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.)