Velocities in inertial and rotating frames of reference

AI Thread Summary
The discussion centers on understanding velocities in inertial and rotating frames of reference, specifically regarding the equation that relates these velocities. The user questions whether the position vector \(\mathbf{r}\) should be interpreted in the inertial or rotating frame and if the velocity derived in the rotating frame is already represented correctly or needs further adjustment. Additionally, the user references a post that clarifies the derivation of the initial equation but raises concerns about the time derivative of a rotation matrix and its implications for matrix multiplication. The user is confused about the non-commutative nature of matrix products and how it affects the transformation of vectors between frames. The conversation highlights the complexities of frame transformations in physics.
ryan88
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Hi,

I have a couple of questions about velocities in inertial and rotating frames of reference, related by the following equation:

\mathbf{v_i} \ \stackrel{\mathrm{def}}{=}\ \frac{d\mathbf{r}}{dt} = <br /> \left( \frac{d\mathbf{r}}{dt} \right)_{\mathrm{r}} + <br /> \boldsymbol\Omega \times \mathbf{r} = <br /> \mathbf{v}_{\mathrm{r}} + \boldsymbol\Omega \times \mathbf{r}

  1. \mathbf{v_i} and \mathbf{v_r} both state which frame of reference they are measured in, however \mathbf{r} does not. Is this supposed to be in the inertial or rotating frame of reference?
  2. If I use the equation to find the velocity in the rotating frame, does this mean that the value is represented in the rotating frame of reference? Or is it that the magnitude of that velocity is correct, but it still needs to be rotated to the rotating frame of reference?

Thanks,

Ryan
 
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After taking a closer look at the post I linked to, I have another question. I thought that the time derivative of a rotation matrix was given by:

\frac{\mathrm{d}R}{\mathrm{d}t} = \tilde{\omega}R

However, in his post, D H states:

\mathbf T&#039;_{R\to I} = \mathbf T_{R\to I}\mathbf X(\mathbf \omega)

Since matrix products are non commutative, doesn't this make the following incorrect?

\mathbf q&#039;_I = <br /> \mathbf T_{R\to I}(\mathbf X(\mathbf \omega)\mathbf q_R + \mathbf q&#039;_R) <br /> = <br /> \mathbf T_{R\to I}(\mathbf \omega\times\mathbf q_R + \mathbf q&#039;_R)
 
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