12.5: Ley del movimiento de Newton en un marco no inercial
- Page ID
- 126952
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)La aceleración del sistema en el marco inercial giratorio se puede derivar diferenciando la relación de velocidad general para\(\mathbf{v}\), Ecuación\(12.4.4\), en la base del marco fijo que da
\[\begin{align} \mathbf{a}_{fix} &= \left(\frac{d\mathbf{v}_{fix}}{dt}\right)_{fixed} \\[4pt] &= \left(\frac{d\mathbf{V}_{fix}}{dt}\right)_{fixed} + \left(\frac{d\mathbf{v}^{\prime\prime}_{rot}}{dt}\right)_{fixed} + \left(\frac{d\omega}{dt}\right)_{fixed} \times \mathbf{r}^{\prime}_{mov} + \omega \times \left(\frac{d\mathbf{r}^{\prime}_{mov}}{dt}\right)_{fixed} \label{12.22} \end{align}\]
Ahora queremos utilizar la transformación general a una base de trama giratoria que requiere la inclusión de la dependencia del tiempo de los vectores unitarios en el marco giratorio, es decir,
\[ \begin{align} \left(\frac{d\mathbf{v}^{\prime\prime}_{rot}}{dt}\right)_{fixed} &= \left(\frac{d\mathbf{v}^{\prime\prime}_{rot}}{dt}\right)_{rotating} + \omega \times \mathbf{v}^{\prime\prime}_{rot} \label{12.23} \\[4pt] \left(\frac{d\omega}{dt}\right)_{fixed} \times \mathbf{r}^{\prime}_{mov} &= \left(\frac{d\omega}{dt}\right)_{rot} \times \mathbf{r}^{\prime}_{mov} \label{12.24} \\[4pt] \omega \times \left(\frac{d\mathbf{r}^{\prime}_{mov}}{dt}\right)_{fixed} &= \omega \times \mathbf{v}^{\prime\prime}_{rot} + \omega \times (\omega \times \mathbf{r}^{\prime}_{mov}) \label{12.25} \end{align}\]
Usando ecuaciones\ ref {12.23},\ ref {12.24},\ ref {12.25} da
\[ \mathbf{a}_{fix} = \mathbf{A}_{fix} + \mathbf{a}^{\prime\prime}_{rot} + 2\omega \times \mathbf{v}^{\prime\prime}_{rot} + \omega \times (\omega \times \mathbf{r}^{\prime}_{mov}) + \dot{\omega} \times \mathbf{r}^{\prime}_{mov} \label{12.26}\]
donde la aceleración en el marco giratorio es\(\mathbf{a}^{\prime\prime}_{rot} = \left(\frac{d\mathbf{v}^{\prime\prime}_{rot}}{dt}\right)_{rot}\) mientras la velocidad es\(\mathbf{v}^{\prime\prime}_{rot} = \left(\frac{\mathbf{r}^{\prime\prime}_{rot}}{dt}\right)_{rot}\) y\(\mathbf{A}_{fix}\) es con respecto al marco fijo.
Las leyes del movimiento de Newton son obedecidas en el marco inercial, es decir
\[ \begin{align} \mathbf{F}_{fix} &= m\mathbf{a}_{fix} \\[4pt] &= m(\mathbf{A}_{fix} + \mathbf{a}^{\prime\prime}_{rot} + 2\omega \times \mathbf{v}^{\prime\prime}_{rot} + \omega \times (\omega \times \mathbf{r}^{\prime}_{mov}) + \dot{\omega} \times \mathbf{r}^{\prime}_{mov}) \label{12.27} \end{align}\]
En el marco de doble cebado, que puede ser tanto giratorio como acelerado en la traslación, se puede atribuir una fuerza efectiva\(\mathbf{F}^{eff}_{rot}\) que obedece a una ley efectiva de Newton para la aceleración\(\mathbf{a}^{\prime\prime}_{rot}\) en el marco giratorio
\[\begin{align}\mathbf{F}^{eff}_{rot} &= m\mathbf{a}^{\prime\prime}_{rot} \\[4pt] &= \mathbf{F}_{fix} - m(\mathbf{A}_{fix} + 2\omega \times \mathbf{v}^{\prime\prime}_{rot} + \omega \times (\omega \times \mathbf{r}^{\prime}_{mov}) + \dot{\omega} \times \mathbf{r}^{\prime}_{mov}) \label{12.28} \end{align}\]
Obsérvese que la fuerza efectiva\(\mathbf{F}^{eff}_{rot}\) comprende la fuerza física\(\mathbf{F}_{fixed}\) menos cuatro fuerzas no inerciales que se introducen para corregir el hecho de que el marco de referencia giratorio es un marco no inercial.