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Drake: /* Velocidad angular */
2024-01-12T17:14:05Z
<p><span dir="auto"><span class="autocomment">Velocidad angular</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Revisión anterior</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revisión del 18:14 12 ene 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34">Línea 34:</td>
<td colspan="2" class="diff-lineno">Línea 34:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obtendremos la velocidad angular instantánea <math>\vec{\omega}=\omega_x\,\vec{\imath}+\omega_y\,\vec{\jmath}+\omega_z\,\vec{k}\,</math> exigiendo el cumplimiento de la ecuación del campo de velocidades de un sólido rígido:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obtendremos la velocidad angular instantánea <math>\vec{\omega}=\omega_x\,\vec{\imath}+\omega_y\,\vec{\jmath}+\omega_z\,\vec{k}\,</math> exigiendo el cumplimiento de la ecuación del campo de velocidades de un sólido rígido:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center><math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center><math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\vec{v}_B=\vec{v}_A<del style="font-weight: bold; text-decoration: none;">\,</del>+\vec{\omega}<del style="font-weight: bold; text-decoration: none;">\,</del>\times<del style="font-weight: bold; text-decoration: none;">\,</del>\overrightarrow{AB}\,\Rightarrow\,\vec{v}_B<del style="font-weight: bold; text-decoration: none;">\,</del>-<del style="font-weight: bold; text-decoration: none;">\,</del>\vec{v}_A=\vec{\omega}<del style="font-weight: bold; text-decoration: none;">\,</del>\times<del style="font-weight: bold; text-decoration: none;">\,</del>\overrightarrow{AB}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\vec{v}_B=\vec{v}_A+\vec{\omega}\times\overrightarrow{AB}<ins style="font-weight: bold; text-decoration: none;">\,</ins>\,\Rightarrow<ins style="font-weight: bold; text-decoration: none;">\,</ins>\,\vec{v}_B-\vec{v}_A=\vec{\omega}\times\overrightarrow{AB}</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\,\Rightarrow\, <del style="font-weight: bold; text-decoration: none;">-</del>\,\Delta y<del style="font-weight: bold; text-decoration: none;">\,</del>\,\vec{\imath}<del style="font-weight: bold; text-decoration: none;">\,</del>+<del style="font-weight: bold; text-decoration: none;">\,</del>(\Delta x+\Delta z)\,\vec{\jmath}<del style="font-weight: bold; text-decoration: none;">\,</del>-<del style="font-weight: bold; text-decoration: none;">\,</del>\Delta y\,\vec{k}=\left|\begin{array}{ccc}\vec{\imath} & \vec{\jmath} & \vec{k} \\ \omega_x & \omega_y & \omega_z \\ \Delta x & \Delta y & \Delta z \end{array}\right|</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\,</ins>\,\Rightarrow\,\, <ins style="font-weight: bold; text-decoration: none;">-</ins>\Delta y\,\vec{\imath}+(\Delta x+\Delta z)\,\vec{\jmath}-\Delta y\,\vec{k}=\left|\begin{array}{ccc}\vec{\imath} & \vec{\jmath} & \vec{k} \\ \omega_x & \omega_y & \omega_z \\ \Delta x & \Delta y & \Delta z \end{array}\right|</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></center></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></center></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Calculando el determinante e igualando componente a componente, se obtiene el siguiente sistema de ecuaciones:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Calculando el determinante e igualando componente a componente, se obtiene el siguiente sistema de ecuaciones:</div></td></tr>
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Drake
http://laplace.us.es/wiki/index.php?title=No_Bolet%C3%ADn_-_Ejemplo_de_campo_de_velocidades_de_un_s%C3%B3lido_II_(Ex.Ene/20)&diff=3180&oldid=prev
Drake: /* Velocidad angular */
2024-01-12T17:12:24Z
