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# Dynamics of a Solid Body

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• 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: $$\vec{\omega^\prime} \times \vec{M} = \vec{N}.$$

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