CLS-05

Mass–spring period

T = 2π √(m/k) for undamped SHM.

Reading speed

OscillationsHooke / Huygens

Governing equation

T=2\pi\sqrt{\dfrac{m}{k}},\quad \omega=\sqrt{k/m}

$m \in \left[0.02, 20\right]$

where

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

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{\dfrac{m}{k}},\quad \omega=\sqrt{k/m}

$m \in \left[0.02, 20\right]$

$m = 0.02$ $T = 0.099$

Along the curve

②

Domain of the variable — blue

m_{\min} = 0.02,\quad m_{\max} = 20

$m \in \left[0.02, 20\right]$

$m = 0.02$ $m = 0.02$

Along the curve

③

Response curve — green

T=2\pi\sqrt{\dfrac{m}{k}},\quad \omega=\sqrt{k/m}

$m \in \left[0.02, 20\right]$

$m = 0.02$ $T = 0.099$

Along the curve

④

Rate of change — amber

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

$m \in \left[0.02, 20\right]$

$m = 0.146$ $T^{\prime} = 1.061$

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-05 — Mass–spring period) is the form associated with Hooke / Huygens. Working symbols: $m$ , $k$ $\rightarrow$ $T$ , $f$ . Hooke's F = −kx and Newton ma = F give ẍ + (k/m)x = 0, whose angular frequency is √(k/m).

Purpose

Purpose: compute

T

f

from

m

k

in Classical mechanics via

T=2\pi\sqrt{\dfrac{m}{k}},\quad \omega=\sqrt{k/m}

T = 2π √(m/k) for undamped SHM. 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 = 0.500\,\mathrm{kg}

k = 80.000\,\mathrm{N/m}

, the governing relation

T=2\pi\sqrt{\dfrac{m}{k}},\quad \omega=\sqrt{k/m}

yields

T = 0.4967\,\mathrm{s}

f = 2.0132\,\mathrm{Hz}

. Linear spring, no damping, 1-D. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass m0.5 kgStiffness k80 N/m

Outputs

Period T0.4967 s
Frequency f2.0132 Hz

Reading speed

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Narration of this film

Linear spring, no damping, 1-D.

Hooke's F = −kx and Newton ma = F give ẍ + (k/m)x = 0, whose angular frequency is √(k/m).

Reading speed

Watch on YouTube