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.
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title Kinetic Flow: Visualizing Rotational Forces
Resulmation: Visualization of four quantities involving the motion of a rigid body
: Centrifugal Potential and Path
: Bead released from rest (relative to rod) on a frictionless rotating rod
: Orbital Motion (Real Space) and Sloshing in the Potential Well
IllustraDemo: Rotation Forces Using Divergence and Curl
Ex-Demo: The Vector Mechanics and Energetics of Rigid Body Rotation
Narr-graphic: The Dynamics of Rotating Rigid Bodies
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%% Proof and Derivation
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