Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass :
(a) Show $\mathbf{p}=\mathbf{p}_{i}^{\prime}+m_{i} \mathbf{V}$
where $\mathbf{p}_{i}$ is the momentum of the ith particle (of mass $m$, and $\mathbf{p}_{i}^{\prime}=m_{l} \mathbf{v}_{i}^{\prime}$. Note $\mathbf{v}_{i}^{\prime}$ is the velocity of the ith particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum \mathbf{p}_{t}^{\prime}=0$
(b) Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles, $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. $7.14$.
(c) Show $\mathbf{L}=\mathbf{L}^{\prime}+\mathbf{R} \times \boldsymbol{M} \mathbf{V}$
where $\mathbf{L}^{\prime}=\sum \mathbf{r}_{t}^{\prime} \times \mathbf{p}_{i}^{\prime}$ is the angular momentum of the system about the centre of mass with
velocities taken relative to the centre of mass. Remember $\mathbf{r}_{i}^{\prime}=\mathbf{r}_{i}-\mathbf{R}$ : rest of the notation is the standard notation used in the chapter. Note $\mathbf{L}^{\prime}$ and $M \mathbf{R} \times \mathbf{V}$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
(d) Show $\frac{d \mathbf{L}^{\prime}}{d t}=\sum \mathbf{r}_{i}^{\prime} \times \frac{d \mathbf{p}^{\prime}}{d t}$
Further, show that $\frac{d \mathbf{L}^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\boldsymbol{\tau}_{\text {eut }}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint : Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)