Chapter 14.1, Problem 14.29P

Consider the frame of reference Ax'y'z' in translation with respect to the newtonian frame of reference Oxyz. We define the angular momentum $H_A^{\text{'}}$ of a system of n particles about A as the sum $H_A^{\text{'}}={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(r_i^{\text{'}}\times m_i{v}_i^{\text{'}}\right)}$ of the moments about A of the momenta $m_i{v}_i^{\text{'}}$ of the particles in their motion relative to the frame Ax'y'z'. Denoting by $H_A$ the $H_A={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(r_i^{\text{'}}\times m_i{v}_i\right)}$ of the moments about A of the momenta $m_i{v}_i$ of the particles in their motion relative to the newtonian frame Oxyz, show that $H_A=H_A^{\text{'}}$ at a given instant if, and only if, one of the following conditions is satisfied at that instant: (a) A has zero velocity with respect to the frame Oxyz, (b) A coincides with the mass center G of the system, (c) the velocity relative to Oxyz is directed along the line AG.