Consider a system of particles with masses $m_{i}$ and position vectors $\mathbf{x}_{i}$. Write down the definition of the position of the centre of mass $\mathbf{R}$ of the system. Let $T$ be the total kinetic energy of the system. Show that

$$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} \sum_{i} m_{i} \dot{\mathbf{y}}_{i} \cdot \dot{\mathbf{y}}_{i}$$

where $M$ is the total mass and $\mathbf{y}_{i}$ is the position vector of particle $i$ with respect to $\mathbf{R}$.

The particles are connected to form a rigid body which rotates with angular speed $\omega$ about an axis $\mathbf{n}$ through $\mathbf{R}$, where $\mathbf{n} \cdot \mathbf{n}=1$. Show that

$$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} I \omega^{2},$$

where $I=\sum_{i} I_{i}$ and $I_{i}$ is the moment of inertia of particle $i$ about $\mathbf{n}$.