# Homework Help: Derivation of non-inertial terms in non-inertial systems using rotation matrices

1. Jan 17, 2010

### andresordonez

(I know how to do this without the rotation matrices)
Any suggestion would be much appreciated.

1. The problem statement, all variables and given/known data
Show that the relationship between the forces in the inertial (S') and non-inertial(S) reference frames, with a coordinate transformation given by

$$\vec{r}=R \vec{r'}$$

is:

$$\vec{F'} = \vec{F} + m(\vec{\omega} \times (\vec{\omega} \times \vec{r}) + 2\vec{\omega} \times \vec{v})$$

2. Relevant equations
$$$R = \left( \begin{array}{ccc} \cos(\omega t) & \sin(\omega t) & 0 \\ -\sin(\omega t) & \cos(\omega t) & 0 \\ 0 & 0 & 1 \end{array} \right).$ $R^{-1} = R^t = \left( \begin{array}{ccc} \cos(\omega t) & -\sin(\omega t) & 0 \\ \sin(\omega t) & \cos(\omega t) & 0 \\ 0 & 0 & 1 \end{array} \right)$ $-\vec{\omega} \times (\vec{\omega} \times \vec{r}) = -\omega^2 \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \vec{r}$ $\vec{\omega} \times \vec{v} = -\omega \left( \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \vec{v}$$$

3. The attempt at a solution

$$\ddot{\vec{r}} = R\ddot{\vec{r'}} + 2\dot{R}\dot{\vec{r'}} + \ddot{R}\vec{r'} = \ddot{R}R^{-1}\vec{r} + 2\dot{R}(\dot{R^{-1}}\vec{r}+R^{-1}\dot{\vec{r}}) + R\ddot{\vec{r'}}$$

$$$\ddot{R}R^{-1}\vec{r} = -\omega^2 \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \vec{r}$$$
$$$\dot{R}(R^{-1}\vec{r} + R^{-1}\dot{\vec{r}}) = \omega^2 \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \vec{r} - \omega \left( \begin{array}{ccc} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0\end{array} \right) \dot{\vec{r}}$$$

then

$$\ddot{\vec{r}} = -\vec{\omega} \times (\vec{\omega} \times \vec{r}) - 2\vec{\omega} \times \dot{\vec{r}} + R\ddot{\vec{r'}}$$
$$R\vec{F'} = \vec{F} + m\vec{\omega}\times(\vec{\omega}\times\vec{r}) + 2m \vec{\omega} \times \dot{\vec{r}}$$

How do I get rid of R??