot {\mathbf {r} }}_{rot}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ec2c883e333c93f507ae083e38c974bcf7e853)
![{\displaystyle \mathbf {v} _{rot}=\mathbf {\omega } \,\left(0,{\vec {r}}\right)=\left(-\omega {\vec {r}},{\dot {\vec {r}}}+{\vec {\omega }}\times {\vec {r}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ade8cbcd35b1ce3fa8ed71b96c69fe647780d1)
![{\displaystyle \mathbf {a} _{rot}=\mathbf {\omega } \,\mathbf {v} _{rot}=\left(-{\dot {\vec {\omega }}}{\vec {r}},{\ddot {\vec {r}}}+2\,{\vec {\omega }}\times {\dot {\vec {r}}}+{\dot {\vec {\omega }}}\times {\vec {r}}-{\vec {\omega }}\,{\vec {r}}\,{\vec {\omega }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90ff136d4c33e72a3498e3b35909c82e7bdf8567)
Die Beschleunigung steht senkrecht zur Geschwindigkeit - dh. die Beschleunigung in Richtung zur Drehachse verursacht die Drehung. Der Beschleunigungsvektor entspricht damit der Zentripetalbeschleunigung:
![{\displaystyle -{\vec {\omega }}\,{\vec {r}}\,{\vec {\omega }}={\vec {\omega }}\times \left({\vec {\omega }}\times {\vec {r}}\right)+\omega ^{2}\,{\vec {r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcfabaf76249cb2d7cb426358ece95382c548db)
![{\displaystyle x^{2}+y^{2}=r^{2}\,\left[\cos ^{2}\left(\omega \,t\right)+\sin ^{2}\left(\omega \,t\right)\right]=r^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26f1b79417169e46bbcf99798487d3465f0e30b1)
![{\displaystyle \mathbf {r} =\left(t,{\vec {r}}\right)=\left(1,{\vec {v}}\right)\,\left(t,{\vec {0}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd8c1ff2ba4e78b3248f1d1a164cf991a4c4678)
- zeitliche Ableitung
![{\displaystyle {\dot {\mathbf {r} }}=\left(\nabla ,{\vec {0}}\right)\,\mathbf {r} =\left({\frac {\partial }{\partial t}},{\vec {0}}\right)\,\mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a42f88281078c55a164f81f6e5a791c6fd718ec)
- räumliche Ableitung
![{\displaystyle \mathbf {r} '=\left(0,{\vec {\nabla }}\right)\,\mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3ebadf5817705fd813bf598c334ba9b59f8bb4)
- Differenzialform
![{\displaystyle \mathbf {a} ={\dot {\mathbf {v} }}={\ddot {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/224196943166a3d7079301e1a9dfbe87fe4f0904)
- Integralform
![{\displaystyle \mathbf {r} =\iint \mathbf {a} \ \mathrm {d} t^{2}=\int \left(\mathbf {v} \,t+\mathbf {v} _{0}\right)\ \mathrm {d} t={\frac {1}{2}}\,\mathbf {a} \,t^{2}+\mathbf {v} _{0}\,t+\mathbf {r} _{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/829a07884fb34012487242740801d76709f8a033)
- mittlere Beschleunigung
![{\displaystyle {\overline {\mathbf {a} }}={\frac {\mathbf {v} \cdot (t+\Delta t)-\mathbf {v} \cdot t}{\Delta t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56616fdcaaf452161d9cbb99c2065d44f16740da)
- momentane Beschleunigung
![{\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\frac {\mathbf {v} \cdot (t+\Delta t)-\mathbf {v} \cdot t}{\Delta t}}={\dot {\mathbf {v} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8db0079454db75c93e25df25e1d799c7b0eb40)
- Rotation
![{\displaystyle \mathbf {v} _{rot}=\mathbf {\omega } \,\mathbf {r} =\left({\frac {\mathrm {d} t}{\mathrm {d} t}}=1,{\vec {\omega }}\right)\,\left(t,{\vec {r}}\right)=\left(1\,t-{\vec {\omega }}\,{\vec {r}},1\,{\vec {r}}+t\,{\vec {\omega }}+{\vec {\omega }}\times {\vec {r}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f387d6dee781ff4d0d09f524eb4d8550a73aaa)
- Freier Fall
![{\displaystyle \mathbf {r} ={\frac {1}{2}}\,\mathbf {g} \,\left(\Delta t\right)^{2}+\mathbf {v} _{0}\,\Delta t+\mathbf {r} _{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ce845e53cf4f3c21943ee71d9cde50c5d5e886)
![{\displaystyle \mathbf {v} =\mathbf {g} \,\Delta t+v_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1d639bc01d9250c6c3a41f79bfaa693f3041b4)
![{\displaystyle g=g_{Erde}-g_{Zentrifugal}=G\,{\frac {m_{1}\,m_{2}}{d_{\overline {12}}^{2}}}-\omega ^{2}\,d_{\overline {12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a094eda19af2b4e074ef121e12afcb5d32974a3)
- Fallzeit
- bei h=0; t=?
- Endgeschwindigkeit
- bei h=0; vvert=?
- Scheitelpunkt
![{\displaystyle {\dot {y}}=0\to \mathbf {v} _{y}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc8e963b1357e0d2a1aed0b8934b93bd4675e850)
![{\displaystyle \mathbf {a} ={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}={\frac {c}{c}}\,{\begin{pmatrix}1\\{\frac {\mathrm {d} x}{dt}}\\{\frac {\mathrm {d} y}{\mathrm {d} t}}\\{\frac {\mathrm {d} z}{\mathrm {d} t}}\end{pmatrix}}=v_{x}\,{\begin{pmatrix}{\frac {\mathrm {d} t}{\mathrm {d} x}}\\1\\{\frac {\mathrm {d} y}{\mathrm {d} x}}\\{\frac {\mathrm {d} z}{\mathrm {d} x}}\end{pmatrix}}=v_{y}\,{\begin{pmatrix}{\frac {\mathrm {d} t}{\mathrm {d} y}}\\{\frac {\mathrm {d} x}{\mathrm {d} y}}\\1\\{\frac {\mathrm {d} z}{\mathrm {d} y}}\end{pmatrix}}=v_{z}\,{\begin{pmatrix}{\frac {\mathrm {d} t}{\mathrm {d} z}}\\{\frac {\mathrm {d} x}{\mathrm {d} z}}\\{\frac {\mathrm {d} y}{\mathrm {d} z}}\\1\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/227e94cd2398128ff52bb045003b992c545e1961)
für x:
![{\displaystyle \mathbf {a} \,\mathrm {d} x=v_{x}\,{\begin{pmatrix}\mathrm {d} t\\\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{pmatrix}}=v_{x}\,\mathrm {d} \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a86a47525c4f8869953868d08b824d4d9d8c6b)
![{\displaystyle \int _{x_{0}}^{x}{\mathbf {a} }\,\mathrm {d} x=\int _{v_{0}}^{v}\,v_{x}\,\mathrm {d} \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8daf77bb6124c6960673ae05111a1c9fb15bca)
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