Kinetic Energy of Particle Systems
Particle System’s Kinetic Energy1
The kinetic energy of a particle system, like the linear momentum and angular momentum we defined before, can also be naturally defined as the sum of the kinetic energy of each particle.
$$ \begin{equation} T=\sum \limits _{i=1} ^{n} \frac{ 1}{ 2 }m_{i}v_{i}^{2} \label{kinetic} \end{equation} $$
Now, we will do the same operation for the particle system’s linear and angular momentum, representing each particle’s position vector with respect to the center of mass as shown in the figure below.
$$ \mathbf{r}_{i}=\mathbf{r}_{cm}+\overline{\mathbf{r}}_{i} $$
Differentiating this with respect to time gives the following result.
$$ \mathbf{v}_{i}=\mathbf{v}_{cm}+\overline{\mathbf{v}}_{i} $$
Substituting this into $\eqref{kinetic}$ yields the following.
$$ \begin{align*} T &= \sum \limits _{i=1} ^{n}\textstyle{\frac{1}{2}}m_{i}(\mathbf{v}_{i}\cdot \mathbf{v}_{i}) \\ &= \sum \limits _{i=1} ^{n}\textstyle{\frac{1}{2}}m_{i}(\mathbf{v}_{cm}+\overline{\mathbf{v}}_{i})\cdot (\mathbf{v}_{cm}+\overline{\mathbf{v}}_{i}) \\ \ &= \sum \limits _{i=1} ^{n} \textstyle{\frac{1}{2}}m_{i}v_{cm}^{2}+\sum \limits _{i=1} ^{n}m_{i}(\mathbf{v}_{cm}\cdot \overline{\mathbf{v}}_{i})+\sum \limits _{i=1} ^{n}\textstyle{\frac{1}{2}}m_{i}\overline{v}_{i}^{2} \\ &= \textstyle{\frac{1}{2}}v_{cm}^{2} \sum \limits _{i=1} ^{n}m_{i}+\mathbf{v}_{cm}\cdot \left( \sum \limits _{i=1} ^{n}m_{i}\overline{\mathbf{v}}_{i}\right)+\sum \limits _{i=1} ^{n}\textstyle{\frac{1}{2}}m_{i}\overline{v}_{i}^{2} \end{align*} $$
Here, the parentheses of the second term refer to $\mathbf{0}$.2 Therefore, the kinetic energy of the particle system is as follows.
$$ T=\textstyle{\frac{1}{2}}mv_{cm}^{2} + \sum \limits _{i=1} ^{n}\frac{1}{2}m_{i}\overline{v}_{i}^{2} $$
The first term is the kinetic energy with respect to the center of mass. The second term is the kinetic energy of each particle with respect to the center of mass. Understanding the kinetic energy in terms of terms related to the center of mass and terms relative to the center of mass when referenced is helpful in many parts of physics.