Angular Velocity & Tangential Velocity

Consider a point $P$ on a rigid body rotating about a fixed axis. If the point is at radius $r$ from the axis and the body rotates through an angle $\theta$ (measured in radians), the arc length $s$ traced by $P$ is given by the definition of the radian: $$s = r\theta$$ This is the fundamental geometric link between linear distance and angle. Note that $\theta$ must be in radians for this equation to hold — degrees will not work directly.

To find velocities, differentiate both sides of $s = r\theta$ with respect to time $t$. Since $r$ is constant for a rigid body (the distance from axis to point does not change), it comes out of the derivative: $$\frac{ds}{dt} = r\frac{d\theta}{dt}$$ The left side $\frac{ds}{dt}$ is the rate at which arc length increases, which is precisely the tangential (linear) speed $v$ of point $P$. The right side contains $\frac{d\theta}{dt}$, which is the angular velocity $\omega$ (omega). We therefore obtain the key equation: $$v = r\omega$$

The speed $v$ gives only the magnitude of the velocity vector. The direction of $\vec{v}$ is always tangential to the circular path — perpendicular to the radius $r$ at every instant. In vector form, using the cross product, $\vec{v} = \vec{\omega} \times \vec{r}$, where $\vec{\omega}$ points along the rotation axis (by the right-hand rule). For most high-school problems it is sufficient to use the scalar form $v = r\omega$ and remember that the velocity is directed tangentially.

If $\omega$ is constant (uniform circular motion), the tangential speed $v$ is also constant, but the velocity direction continuously changes — producing centripetal acceleration $a_c = \frac{v^2}{r} = r\omega^2$ directed inward. If $\omega$ changes (non-uniform rotation), there is additionally a tangential acceleration $a_t = r\alpha$, where $\alpha = \frac{d\omega}{dt}$ is the angular acceleration. The total acceleration is the vector sum of these two perpendicular components.

For a rigid body, every point shares the same $\omega$ at any instant. However, because $v = r\omega$, a point at radius $2r$ moves at twice the tangential speed of a point at radius $r$. This explains why the outer edge of a vinyl record moves past the stylus much faster than the inner grooves, even though the disc spins at the same number of revolutions per minute everywhere. The mapping $v = r\omega$ is linear in $r$: tangential speed increases proportionally with distance from the axis.

• v — Tangential (linear) speed (m s⁻¹): The instantaneous speed of the point along the direction tangent to its circular path.
• r — Radius (m): The perpendicular distance from the rotation axis to the point of interest.
• \omega — Angular velocity (rad s⁻¹): The rate of change of angle of the rigid body; the same value for every point on the body at a given instant.

Problem. A bicycle wheel has a radius of 0.35 m and spins at 120 revolutions per minute (rpm). (a) Convert the angular speed to rad s⁻¹. (b) Find the tangential speed of a point on the rim. (c) Find the tangential speed of a reflector clipped to a spoke 0.20 m from the axle.

Answer. The rim moves at approximately 4.40 m/s; the reflector moves at approximately 2.51 m/s. Both share the same angular speed of 4π ≈ 12.57 rad/s.

• Angular Displacement and Radians — The derivation of $v = r\omega$ begins with the arc-length formula $s = r\theta$, which requires angles in radians. Understanding angular displacement is the essential prerequisite.
• Centripetal Acceleration — Once tangential speed $v$ is known, centripetal acceleration $a_c = v^2/r = r\omega^2$ follows directly, explaining why rotating points continuously accelerate inward even at constant speed.
• Angular Acceleration and Tangential Acceleration — The parallel relation $a_t = r\alpha$ extends the linear–rotational mapping from velocity to acceleration, completing the kinematic picture for non-uniform rotation.
• Moment of Inertia — Moment of inertia $I = \sum m_i r_i^2$ uses the same radial distances $r_i$ as the $v = r\omega$ relation, weighting each mass element by its position — showing how the distribution of mass affects rotational dynamics.
• Uniform Circular Motion — Uniform circular motion is the special case of rotation at constant $\omega$, where $v = r\omega$ gives a constant tangential speed and only direction changes, producing centripetal acceleration.