<p><span dir="auto"><span class="autocomment">Velocidad angular</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Revisión anterior</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revisión del 18:12 12 ene 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34">Línea 34:</td>
<td colspan="2" class="diff-lineno">Línea 34:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obtendremos la velocidad angular instantánea <math>\vec{\omega}=\omega_x\,\vec{\imath}+\omega_y\,\vec{\jmath}+\omega_z\,\vec{k}\,</math> exigiendo el cumplimiento de la ecuación del campo de velocidades de un sólido rígido:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obtendremos la velocidad angular instantánea <math>\vec{\omega}=\omega_x\,\vec{\imath}+\omega_y\,\vec{\jmath}+\omega_z\,\vec{k}\,</math> exigiendo el cumplimiento de la ecuación del campo de velocidades de un sólido rígido:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center><math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><center><math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\vec{v}_B=\vec{v}_A\,+<del style="font-weight: bold; text-decoration: none;">\,\,</del>\vec{\omega}<del style="font-weight: bold; text-decoration: none;">\,</del>\,\times\,\overrightarrow{AB}\,\<del style="font-weight: bold; text-decoration: none;">,\,\,\,\Longrightarrow\,\,\,\,</del>\,\vec{v}_B\,-<del style="font-weight: bold; text-decoration: none;">\,</del>\,\vec{v}_A=\vec{\omega}<del style="font-weight: bold; text-decoration: none;">\,</del>\,\times\,\overrightarrow{AB}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\vec{v}_B=\vec{v}_A\,+\vec{\omega}\,\times\,\overrightarrow{AB}\,\<ins style="font-weight: bold; text-decoration: none;">Rightarrow</ins>\,\vec{v}_B\,-\,\vec{v}_A=\vec{\omega}\,\times\,\overrightarrow{AB}</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\,\<del style="font-weight: bold; text-decoration: none;">,\,\,\,\Longrightarrow\,\,\,\,</del>\, -\,\Delta y\,\,\vec{\imath}<del style="font-weight: bold; text-decoration: none;">\,</del>\,+\,(\Delta x<del style="font-weight: bold; text-decoration: none;">\,\,</del>+<del style="font-weight: bold; text-decoration: none;">\,\,</del>\Delta z)\,\vec{\jmath}<del style="font-weight: bold; text-decoration: none;">\,</del>\,-\,\Delta y\,\vec{k}=\left|\begin{array}{ccc}\vec{\imath} & \vec{\jmath} & \vec{k} \\ \omega_x & \omega_y & \omega_z \\ \Delta x & \Delta y & \Delta z \end{array}\right|</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\,\<ins style="font-weight: bold; text-decoration: none;">Rightarrow</ins>\, -\,\Delta y\,\,\vec{\imath}\,+\,(\Delta x+\Delta z)\,\vec{\jmath}\,-\,\Delta y\,\vec{k}=\left|\begin{array}{ccc}\vec{\imath} & \vec{\jmath} & \vec{k} \\ \omega_x & \omega_y & \omega_z \\ \Delta x & \Delta y & \Delta z \end{array}\right|</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></center></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></center></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Calculando el determinante e igualando componente a componente, se obtiene el siguiente sistema de ecuaciones:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Calculando el determinante e igualando componente a componente, se obtiene el siguiente sistema de ecuaciones:</div></td></tr>
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Drake
http://laplace.us.es/wiki/index.php?title=No_Bolet%C3%ADn_-_Ejemplo_de_campo_de_velocidades_de_un_s%C3%B3lido_II_(Ex.Ene/20)&diff=3080&oldid=prev
Drake en 21:42 11 ene 2024
2024-01-11T21:42:28Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Revisión anterior</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revisión del 22:42 11 ene 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8">Línea 8:</td>
