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Mechanical Engineering QUESTION #2530
Question 1
The transport theorem relates the total time derivative of a vector \(\mathbf{r}\) observed from a fixed frame to the local derivative in a rotating frame with angular velocity \(\boldsymbol{\omega}\) as:
  • \(\frac{d\mathbf{r}}{dt}=\frac{\partial\mathbf{r}}{\partial t}-\boldsymbol{\omega}\times\mathbf{r}\)
  • \(\frac{d\mathbf{r}}{dt}=\frac{\partial\mathbf{r}}{\partial t}+\boldsymbol{\omega}\times\mathbf{r}\)✔️
  • \(\frac{d\mathbf{r}}{dt}=\boldsymbol{\omega}\times\mathbf{r}\) only
  • \(\frac{d\mathbf{r}}{dt}=\frac{\partial^2\mathbf{r}}{\partial t^2}+\boldsymbol{\omega}\times\mathbf{r}\)
Correct Answer Explanation
The transport theorem: \(\frac{d\mathbf{r}}{dt}=\frac{\partial\mathbf{r}}{\partial t}+\boldsymbol{\omega}\times\mathbf{r}\). The \(\partial/\partial t\) term is the rate as seen from the rotating frame; \(\boldsymbol{\omega}\times\mathbf{r}\) accounts for the frame rotation.