# Co-rotating body frame

I have a question that has been bothering me for some time. Imagine we have a bunch of point masses and two frames, the laboratory frame and a frame wich is located at the center of mass of the bunch of point masses. Now imagine this entire bunch of point masses rotate (the point masses somehow interact with each other such that they cannot escape) and the frame, which is located at the center of mass, co-rotate with the entire bunch. The angular velocity of each point mass is the same (they rotate collectively).

My question: is it valid to say that the total angular momentum $$\sum_i m_i x_i\times\dot{x}_i$$ vanishes relative to such a frame, that is $$\sum_i m_i x_i\times\dot{x}_i=0$$ ? $$x_i$$ is the position vector of the mass $$m_i$$ relative to the co-rotating reference frame.

Since the frame you are speaking of co-rotates with all of the masses, these masses are at rest with respect to this frame, so $$\dot{\vec{x_i}} = 0$$, and hence the total angular momentum is zero in this frame.
• Ok this seem logic, but if we assume that $x_i$ can fluctuate, described by $u_i$, about an equilibrium $R_i$, that is $x_i=u_i+R_i$ then this statement does not hold, right? Jul 27, 2019 at 9:58