<td colspan="2" class="diff-lineno">Línea 8:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Verifique la equiproyectividad del campo de velocidades.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Verifique la equiproyectividad del campo de velocidades.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># ¿Cuál es la velocidad angular instantánea del sólido rígido?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># ¿Cuál es la velocidad angular instantánea del sólido rígido?</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Verificación de la equiproyectividad==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Sean dos puntos arbitrarios <math>A\,(x_A,y_A,z_A)\,\,</math> y <math>\,B\,(x_B,y_B,z_B)\,\,</math>. Sus velocidades <math>\vec{v}_A\,\,</math> y <math>\,\vec{v}_B\,</math> se obtienen sustituyendo sus respectivas coordenadas en la expresión del campo:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><center><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\vec{v}_A=(2\,-\,y_A)\,\vec{\imath}\,+\,(1\,+\,x_A\,+\,z_A)\,\vec{\jmath}\,-\,y_A\,\vec{k}\,;\,\,\,\,\,\,\,\,</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\vec{v}_B=(2\,-\,y_B)\,\vec{\imath}\,+\,(1\,+\,x_B\,+\,z_B)\,\vec{\jmath}\,-\,y_B\,\vec{k}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></center></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">La equiproyectividad del campo de velocidades quedará verificada si se cumple la siguiente igualdad:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><center><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\vec{v}_A\cdot\overrightarrow{AB}=\vec{v}_B\cdot\overrightarrow{AB}\,\,\,\,\,\Longrightarrow\,\,\,\,\,(\vec{v}_B-\vec{v}_A)\cdot\overrightarrow{AB}=0</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></center></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Comprobemos, pues, que es nulo el producto escalar de los siguientes vectores:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><center><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\begin{array}{l}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\overrightarrow{AB}=\underbrace{(x_B-x_A)}_{\displaystyle=\Delta x}\,\vec{\imath}\,+\underbrace{(y_B-y_A)}_{\displaystyle=\Delta y}\,\vec{\jmath}\,+\underbrace{(z_B-z_A)}_{\displaystyle=\Delta z}\,\vec{k}=\Delta x\,\vec{\imath}\,+\Delta y\,\vec{\jmath}\,+\Delta z\,\vec{k} \\ \\</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\vec{v}_B-\vec{v}_A=-\Delta y\,\,\vec{\imath}\,+(\Delta x+\Delta z)\,\vec{\jmath}\,-\Delta y\,\vec{k}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\end{array}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></center></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">En efecto, se verifica la equiproyectividad porque:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><center><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(\vec{v}_B-\vec{v}_A)\cdot\overrightarrow{AB}=-\Delta y\,\Delta x\,+(\Delta x+\Delta z)\,\Delta y\,-\Delta y\,\Delta z=0</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></center></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Velocidad angular==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Obtendremos la velocidad angular instantánea <math>\vec{\omega}=\omega_x\,\vec{\imath}+\omega_y\,\vec{\jmath}+\omega_z\,\vec{k}\,</math> exigiendo el cumplimiento de la ecuación del campo de velocidades de un sólido rígido:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><center><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\vec{v}_B=\vec{v}_A\,+\,\,\vec{\omega}\,\,\times\,\overrightarrow{AB}\,\,\,\,\,\Longrightarrow\,\,\,\,\,\vec{v}_B\,-\,\,\vec{v}_A=\vec{\omega}\,\,\times\,\overrightarrow{AB}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\,\,\,\,\,\Longrightarrow\,\,\,\,\, -\,\Delta y\,\,\vec{\imath}\,\,+\,(\Delta x\,\,+\,\,\Delta z)\,\vec{\jmath}\,\,-\,\Delta y\,\vec{k}=\left|\begin{array}{ccc}\vec{\imath} & \vec{\jmath} & \vec{k} \\ \omega_x & \omega_y & \omega_z \\ \Delta x & \Delta y & \Delta z \end{array}\right|</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></center></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Calculando el