Statement 1 If a rigid body is in general plane motion, no...

Statement 1 If a rigid body is in general plane motion, no two points of the body can have the same velocity vector. and Statement 2 Velocity of any point of the body whose position vector with respect to $\mathrm{CM}$ is $\overline{\mathrm{r}}$ is given by $\overline{\mathrm{a}}+\overline{\mathrm{r}} \times \overline{\mathrm{b}}$ where $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are constant vectors.

Statement 1
If a rigid body is in general plane motion, no two points of the body can have the same velocity vector.
and
Statement 2 Velocity of any point of the body whose position vector with respect to $\mathrm{CM}$ is $\overline{\mathrm{r}}$ is given by $\overline{\mathrm{a}}+\overline{\mathrm{r}} \times \overline{\mathrm{b}}$ where $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are constant vectors.

Key Concepts

Rigid Body Kinematics

This concept deals with the motion of bodies where the distances between all points remain constant. In such systems, the overall movement is described by translations (movement of a reference point) and rotations (movement about an axis or center), ensuring that the body’s shape does not change during motion.

Decomposition of Motion into Translation and Rotation

Any motion of a rigid body can be divided into a translational component (common to all points) and a rotational component (varying with position relative to a reference point). This decomposition helps in expressing the velocity of any point on the body as the vector sum of the translational velocity and a term involving the cross product of the angular velocity with the position vector.

Velocity Field Representation in Rigid Bodies

The velocity field of a rigid body describes how the velocity varies from point to point within the body. It is typically expressed as the velocity at a reference point plus the cross product of the angular velocity vector with the relative position vector. This representation captures the spatial variation in velocity due to rotational motion.

Angular Velocity and Cross Product Formulation

Angular velocity is a vector quantity that represents the rate and axis of rotation. In the cross product formulation, it is used to determine the rotational contribution to the velocity at any point on the body by taking the cross product with the position vector relative to the center of rotation. This fundamentally links the rotational motion to the linear velocities observed in different parts of the body.

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