CLS-27

Circular period

Kepler circular orbit.

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GoverningCircular period

Governing equation

T=2\pi\sqrt{r^3/GM}

$r \in \left[1000, 4\times 10^{5}\right]$

where

r: r (km)
M: M (Mearth)
T: Circular period (min)

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{r^3/GM}

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 1000$ $T = 5.245$

Along the curve

②

Domain of the variable — blue

r_{\min} = 1000,\quad r_{\max} = 4\times 10^{5}

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 1000$ $r = 1000$

Along the curve

③

Response curve — green

T=2\pi\sqrt{r^3/GM}

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 1000$ $T = 5.245$

Along the curve

④

Rate of change — amber

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

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 3509.4$ $T^{\prime} = 0.014$

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-27 — Circular period) is the form associated with Circular period. Working symbols: $r$ , $M$ $\rightarrow$ $T$ . Kepler circular orbit. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

T

from

r

M

in Classical mechanics via

T=2\pi\sqrt{r^3/GM}

Kepler circular orbit. 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

r = 7000.000\,\mathrm{km}

M = 1.000\,\mathrm{Mearth}

, the governing relation

T=2\pi\sqrt{r^3/GM}

yields

T = 97.145\,\mathrm{min}

. 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

r r7000 kmM M1 Mearth

Outputs

Circular period T97.145 min

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Narration of this film

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

Kepler circular orbit. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube