Why is Term 4 Non-Zero in the Center of Mass Frame?

[aim1732]

For general motion of a rigid body :
[Vectors in bold]
L = Σ mir_i × v_i
= Σ mi(r₀+r_i,cm)×(v_o+v_i,cm)
= Σmir₀×v_o + Σr₀×(m_iv_i,cm) + Σ(m_ir_i,cm)×v_o + Σm_ir_i,cm×v_i,cm ...[1]
If the centre of coordinate system is at the centre of mass then by definition of centre of mass: Σm_ir_i=0 and Σm_iv_i,cm=0.
Now here's the the problem: terms 3 and 4 in [1] are zero in the centre of mass frame but the term 4 is not.But then using the same arguments I can say it is zero too. I am missing a point but can not point out what.

 

[betel]

I'm sure you mean term 2 and 3 are zero in [1].
Term 4 is not zero because you have to sum over both, r and v simultaneously.

 

[aim1732]

Heck that was easy.Thank you very much!