Angular Velocity and its Relation with Linear Velocity

The figure shows a rigid body rotating about a fixed vertical axis — the $z$-axis. The origin $O$ is placed somewhere on that axis. The $y$-axis points to the right, and the $x$-axis points down and to the left (a standard right-handed coordinate system). A dashed horizontal circle is drawn with its centre $C$ on the $z$-axis; this circle lies in a plane perpendicular to the axis of rotation.

A particle $P$ of the rigid body is shown on that circle. The radius of the circle — the perpendicular distance from $P$ to the axis — is labelled $r$. A nearby position $P'$ is also marked, separated from $P$ by a small angular displacement $\Delta \theta$. The velocity vector $\mathbf{v}$ of the particle is drawn tangent to the circle at $P$, perpendicular to the radius $CP$.

The physical idea is straightforward: when a rigid body rotates about a fixed axis, every particle moves in a circle whose centre lies on the axis. The particle's linear velocity is not the same for all particles — it depends on how far the particle is from the axis. But every particle shares the same angular velocity $\boldsymbol{\omega}$, which points along the axis of rotation (upward for counterclockwise rotation, by the right-hand rule).

The key relation the textbook develops from this figure is the connection between angular velocity and linear velocity:

Here $\mathbf{r}$ is the position vector of the particle taken from any point on the axis (often from $C$ or from $O$). The magnitude of this cross product gives $v = \omega r \sin\phi$, where $\phi$ is the angle between $\boldsymbol{\omega}$ and $\mathbf{r}$. Since $\boldsymbol{\omega}$ is along the axis and $\mathbf{r}$ is perpendicular to the axis for a particle on the circle, $\phi = 90^\circ$ and $\sin\phi = 1$, so:

The direction of $\mathbf{v}$ is given by the right-hand rule: curl the fingers of your right hand in the direction of rotation; your thumb points along $\boldsymbol{\omega}$. For a particle, $\mathbf{v}$ is tangent to the circle, perpendicular to both $\boldsymbol{\omega}$ and $\mathbf{r}$.

The figure has two panels, (a) and (b), and they serve two different but connected purposes.

Panel (a) establishes the direction of angular velocity $\boldsymbol{\omega}$. It shows two shaded disks, each rotating in opposite senses. For each disk, the vector $\boldsymbol{\omega}$ is drawn along the axis of rotation. The key idea is the right-handed screw rule: if you curl the fingers of your right hand in the direction of rotation, your thumb points along $\boldsymbol{\omega}$. So one disk rotates clockwise when viewed from above, and its $\boldsymbol{\omega}$ points downward; the other rotates anticlockwise, and its $\boldsymbol{\omega}$ points upward. The figure makes it clear that $\boldsymbol{\omega}$ is not a "spin" that lives on the rim — it is an axial vector that lies along the axis of rotation, with its sense determined by the rotation sense.

Panel (b) moves from the axis to a single point on the rotating body. It shows a $z$-axis with $\boldsymbol{\omega}$ drawn along it. A point $P$ is located on the body. The centre $C$ is the foot of the perpendicular from $P$ to the axis — so $C$ lies on the axis, and the line $CP$ is perpendicular to the axis. The distance $CP$ is labelled $r_\perp$ (the perpendicular distance from the axis to the point). The vector from the origin $O$ (some fixed point on the axis) to $P$ is $\mathbf{r}$. A dashed circle is drawn in the plane perpendicular to the axis, centred at $C$ and passing through $P$ — this is the circular path that $P$ follows as the body rotates. Finally, the velocity $\mathbf{v}$ is shown as a vector tangent to this circle at $P$.

The physical idea is that every point in a rigid body rotating with angular velocity $\boldsymbol{\omega}$ moves in a circle around the axis. The linear velocity of a point is not simply $\omega r$; it depends on the perpendicular distance from the axis, not on the distance from the origin.

Here $\boldsymbol{\omega}$ is the angular velocity vector (direction along the axis, magnitude $\omega$), $\mathbf{r}$ is the position vector of the point from any origin on the axis, and $\mathbf{v}$ is the instantaneous linear velocity of that point. The cross product ensures three things: (1) $\mathbf{v}$ is perpendicular to both $\boldsymbol{\omega}$ and $\mathbf{r}$; (2) its magnitude is $v = \omega \, r \sin\theta$, where $\theta$ is the angle between $\boldsymbol{\omega}$ and $\mathbf{r}$. But since $\boldsymbol{\omega}$ is along the axis and $\mathbf{r}$ makes an angle $\theta$ with it, $r \sin\theta$ is exactly the perpendicular distance $r_\perp$ from the axis to the point. So $v = \omega \, r_\perp$, which matches the geometry of the dashed circle in panel (b). (3) The direction of $\mathbf{v}$ is tangent to the circle, as shown. …