3.10: Kinetic energy

We remind ourselves that we are discussing particles, and that all kinetic energy is translational kinetic energy.

Notation:

• $T_{C}$ = kinetic energy with respect to the centre of mass C.
• $T$ = kinetic energy with respect to the origin O.

Theorem:

$$T = T_{C} + \frac{1}{2}M\overline{v}^{2}\tag{3.10.1}\label{eq:3.10.1}$$

Thus:

$T = \frac{1}{2}\sum m_{i}{v}^{2}_{i} = \frac{1}{2} \sum m_{i} ({\bf v} ^{\prime}_{i} + \overline{{\bf v} })\cdot ({\bf v} ^{\prime}_{i} + \overline{{\bf v} })$

$= \frac{1}{2}\sum m{v}^{\prime 2}_{i} \times \overline{{\bf v} } \sum m{{\bf v} }^{\prime}_{i} + \frac{1}{2} v^{-2} \sum m_{i}$.

$\therefore \qquad T = T_{C} + \frac{1}{2}M\overline{v}^{2}$.

Corollary:

If $\overline{{\bf v} } = 0, T = T_{C}$. (Think about what this means.)

Corollary:

Corollary: For a non-rotating rigid body, $T_{C}$ = 0, and therefore $T = \frac{1}{2}M\overline{v}^{2}$

(Think about what this means.)