Integration in angular momentum

AI Thread Summary
The discussion centers on the integration of angular momentum, specifically the equation ##\int R dm = M R##. There is a concern about the variable nature of R, which changes over time, leading to questions about the validity of the equation. However, it is clarified that the integral represents a snapshot of mass distribution at a specific instant, thus not involving any dynamic changes. The conversation also touches on the preference for expressing angular momentum in terms of momentum density and mass density. Overall, the integration technique discussed is valid when considering instantaneous values rather than time-dependent changes.
Rikudo
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https://www.physicsforums.com/threa...f-a-translating-and-rotating-pancake.1005990/
So,I think I posted this in the wrong place. So, I will move it to here.
Here, in post #6, it is stated that ##\int R dm = M R##. As far as I know, R change from time to time and it is not constant. Hence, isn't it incorrect to say that ##\int R dm = M R##? Or, are there any techniques that are skipped?
 
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Though it is not an answer to your question but a matter of taste, I am not familiar with expression dm. I prefer to say it
\mathbf{L}=\int \mathbf{r} \times \mathbf{\pi}(\mathbf{r}) d^3\mathbf{r} =\int \mathbf{r} \times \rho(\mathbf{r}) \mathbf{v}(\mathbf{r}) d^3\mathbf{r}
where ##\pi(r)## is momentum density and ##\rho(r)## is mass density.
So now I know
dm = \rho(\mathbf{r}) d^3\mathbf{r}
 
Rikudo said:
Here, in post #6, it is stated that ##\int R dm = M R##. As far as I know, R change from time to time and it is not constant.
Right, but that integral has no moving parts. It is the integral over the mass at some instant.
 
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