CLS-11

Simple-pendulum period

T = 2π √(ℓ/g) for small angles.

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OscillationsHuygens 1673

Governing equation

T=2\pi\sqrt{\dfrac{\ell}{g}}

$\ell \in \left[0.1, 20\right]$

where

\ell: Length (m)
g: Gravity (m/s²)
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{\dfrac{\ell}{g}}

$\ell \in \left[0.1, 20\right]$

$\ell = 0.1$ $T = 0.634$

Along the curve

②

Domain of the variable — blue

\ell_{\min} = 0.1,\quad \ell_{\max} = 20

$\ell \in \left[0.1, 20\right]$

$\ell = 0.1$ $\ell = 0.1$

Along the curve

③

Response curve — green

T=2\pi\sqrt{\dfrac{\ell}{g}}

$\ell \in \left[0.1, 20\right]$

$\ell = 0.1$ $T = 0.634$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T}{\mathrm{d}\ell}

$\ell \in \left[0.1, 20\right]$

$\ell = 0.225$ $T^{\prime} = 2.209$

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-11 — Simple-pendulum period) is the form associated with Huygens 1673. Working symbols: $\ell$ , $g$ $\rightarrow$ $T$ . Linearising ẍ + (g/ℓ) sin θ ≈ (g/ℓ) θ gives SHM of period 2π √(ℓ/g), Huygens' clock law.

Purpose

Purpose: compute

T

from

\ell

g

in Classical mechanics via

T=2\pi\sqrt{\dfrac{\ell}{g}}

T = 2π √(ℓ/g) for small angles. 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

\ell = 1.000\,\mathrm{m}

g = 9.810\,\mathrm{m/s^{2}}

, the governing relation

T=2\pi\sqrt{\dfrac{\ell}{g}}

yields

T = 2.0061\,\mathrm{s}

. Massless rod, point bob, θ ≲ 15°, vacuum. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Length \ell1 mGravity g9.81 m/s²

Outputs

Period T2.0061 s

Reading speed

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Narration of this film

Massless rod, point bob, θ ≲ 15°, vacuum.

Linearising ẍ + (g/ℓ) sin θ ≈ (g/ℓ) θ gives SHM of period 2π √(ℓ/g), Huygens' clock law.

Reading speed

Watch on YouTube