Propagación de errores 02

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Mar 022013
 

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Ejercicio 2. Para cierto sistema dinámico, la aceleración se calcula con la ecuación:

(1)   \begin{eqnarray*} a=\frac{m_{2}-\mu\cdot m_{1}}{m_{1}+m_{2}}\cdot g \nonumber\\ \end{eqnarray*}

considerando a las masas m_{1}y m_{2} como variables, propague la ecuación para el error relativo en la aceleración a.
Respuesta 1.
Aplicando logaritmos naturales a la ecuación (1) obtenemos:

    \begin{eqnarray*} \ln a & = & \ln\left\{ \frac{m_{2}-\mu\cdot m_{1}}{m_{1}+m_{2}}\right\} \cdot g\\ \ln a & = & \ln\left\{ \mbox{\ensuremath{m_{2}}-\ensuremath{\mu\cdot m_{1}}}\right\} -\ln\left\{ m_{1}+m_{2}\right\} +\ln g\\ d\left(\ln a\right) & = & d\left(\ln\left\{ \mbox{\ensuremath{m_{2}}-\ensuremath{\mu\cdot m_{1}}}\right\} \right)-d\left(\ln\left\{ m_{1}+m_{2}\right\} \right)+d\left(\ln g\right)\\ \frac{da}{a} & = & \frac{d\left\{ \mbox{\ensuremath{m_{2}}-\ensuremath{\mu\cdot m_{1}}}\right\} }{\left\{ \mbox{\ensuremath{m_{2}}-\ensuremath{\mu\cdot m_{1}}}\right\} }-\frac{d\left\{ m_{1}+m_{2}\right\} }{\left\{ m_{1}+m_{2}\right\} }+\frac{dg}{g}\\ \frac{da}{a} & = & \frac{dm_{2}-\mu\cdot dm_{1}}{m_{2}-\mu\cdot m_{1}}-\frac{dm_{1}+dm_{2}}{m_{1}+m_{2}}+0\\ \frac{da}{a} & = & \frac{dm_{2}}{m_{2}-\mu\cdot m_{1}}-\frac{\mu\cdot dm_{1}}{m_{2}-\mu\cdot m_{1}}-\frac{dm_{1}}{m_{1}+m_{2}}+\frac{dm_{2}}{m_{1}+m_{2}}\\ \frac{E_{<a>}}{<a>} & = & \frac{E_{<m_{2}>}}{<m_{2}>-\mu\cdot<m_{1}>}-\frac{\mu\cdot E_{<m_{1}>}}{<m_{2}>-\mu\cdot<m_{1}>}-\frac{E_{<m_{1}>}}{<m_{2}>+<m_{1}>}+\frac{E_{<m_{2}>}}{<m_{2}>+<m_{1}>}\\ \left|\frac{E_{<a>}}{<a>}\right| & = & \left|\frac{E_{<m_{2}>}}{<m_{2}>-\mu\cdot<m_{1}>}\right|+\left|-\frac{\mu\cdot E_{<m_{1}>}}{<m_{2}>-\mu\cdot<m_{1}>}\right|+\left|-\frac{E_{<m_{1}>}}{<m_{2}>+<m_{1}>}\right|+\left|\frac{E_{<m_{2}>}}{<m_{2}>+<m_{1}>}\right|\\ \frac{E_{<a>}}{<a>} & = & \frac{E_{<m_{2}>}}{<m_{2}>-\mu\cdot<m_{1}>}+\frac{\mu\cdot E_{<m_{1}>}}{<m_{2}>-\mu\cdot<m_{1}>}+\frac{E_{<m_{1}>}}{<m_{2}>+<m_{1}>}+\frac{E_{<m_{2}>}}{<m_{2}>+<m_{1}>}\\ \frac{E_{<a>}}{<a>} & = & \frac{E_{<m_{2}>}+\mu\cdot E_{<m_{1}>}}{<m_{2}>-\mu\cdot<m_{1}>}+\frac{E_{<m_{1}>}+E_{<m_{2}>}}{<m_{2}>+<m_{1}>} \end{eqnarray*}

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