Which of the following is the correct relation between...

Which of the following is the correct relation between linear relocity $\bar{v}$ and angular velocity $\vec{\omega}$ of a particle? (a) $\overline{\boldsymbol{v}}=\overline{\bar{F}} \times \bar{\omega}$ (b) $\vec{v}=\overrightarrow{(\dot{a}} \times \overline{\boldsymbol{r}}$ (c) $\overrightarrow{\mathbf{u}}=\bar{r} \times \vec{v}$ (d) $\overrightarrow{\mathbf{g}}=\overline{\bar{v}} \times \vec{r}$

Which of the following is the correct relation between linear relocity $\bar{v}$ and angular velocity $\vec{\omega}$ of a particle?
(a) $\overline{\boldsymbol{v}}=\overline{\bar{F}} \times \bar{\omega}$
(b) $\vec{v}=\overrightarrow{(\dot{a}} \times \overline{\boldsymbol{r}}$
(c) $\overrightarrow{\mathbf{u}}=\bar{r} \times \vec{v}$
(d) $\overrightarrow{\mathbf{g}}=\overline{\bar{v}} \times \vec{r}$

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Key Concepts

Vector Cross Product Relation

The relation between linear velocity and angular velocity in rotational motion is given by the cross product, expressed as v = ? x r. This mathematical operation indicates that the linear velocity is perpendicular to both the angular velocity vector and the position vector from the axis of rotation. The cross product encapsulates both the magnitude and the directional properties necessary for proper description of rotational kinematics.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object about an axis. It gives both the magnitude of the rotational speed and the direction of the axis of rotation, following the right-hand rule. This concept is fundamental in dynamics and kinematics when analyzing rotational motion in both rigid bodies and particles moving along curved paths.

Linear Velocity

Linear velocity is a vector that denotes the rate of change of an object's position with respect to time. In the context of rotational motion, it represents the speed at which a point on a rotating body moves along its circular path. Its direction is tangential to the path, reflecting the instantaneous motion of the particle.

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