Weighing the Earth by the Moon

solved EarthSatellites

Q. Weighing the Earth: the distance to the Moon is 3.84 × 10⁸ m $r_{o} = 3.84 \times 10^{8}$ m and the time period of the Moon's revolution $m_{e} = M_{o}$ is 27.3 d $T = 27.3$ d. What is the mass of the Earth, in kg?

Inputs

distance

r_{o}

384000000.00 m

period

T

27.3 d

Outputs

mass

m_{e}

How it's solved

Solved with EarthSatellites

Given

$r_{o} = 3.84 \times 10^{8}$ m
$T = 27.3$ d
$m_{e} = M_{o}$ from “the Moon's revolution”

Find $m_{e}$ .

Working

$T = 27.3$ d $= 2.35872 \times 10^{6}$ s
once round a period(round) $ω_{m} = 2 \cdot \frac{π}{T}$
$ω_{m} = 2 \cdot \frac{3.14159265}{2.35872 \times 10^{6}}$
$ω_{m} = 2.66381143 \times 10^{−6}$ rad/s
speed round the circle(round) $v_{m} = ω_{m} \cdot r_{o}$
$v_{m} = 2.66381143 \times 10^{−6} \cdot 3.84 \times 10^{8}$
$v_{m} = 1022.90359$ m/s
held in orbit by gravity(round) ${v_{m}}^{2} = \frac{G \cdot M_{o}}{r_{o}}$
${1022.90359}^{2} = \frac{6.674 \times 10^{−11} \cdot M_{o}}{3.84 \times 10^{8}}$
$M_{o} = 6.02024864 \times 10^{24}$ kg
the Moon's revolution $m_{e} = M_{o}$
$m_{e} = 6.02024864 \times 10^{24}$
$m_{e} = 6.02 \times 10^{24}$ kg
$m_{e} \approx 6.02 \times 10^{24}$ kgto 3 significant figures

Conclusion

The Moon is a satellite of the Earth, so Kepler's law of periods for its orbit holds the Earth's mass; it agrees with the surface weighing to within a percent.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :weighing_the_earth_by_the_moon do4  kind :numerical5  describes "Weighing the Earth: the distance to the Moon is %{distance} and the time period of the Moon's revolution " \6            "is %{period}. %{mass}, in kg?"7  given distance: "3.84e8 m"8  given period: "27.3 d"9  asks mass: "What is the mass of the Earth"10end

solution.rb

1require_relative "../questions/weighing_the_earth_by_the_moon"2require_relative "../science/gravitation/earth_satellites"34solve :weighing_the_earth_by_the_moon, using: EarthSatellites do5  objects :moon, :earth6  paths :orbit7  moon.goes(:round, on: orbit)89  take distance: orbit.radius10  take period: round.period1112  deduce_from "the Moon's revolution", earth.mass => orbit.central_mass1314  mass = earth.mass15  answer mass:, in: "kg", figures: 31617  conclusion "The Moon is a satellite of the Earth, so Kepler's law of periods for its orbit holds the " \18             "Earth's mass; it agrees with the surface weighing to within a percent."19end

