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When considering a system of particles with respect to a fixed point O, then the total angular momentum around that point can be writtin as

[tex] \textbf{L}_O = \sum_i \textbf{r}_i\times m_i\textbf{v}_i [/tex]

However in my book they add and substract the vektor [itex] \sum_i \textbf{R}_{CM}\times m_i\textbf{v}_i [/itex], which gives,

[tex] \textbf{L}_O = \sum_i (\textbf{r}_i-\textbf{R}_{CM})\times m_i\textbf{v}_i + \sum_i \textbf{R}_{CM}\times m_i\textbf{v}_i[/tex]

It is then stated that the first term in the above equation, resembles the angular momentum around CM, and the second term "the angular momentum of CM", i.e.

[tex] \textbf{L}_O = \textbf{L}_{CM} + \textbf{R}_{CM}\times \textbf{P} [/tex]

I understand the second term, but the first term seems a little confusing. How can the 1st term be considered as the angular momentum around CM? Indeed the vector [itex] \textbf{r}_i -\textbf{R}_{CM} [/itex] points from CM to the particle

[tex] \textbf{L}_O = \sum_i \textbf{r}_i\times m_i\textbf{v}_i [/tex]

However in my book they add and substract the vektor [itex] \sum_i \textbf{R}_{CM}\times m_i\textbf{v}_i [/itex], which gives,

[tex] \textbf{L}_O = \sum_i (\textbf{r}_i-\textbf{R}_{CM})\times m_i\textbf{v}_i + \sum_i \textbf{R}_{CM}\times m_i\textbf{v}_i[/tex]

It is then stated that the first term in the above equation, resembles the angular momentum around CM, and the second term "the angular momentum of CM", i.e.

[tex] \textbf{L}_O = \textbf{L}_{CM} + \textbf{R}_{CM}\times \textbf{P} [/tex]

I understand the second term, but the first term seems a little confusing. How can the 1st term be considered as the angular momentum around CM? Indeed the vector [itex] \textbf{r}_i -\textbf{R}_{CM} [/itex] points from CM to the particle

*i*, but how about the velocity vector term [itex] \textbf{v}_i [/itex] of particle*i*in [itex] \textbf{L}_{CM} [/itex], doesn't that need to be the velocity of the particle*i*__relative__to the CM, before it can be considered as the angular momentum around CM?
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