Kinematics and Vector Calculus of a Rotating Rigid Body

The motion of a rotating rigid body is characterized by a velocity field that is solenoidal ( $\nabla$. $v=0$ ), reflecting the incompressible nature of rigid motion, while its vorticity is exactly twice the angular velocity ( $\nabla \times v=2 \omega$ ). The acceleration field consists of both Euler and centripetal components, resulting in a constant negative divergence proportional to the square of the angular speed ( $\nabla \cdot a=-2 \omega^2$ ) and a curl that tracks the rate of change of the rotation $(\nabla \times a=2 \dot{\omega})$. Together, these results demonstrate that while the velocity describes the instantaneous rotation, the acceleration captures both the change in that rotation and the inward "pull" required to maintain the circular paths of the body's constituent points.

🎬Narrated Video

https://youtu.be/YLvijYirS1M

🏗️Structural clarification of Poof and Derivation

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