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3.10: Kinetic energy

  • Page ID
    8382
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    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.)


    This page titled 3.10: Kinetic energy is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.