Lessons · shm

A mass on a spring

Regime 2 — calculus does what algebra cannot. The acceleration isn't constant, so the algebra formulas don't apply; calculus is the only road in, and SymPy proves the closed form solves the equation of motion.

Regime 2 · calculus does more Math machine-derived & checked by SymPy 3 modeling assumptions (author-asserted)

A 1 kg mass on a spring (k = 4 N/m) is pulled 0.3 m from equilibrium and released from rest. How does it move — and where does the period the algebra course made you memorize actually come from?

Angular frequency

\omega = \sqrt{k/m}

2 rad/s

Period

T = 2\pi/\omega

3.142 s

Amplitude

0.3 m

Maximum speed

\omega A

0.6 m/s

Newton's second law with Hooke's law

m\,x'' = -k\,x \;\;\Longrightarrow\;\; x'' = -\omega^2 x,\quad \omega^2 = \tfrac{k}{m}

The spring force is proportional to displacement and points back toward equilibrium. This is the equation of motion — a differential equation, not an algebra formula.

back-substitute $x(t)$ into $x'' = -\omega^2 x$ ; check the initial conditions; check $\tfrac{d}{dt}\,\mathrm{energy} = 0$ ; check periodicity

✓ The closed form solves the equation of motion $x'' = -\omega^2 x$ . [structural]
✓ It matches the initial position $x(0) = x_0$ . [structural]
✓ It matches the initial velocity $x'(0) = v_0$ . [structural]
✓ Total energy $\tfrac12 v^2 + \tfrac12 \omega^2 x^2$ is conserved (its time-derivative is zero). [structural]
✓ The memorized period $T = 2\pi/\omega$ falls out: $x(t+T) = x(t)$ . [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “The mass moves fastest at the ends of its swing (the turning points).”

At a turning point the speed is zero — that is where it reverses. It moves fastest through equilibrium ( $x = 0$ ), where the v-t graph peaks a quarter-period out of phase with x-t. Acceleration, $a = -\omega^2 x$ , is largest at the ends and zero in the middle.

Modeling assumptions — author-asserted, disclosed not discharged

Ideal spring obeying Hooke's law: the restoring force is proportional to displacement.
No friction or air resistance (undamped motion).
The mass is a point and the spring is massless.

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):