Angular Displacement & Arc Length

A radian is the angle subtended at the center of a circle when the arc length equals the radius. Formally, $\theta = s/r$, so $\theta = 1$ rad when $s = r$. Because a full circle has circumference $2\pi r$, a complete revolution equals $2\pi r / r = 2\pi$ rad $\approx 6.283$ rad. Converting: $360^\circ = 2\pi$ rad, so $1^\circ = \pi/180$ rad. Always convert degrees to radians before using rotational equations.

Angular displacement $\Delta\theta = \theta_f - \theta_i$ is positive for counterclockwise rotation and negative for clockwise rotation (by convention in a standard coordinate system). It is measured in radians and, unlike arc length, does not depend on which point on the body you observe — every point on a rigid body shares the same $\Delta\theta$ at every instant. This is the fundamental advantage of the rotational description.

From the definition of a radian, $\theta = s/r$, we immediately rearrange to get $s = r\theta$. Here $s$ is the arc length (in meters) traveled by a point at perpendicular distance $r$ from the rotation axis, and $\theta$ must be in radians. This is not an approximation — it is an exact geometric identity. For example, if a disk of radius $0.5$ m rotates by $4$ rad, the rim travels $s = (0.5)(4) = 2.0$ m exactly.

The $s = r\theta$ relationship is the first entry in a systematic analogy between linear and rotational quantities. Linear displacement $x$ maps to angular displacement $\theta$; linear velocity $v$ maps to angular velocity $\omega = d\theta/dt$; linear acceleration $a$ maps to angular acceleration $\alpha = d\omega/dt$. Every linear kinematic formula has a rotational twin: e.g., $v = r\omega$ (derived by differentiating $s = r\theta$ with respect to time), and $a_t = r\alpha$ for the tangential component of linear acceleration.

The formula $s = r\theta$ holds only when $\theta$ is expressed in radians. The radian is a dimensionless ratio (meters/meters), which is why the equation is dimensionally consistent: $[s] =$ m, $[r] =$ m, $[\theta] =$ m/m (dimensionless). If you mistakenly use degrees, you introduce a factor of $\pi/180$ error. All subsequent rotational formulas — $v = r\omega$, kinetic energy $KE = \frac{1}{2}I\omega^2$ — inherit this requirement.

s_{\text{rim}} \approx 0.916\text{ m}, \quad s_{\text{inner}} \approx 0.524\text{ m}

• Angular Velocity & Linear Speed — Differentiating $s = r\theta$ with respect to time gives $v = r\omega$, the direct link between angular velocity and the linear speed of a point on a rotating body.
• Angular Acceleration & Tangential Acceleration — A second time-derivative of $s = r\theta$ yields $a_t = r\alpha$, connecting angular acceleration to the tangential linear acceleration of points on a rigid body.
• Centripetal Acceleration — Points moving along circular arcs experience centripetal acceleration $a_c = v^2/r = r\omega^2$, which depends on the same radius r that appears in the arc length formula.
• Rotational Kinematics Equations — The kinematic equations for constant angular acceleration ($\theta = \omega_0 t + \frac{1}{2}\alpha t^2$, etc.) are the rotational twins of the linear kinematic equations, built on the same angular displacement concept.
• Rigid Body Rotation — The fact that every point in a rigid body shares the same angular displacement is the defining property of rigid body motion, making $s = r\theta$ universally applicable within such objects.