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rotating frame

 Definition/Summary Often in physics we need to consider frames of reference that are non-inertial (the Earth spinning on its axis for example). We must therefore see how these rotating reference frames relate to an inertial reference frame.

 Equations $$\frac{d^2\mathbf{r}}{dt^2} = \ddot{\mathbf{r}} + 2(\mathbf{\Omega} \times \dot{\mathbf{r}}) +\mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r})$$

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 Breakdown Physics > Classical Mechanics >> Newtonian Dynamics

 See Also centripetal acceleratioCoriolis forceEuler's equationsinertial observercentrifugal force

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 Extended explanation Effect on 1st derivatives: Consider a rotating frame with an instantaneous angular velocity $\mathbf{\Omega}$. A unit vector $\mathbf{e}_i$ traces a circle about $\mathbf{\Omega}$ at a rate: $$\frac{d\mathbf{e}_i}{dt} = \mathbf{\Omega} \times \mathbf{e}_i$$ A particle will have a position in the rotating frame given by $\mathbf{r} = x_i\mathbf{e}_i$ (where $i$ is summed from 1 to 3) and thus the velocity in an inertial frame is then: $$\frac{d\mathbf{r}}{dt} = \frac{d}{dt}( x_i\mathbf{e}_i) = \frac{dx_i}{dt}\mathbf{e}_i + x_i\frac{d\mathbf{e}_i}{dt} = \frac{dx_i}{dt}\mathbf{e}_i + x_i(\mathbf{\Omega} \times \mathbf{e}_i)$$ Example: torque equation: For example, in a fixed frame of reference, the equation relating net torque on a body to its rate of change of angular momentum is: $$\mathbf{\tau}_{net}\ =\ \frac{d}{dt}\left(\tilde{I}\,\mathbf{\Omega}\right)\ =\ \tilde{I}\,\frac{d}{dt}\left(\mathbf{\Omega}\right)\ +\ \frac{d\tilde{I}}{dt}\left(\mathbf{\Omega}\right)$$ but in a frame rotating with the body, it is: $$\mathbf{\tau}_{net}\ =\ \frac{d}{dt}\left(\tilde{I}\,\mathbf{\Omega}\right)\ \ +\ \ \mathbf{\Omega}\,\times \left(\tilde{I}\,\mathbf{\Omega}\right)\ =\ \tilde{I}\,\frac{d}{dt}\left(\mathbf{\Omega}\right)\ \ +\ \ \mathbf{\Omega}\,\times \left(\tilde{I}\,\mathbf{\Omega}\right)$$ Effect on 2nd derivatives: The acceleration is then (assuming $\mathbf{\Omega}$ is constant): $$\frac{d^2\mathbf{r}}{dt^2} = \frac{d^2r}{dt^2}+ 2\frac{dx_i}{dt}(\mathbf{\Omega}\times\mathbf{e}_i) + x_i(\mathbf{\Omega}\times(\mathbf{\Omega}\times \mathbf{e}_i))$$ Tidying up a bit we have: $$\frac{d^2\mathbf{r}}{dt^2} = \ddot{\mathbf{r}} + 2(\mathbf{\Omega} \times \dot{\mathbf{r}}) +\mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r})$$ The $2(\mathbf{\Omega} \times \dot{\mathbf{r}})$ term is called the Coriolis acceleration and the $\mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r})$ term is called the centripetal acceleration. What we have essentially is: Acceleration seen by inertial observer = Acceleration seen by rotating observer + extra terms

Commentary

 tiny-tim @ 04:48 PM Dec20-08 Added example of torque equation in a rotating frame.

 tiny-tim @ 01:06 PM Jul3-08 Changed title from "rotating reference frames" to "rotating frame" to take advantage of the autolinking facility.