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Dynamics of a Solid Body

  • Equation of dynamics of a solid body rotating about a stationary axis $ z $:
    $$I \beta_z = N_z,$$

    where $ N_z $ is the algebraic sum of the moments of external forces relative to the $ z $ axis.

  • According to Steiner's theorem:
    $$ I = I_C + m a^2.$$
  • Kinetic energy of a solid body rotating about a stationary axis:
    $$T = \frac{1}{2}I \omega^2.$$
  • Work performed by external forces during the rotation of a solid body about a stationary axis:
    $$A = \int N_z d\phi.$$
  • Kinetic energy of a solid body in plane motion:
    $$T = \frac{I_C \omega^2}{2} + \frac{mv_C^2}{2}.$$
  • Relationship between the angular velocity $ \vec{\omega^\prime} $ of a gyroscope precession, its angular momentum $ \vec{M} $ equal to $ I \vec{\omega} $, and the moment $ \vec{N} $ of the external forces:
    $$\vec{\omega^\prime} \times \vec{M} = \vec{N}.$$
Keywords: 
dynamics, solid body, moment,
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