determinante e igualando componente a componente, se obtiene el siguiente sistema de ecuaciones:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><center><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\left\{\begin{array}{rcl}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">-\Delta y & = & \omega_y\,\Delta z-\omega_z\,\Delta y \\</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\Delta x+\Delta z & = & \omega_z\,\Delta x-\omega_x\,\Delta z \\</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">-\Delta y & = & \omega_x\,\Delta y-\omega_y\,\Delta x</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\end{array}\right.\,\,\,\,\,\Longrightarrow\,\,\,\,\,</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\left\{\begin{array}{rcl}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(-1+\omega_z)\,\Delta y-\omega_y\,\Delta z & = & 0 \\</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(1-\omega_z)\,\Delta x+(1+\omega_x)\,\Delta z & = & 0 \\</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\omega_y\,\Delta x-(1+\omega_x)\,\Delta y & = & 0</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\end{array}\right.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></center></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Dado que este sistema de ecuaciones debe verificarse para todo valor de <math>\Delta x\,</math>, <math>\Delta y\,\,</math> y <math>\,\Delta z\,</math> (ya que los puntos <math>A\,\,</math> y <math>\,B\,</math> son cualesquiera), es necesario que los coeficientes de <math>\Delta x\,</math>, <math>\Delta y\,\,</math> y <math>\,\Delta z\,</math> en estas ecuaciones sean todos nulos, llegándose a la conclusión de que:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><center><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\left.\begin{array}{l} \omega_x=-1 \\ \omega_y=0 \\ \omega_z=1 \end{array}\right\}\,\,\,\,\,\Longrightarrow\,\,\,\,\,\vec{\omega}=(-\,\vec{\imath}+\vec{k}\,)\,\,\mathrm{rad/s}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></center></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Categoría:Problemas de Cinemática del Sólido Rígido (GITI)]]</ins></div></td></tr>
</table>
Drake
http://laplace.us.es/wiki/index.php?title=No_Bolet%C3%ADn_-_Ejemplo_de_campo_de_velocidades_de_un_s%C3%B3lido_II_(Ex.Ene/20)&diff=3050&oldid=prev
Drake: Página creada con «==Enunciado== Sea un sólido rígido en movimiento respecto a un triedro cartesiano OXYZ. En cierto instante, el campo de velocidades del sólido tiene la siguiente expresión (unidades del SI): <center><math> \vec{v}=(2-y)\,\vec{\imath}+(1+x+z)\,\vec{\jmath}-y\,\vec{k} </math></center> donde <math>(x,y,z)\,</math> son las coordenadas cartesianas de cada punto del sólido. # Verifique la equiproyectividad del campo de velocidades. # ¿Cuál es la velocidad angular in…»
2024-01-11T18:41:00Z
<p>Página creada con «==Enunciado== Sea un sólido rígido en movimiento respecto a un triedro cartesiano OXYZ. En cierto instante, el campo de velocidades del sólido tiene la siguiente expresión (unidades del SI): <center><math> \vec{v}=(2-y)\,\vec{\imath}+(1+x+z)\,\vec{\jmath}-y\,\vec{k} </math></center> donde <math>(x,y,z)\,</math> son las coordenadas cartesianas de cada punto del sólido. # Verifique la equiproyectividad del campo de velocidades. # ¿Cuál es la velocidad angular in…»</p>
<p><b>Página nueva</b></p><div>==Enunciado==<br />
Sea un sólido rígido en movimiento respecto a un triedro cartesiano OXYZ. En cierto instante, el campo de velocidades del sólido tiene la siguiente expresión (unidades del SI):<br />
<center><math><br />
\vec{v}=(2-y)\,\vec{\imath}+(1+x+z)\,\vec{\jmath}-y\,\vec{k}<br />
</math></center><br />
donde <math>(x,y,z)\,</math> son las coordenadas cartesianas de cada punto del sólido.<br />
<br />
# Verifique la equiproyectividad del campo de velocidades.<br />
# ¿Cuál es la velocidad angular instantánea del sólido rígido?</div>
Drake