Diferencia entre las páginas «Archivo:F1 GIA ventiladores vectores.png» y «Disco deslizando sobre hilo rotante, Noviembre 2015 (MR G.I.C.)»

${\displaystyle {\vec {\omega }}_{01}=\Omega _{0}\,{\vec {k}}_{0},\qquad \qquad {\vec {v}}_{01}^{\,O}={\vec {0}}.}$

${\displaystyle {\vec {\alpha }}_{01}={\vec {0}},\qquad \qquad {\vec {a}}_{01}^{\,O}={\vec {0}}.}$

${\displaystyle {\vec {\omega }}_{20}=\omega _{0}\,{\vec {k}}_{2}\qquad \qquad {\vec {v}}_{20}=-v_{0}\,{\vec {k}}_{2}}$

${\displaystyle {\vec {\alpha }}_{20}={\vec {0}},\qquad \qquad {\vec {a}}_{20}={\vec {0}}.}$

${\displaystyle {\vec {\omega }}_{21}={\vec {\omega }}_{20}+{\vec {\omega }}_{01}=\Omega _{0}\,{\vec {k}}_{0}+\omega _{0}\,{\vec {k}}_{2}}$

${\displaystyle {\vec {k}}_{2}=-{\dfrac {1}{\sqrt {2}}}\,{\vec {\imath }}_{0}+{\dfrac {1}{\sqrt {2}}}\,{\vec {k}}_{0}}$

${\displaystyle {\vec {\omega }}_{21}=-{\dfrac {\omega _{0}}{\sqrt {2}}}\,{\vec {\imath }}_{0}+\left(\Omega _{0}+{\dfrac {\omega _{0}}{\sqrt {2}}}\right)\,{\vec {k}}_{0}.}$

${\displaystyle {\vec {\alpha }}_{21}={\vec {\alpha }}_{20}+{\vec {\alpha }}_{01}+{\vec {\omega }}_{01}\times {\vec {\omega }}_{20}=-{\dfrac {1}{\sqrt {2}}}\,\omega _{0}\Omega _{0}\,{\vec {\jmath }}_{0}.}$

${\displaystyle {\vec {v}}_{21}^{\,G}={\vec {v}}_{20}^{\,G}+{\vec {v}}_{01}^{\,G}.}$

${\displaystyle {\vec {v}}_{01}^{\,G}(t=0)={\vec {v}}^{\,G}+{\vec {\omega }}_{01}\times {\overrightarrow {OA}}={\vec {0}}.}$

${\displaystyle {\vec {v}}_{21}^{\,G}(t=0)={\vec {v}}_{20}^{\,G}=-v_{0}\,{\vec {k}}_{2}={\dfrac {v_{0}}{\sqrt {2}}}\,\left({\vec {\imath }}_{0}-{\vec {k}}_{0}\right).}$

${\displaystyle {\vec {v}}_{21}^{\,G}\cdot {\vec {\omega }}_{21}=0\Longrightarrow {\sqrt {2}}\,\omega _{0}+\Omega _{0}=0.}$

${\displaystyle {\vec {\omega }}_{21}=\Omega _{0}\,{\vec {k}}_{0}={\dfrac {\Omega _{0}}{\sqrt {2}}}\,{\vec {\imath }}_{2}+{\dfrac {\Omega _{0}}{\sqrt {2}}}\,{\vec {k}}_{2}.}$

${\displaystyle {\vec {L}}_{G}={\overset {\leftrightarrow }{I}}_{G}\cdot {\vec {\omega }}_{21}}$

${\displaystyle {\overset {\leftrightarrow }{I}}_{G}=I\left[{\begin{array}{ccc}1/2&0&0\\0&1/2&0\\0&0&1\end{array}}\right]_{2}}$

${\displaystyle {\vec {L}}_{G}=I\left[{\begin{array}{ccc}1/2&0&0\\0&1/2&0\\0&0&1\end{array}}\right]_{2}\left[{\begin{array}{c}\Omega _{0}/{\sqrt {2}}\\0\\\Omega _{0}/{\sqrt {2}}\end{array}}\right]_{2}={\dfrac {I\Omega _{0}}{\sqrt {2}}}\,\left({\dfrac {1}{2}}\,{\vec {\imath }}_{2}+{\vec {k}}_{2}\right).}$

${\displaystyle {\begin{array}{l}{\vec {\imath }}_{2}={\dfrac {1}{\sqrt {2}}}\,{\vec {\imath }}_{0}+{\dfrac {1}{\sqrt {2}}}\,{\vec {k}}_{0}\\{\vec {k}}_{2}=-{\dfrac {1}{\sqrt {2}}}{\vec {\imath }}_{0}+{\dfrac {1}{\sqrt {2}}}{\vec {k}}_{0}\end{array}}}$

${\displaystyle {\vec {L}}_{G}=-{\dfrac {1}{8}}mR^{2}\Omega _{0}\,({\vec {\imath }}_{0}-3{\vec {k}}_{0}).}$

${\displaystyle T=T_{tras}+T_{rot}.}$

${\displaystyle T_{tras}={\dfrac {1}{2}}m|{\vec {v}}_{21}^{\,G}|^{2}.}$

${\displaystyle {\vec {v}}_{21}^{\,G}={\vec {v}}_{21}^{\,B}={\vec {v}}_{20}^{\,B}+{\vec {v}}_{01}^{\,B}}$

${\displaystyle {\vec {v}}_{20}^{B}=-v_{0}\,{\vec {k}}_{2}}$

${\displaystyle {\vec {v}}_{01}^{\,B}={\vec {v}}_{01}^{\,O}+{\vec {\omega }}_{01}\times {\overrightarrow {OB}}={\dfrac {L\Omega _{0}}{\sqrt {2}}}\,{\vec {\jmath }}_{0}.}$

${\displaystyle {\vec {v}}_{21}^{\,B}={\dfrac {v_{0}}{\sqrt {2}}}\,{\vec {\imath }}_{0}+{\dfrac {L\Omega _{0}}{\sqrt {2}}}\,{\vec {\jmath }}_{0}-{\dfrac {v_{0}}{\sqrt {2}}}\,{\vec {k}}_{0}.}$

${\displaystyle T_{tras}={\dfrac {1}{2}}m\,\left(v_{0}^{2}+{\dfrac {L^{2}\Omega _{0}^{2}}{2}}\right).}$

${\displaystyle T_{rot}={\dfrac {1}{2}}{\vec {L}}_{G}\cdot {\vec {\omega }}_{21}={\dfrac {3}{16}}mR^{2}\Omega _{0}^{2}.}$

${\displaystyle T={\dfrac {1}{16}}m\,\left(8v_{0}^{2}+(4L^{2}+3R^{2})\Omega _{0}^{2}\right).}$