CLS-15

Simple harmonic period

T = 2π √(m/k) for a mass on a Hookean spring.

Reading speed

NewtonianSHM

Governing equation

T=2\pi\sqrt{m/k}

$m \in \left[0.05, 8\right]$

where

m: Mass (kg)
k: Stiffness (N/m)
T: Period (s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

T=2\pi\sqrt{m/k}

$m \in \left[0.05, 8\right]$

$m = 0.05$ $T = 0.222$

Along the curve

②

Domain of the variable — blue

m_{\min} = 0.05,\quad m_{\max} = 8

$m \in \left[0.05, 8\right]$

$m = 0.05$ $m = 0.05$

Along the curve

③

Response curve — green

T=2\pi\sqrt{m/k}

$m \in \left[0.05, 8\right]$

$m = 0.05$ $T = 0.222$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T}{\mathrm{d}m}

$m \in \left[0.05, 8\right]$

$m = 0.1$ $T^{\prime} = 1.626$

Along the curve

Lecture brief

Historical brief

Newtonian mechanics (1687) plus energy, angular momentum, Kepler and the ballistic parabola remain the first language of motion. Every sheet here is a closed-form orbit, throw, spin or oscillator. This sheet (CLS-15 — Simple harmonic period) is the form associated with SHM. Working symbols: $m$ , $k$ $\rightarrow$ $T$ . The ODE is ẍ + (k/m) x = 0. ω = √(k/m) is independent of amplitude.

Purpose

Purpose: compute

T

from

m

k

in Classical mechanics via

T=2\pi\sqrt{m/k}

T = 2π √(m/k) for a mass on a Hookean spring. Use it when a real classical mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

m = 0.400\,\mathrm{kg}

k = 40.000\,\mathrm{N/m}

, the governing relation

T=2\pi\sqrt{m/k}

yields

T = 0.628\,\mathrm{s}

. A spring, a mass, a ticking period. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass m0.4 kgStiffness k40 N/m

Outputs

Period T0.628 s

Reading speed

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Narration of this film

A spring, a mass, a ticking period.

The ODE is ẍ + (k/m) x = 0. ω = √(k/m) is independent of amplitude.

Reading speed

Watch on YouTube