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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$\gg$Mathematical Structures Underlying Physical Laws
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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.
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.
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.
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.
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
visualize and analyze the motion of a three-dimensional rigid body
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Proving the Cross Product Rules with the Levi-Civita Symbol
Proving the Epsilon-Delta Relation and the Bac-Cab Rule
Simplifying Levi-Civita and Kronecker Delta Identities
Why a Cube's Diagonal Angle Never Changes
How the Cross Product Relates to the Sine of an Angle
Finding the Shortest Distance and Proving Orthogonality for Skew Lines
A Study of Helical Trajectories and Vector Dynamics
The Power of Cross Products: A Visual Guide to Precessing Vectors
Divergence and Curl Analysis of Vector Fields
Unpacking Vector Identities: How to Apply Divergence and Curl Rules
Commutativity and Anti-symmetry in Vector Calculus Identities
Double Curl Identity Proof using the epsilon-delta Relation
The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
Surface Parametrisation and the Verification of the Gradient-Normal Relationship
Proof and Implications of a Vector Operator Identity
Conditions for a Scalar Field Identity
Solution and Proof for a Vector Identity and Divergence Problem
Kinematics and Vector Calculus of a Rotating Rigid Body
Work Done by a Non-Conservative Force and Conservative Force
The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
Divergence Theorem Analysis of a Vector Field with Power-Law Components
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