CLS-31

Physical pendulum

Compound pendulum period.

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GoverningPhysical pendulum

Governing equation

T=2\pi\sqrt{I/(m g d)}

$d \in \left[0.02, 1\right]$

where

I: I (kg·m²)
m: m (kg)
d: d (m)
T: Physical pendulum (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{I/(m g d)}

$d \in \left[0.02, 1\right]$

$d = 0.02$ $T = 6.344$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0.02,\quad d_{\max} = 1

$d \in \left[0.02, 1\right]$

$d = 0.02$ $d = 0.02$

Along the curve

③

Response curve — green

T=2\pi\sqrt{I/(m g d)}

$d \in \left[0.02, 1\right]$

$d = 0.02$ $T = 6.344$

Along the curve

④

Rate of change — amber

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

$d \in \left[0.02, 1\right]$

$d = 0.026$ $T^{\prime} = -109.8$

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-31 — Physical pendulum) is the form associated with Physical pendulum. Working symbols: $I$ , $m$ , $d$ $\rightarrow$ $T$ . Compound pendulum period. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

T

from

I

m

d

in Classical mechanics via

T=2\pi\sqrt{I/(m g d)}

Compound pendulum period. 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

I = 0.400\,\mathrm{kg·m^{2}}

m = 2.000\,\mathrm{kg}

d = 0.150\,\mathrm{m}

, the governing relation

T=2\pi\sqrt{I/(m g d)}

yields

T = 2.316\,\mathrm{s}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

I I0.4 kg·m²m m2 kgd d0.15 m

Outputs

Physical pendulum T2.316 s

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Compound pendulum period. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube