Lessons · rotation

Rotational kinematics: the same calculus, angular labels

Regime 1 — the algebra is the calculus, evaluated. Step the algebra, step the calculus, and watch the algebra formula fall out of the integral. SymPy proves the two registers agree.

Regime 1 · algebra is calculus, evaluated Math machine-derived & checked by SymPy 2 modeling assumptions (author-asserted)

A flywheel starts from rest and spins up under a constant angular acceleration $\alpha = 3$ rad/s² for $4$ s. How fast is it turning, and how many revolutions has it made? The rotational kinematics are the straight-line equations with angular symbols — step them and watch the timeless equation fall out of the integrals.

Angle turned

\theta(T)

24 rad

Final angular velocity

\omega(T)

12 rad/s

Revolutions

\theta/2\pi

3.82 rev

Constant angular acceleration

\alpha = \text{const} \quad\Longrightarrow\quad \omega(t) = \omega_0 + \int_0^t \alpha\,dt' = \omega_0 + \alpha t

A constant net torque gives a constant angular acceleration. Integrate it once for the angular velocity — exactly as constant linear acceleration integrates to $v$ .

check $\theta' = \omega$ and $\omega' = \alpha$ ; show $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$ and $\Delta\theta = \tfrac12(\omega_0+\omega)t$ fall out

✓ Angular velocity is the integral of $\alpha$ : $\omega = d\theta/dt$ . [structural]
✓ Angular acceleration is constant: $\alpha = d\omega/dt$ . [structural]
✓ The memorized timeless equation $\omega^2 = \omega_0^2 + 2\alpha\,\Delta\theta$ falls out. [structural]
✓ The average-velocity relation $\Delta\theta = \tfrac12(\omega_0+\omega)\,t$ holds. [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “Rotational motion needs its own special set of equations to memorize.”

The rotational kinematics equations are the straight-line ones with angular symbols: $x\to\theta$ , $v\to\omega$ , $a\to\alpha$ . They come from the same two integrations of a constant acceleration — the slope of the $\omega$ – $t$ line is $\alpha$ , and the area under it is $\Delta\theta$ . Learn one set, relabel, and you have the other.

Modeling assumptions — author-asserted, disclosed not discharged

Constant angular acceleration (a constant net torque on a rigid body of fixed moment of inertia).
Rigid body; angles in radians; the axis of rotation is fixed.

The stacked graph, fully annotated

A static rendering (Matplotlib) at the default parameters — the interactive version is in the Graph tab above.

Formulas used

Hover a formula to preview its reference entry; click to open it in the reference (or the concept graph):