Kinematics and Vector Calculus of a Rotating Rigid Body

Rigid body motion is characterized by a total acceleration composed of tangential and centripetal components. A key property of rigid bodies is that the divergence of their velocity field is always zero, indicating incompressible motion where the body doesn't expand or contract. The curl of the velocity field is twice the angular velocity, illustrating the relationship between overall angular motion and the local swirling of particles within the body. Additionally, the divergence of the acceleration field is also always zero, directly stemming from its cross-product derivation.

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✍️Mathematical Proof

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The Total Acceleration has Two Components

A rigid body's total acceleration is a sum of two distinct parts: tangential acceleration, which arises from a change in the angular velocity ( $d \omega / d t$ ), and centripetal acceleration, which is always directed towards the axis of rotation and is responsible for the object's circular path.

Divergence of Velocity is Always Zero

The divergence ( $\nabla \cdot v$ ) of a rigid body's velocity field is always zero. This is a fundamental property of rigid body motion and signifies that the body is incompressible-it's not expanding or contracting at any point.

Curl of Velocity is Twice the Angular Velocity

The curl of the velocity field ( $\nabla \times v$ ) is always equal to twice the angular velocity ( $2 \omega$ ). This quantity measures the local rotation of the fluid or body and provides a direct link between the overall angular motion of the body and the microscopic swirling motion of its constituent particles.

Divergence of Acceleration is Always Zero

The divergence of the acceleration field ( $\nabla \cdot a$ ) is also always zero. This is a direct consequence of the fact that the acceleration is derived from cross products, similar to the velocity field.

🎬Demonstration

The divergence of a rigid body's velocity field ( $\nabla \cdot v$ ) is always zero. This demonstrates that the body's motion is incompressible, meaning it's not expanding or contracting. The curl of the velocity field ( $\nabla \times v$ ) is always equal to twice the angular velocity vector ( $2 \omega$ ). This shows that the local rotation at any point in the body is directly proportional to the overall rotation rate.

visualize and analyze the motion of a three-dimensional rigid body