playground.rb

1require_relative "../solutions/weighing_the_earth_by_the_moon"23play_with :weighing_the_earth_by_the_moon do4  title "Weighing the Earth by the Moon"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	UniformCircularMotion
gravitational constant	$G$	N m² kg⁻²	—	UniversalGravitation
mass	$m$	kg	0 … ∞	UniversalGravitation
radius of the body	$R$	m	0 … ∞	UniversalGravitation
surface gravity	$g_{s}$	m/s²	—	UniversalGravitation
separation	$r_{g}$	m	0 … ∞	UniversalGravitation
gravitational intensity	$E_{g}$	m/s²	—	UniversalGravitation
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
radius	$r$	m	0 … ∞	UniformCircularMotion
speed	$v$	m/s	0 … ∞	UniformCircularMotion
angular speed	$ω$	rad/s	—	UniformCircularMotion
revolutions	$n_{rev}$	—	—	UniformCircularMotion
frequency	$ν$	Hz	0 … ∞	UniformCircularMotion
centripetal acceleration	$a_{c}$	m/s²	—	UniformCircularMotion
against gravity	$q$	—	—	UniformCircularMotion
period	$T$	s	0 … ∞	UniformCircularMotion
gravity	$g$	m/s²	—	UniformCircularMotion
centripetal force	$F_{c}$	N	—	UniformCircularMotion
central mass	$M$	kg	—	EarthSatellites
central radius	$R_{c}$	m	—	EarthSatellites
height of the orbit	$h$	m	—	EarthSatellites
keplers constant	$k_{K}$	s²/m³	—	EarthSatellites
binding energy	$E_{b}$	J	—	EarthSatellites
displacement of moon during round	$Δ x_{m}$	m	—
position of moon at end of round	$x_{1,m}$	m	—
position of moon at start of round	$x_{0,m}$	m	—
average velocity of moon during round	$v_{av,m}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of moon during round	$c_{av,m}$	m/s	—
path length of moon during round	$d_{m}$	m	—
speed of moon during round	$v_{m}$	m/s	0 … ∞
angular speed of moon during round	$ω_{m}$	rad/s	—
radius of orbit	$r_{o}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of moon during round	$ν_{m}$	Hz	0 … ∞
revolutions of moon during round	$n_{rev,m}$	—	—
centripetal acceleration of moon during round	$a_{c,m}$	m/s²	—
against gravity of moon during round	$q_{m}$	—	—
centripetal force on moon during round	$F_{c,m}$	N	—
mass of moon	$m_{m}$	kg	0 … ∞
central mass of orbit	$M_{o}$	kg	—
keplers constant of orbit	$k_{K,o}$	s²/m³	—
central radius of orbit	$R_{c,o}$	m	—
height of the orbit of orbit	$h_{o}$	m	—
binding energy of moon during round	$E_{b,m}$	J	—
surface gravity of ground	$g_{s,g}$	m/s²	—
mass of ground	$m_{g}$	kg	0 … ∞
radius of the body of ground	$R_{g}$	m	0 … ∞
surface gravity of moon	$g_{s,m}$	m/s²	—
radius of the body of moon	$R_{m}$	m	0 … ∞
surface gravity of earth	$g_{s,e}$	m/s²	—
mass of earth	$m_{e}$	kg	0 … ∞
radius of the body of earth	$R_{e}$	m	0 … ∞
gravitational intensity of ground by moon	$E_{g,gm}$	m/s²	—
separation of ground from moon	$r_{g,gm}$	m	0 … ∞
gravitational intensity of ground by earth	$E_{g,ge}$	m/s²	—
separation of ground from earth	$r_{g,ge}$	m	0 … ∞
gravitational intensity of moon by ground	$E_{g,mg}$	m/s²	—
separation of moon from ground	$r_{g,m}$	m	0 … ∞
gravitational intensity of moon by earth	$E_{g,me}$	m/s²	—
separation of moon from earth	$r_{g,me}$	m	0 … ∞
gravitational intensity of earth by ground	$E_{g,eg}$	m/s²	—
separation of earth from ground	$r_{g,e}$	m	0 … ∞
gravitational intensity of earth by moon	$E_{g,em}$	m/s²	—
separation of earth from moon	$r_{g,em}$	m	0 … ∞

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

g_{s,b} = \frac{G \cdot m_{b}}{{R_{b}}^{2}}

gravity at the surface

E_{g,oo} = \frac{G \cdot m_{o}}{{r_{g,oo}}^{2}}

law of gravitation

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

Δ x_{m} = x_{1,m} - x_{0,m}

change of position(round)

v_{av,m} = \frac{Δ x_{m}}{Δ t}

displacement a time(round)

c_{av,m} = \frac{d_{m}}{Δ t}

path a time(round)

v_{m} = ω_{m} \cdot r_{o}

speed round the circle(round)

ω_{m} = 2 \cdot \frac{π}{T}

once round a period(round)

ν_{m} = \frac{1}{T}

rounds a second(round)

ν_{m} = \frac{n_{rev,m}}{Δ t}

rounds over the time(round)

a_{c,m} = \frac{{v_{m}}^{2}}{r_{o}}

towards the centre(round)

q_{m} = \frac{a_{c,m}}{g}

measured in gs(round)

F_{c,m} = m_{m} \cdot a_{c,m}

mass times the turning(round)

{v_{m}}^{2} = \frac{G \cdot M_{o}}{r_{o}}

held in orbit by gravity(round)

k_{K,o} = \frac{4 \cdot π^{2}}{G \cdot M_{o}}

keplers constant of an orbit(round)

T^{2} = k_{K,o} \cdot {r_{o}}^{3}

law of periods(round)

r_{o} = R_{c,o} + h_{o}

orbit radius from the centre(round)

E_{b,m} = \frac{G \cdot M_{o} \cdot m_{m}}{2 \cdot r_{o}}

binding energy of an orbit(round)

g_{s,g} = \frac{G \cdot m_{g}}{{R_{g}}^{2}}

gravity at the surface(ground)

g_{s,m} = \frac{G \cdot m_{m}}{{R_{m}}^{2}}

gravity at the surface(moon)

g_{s,e} = \frac{G \cdot m_{e}}{{R_{e}}^{2}}

gravity at the surface(earth)

E_{g,gm} = \frac{G \cdot m_{m}}{{r_{g,gm}}^{2}}

law of gravitation(ground, moon)

E_{g,ge} = \frac{G \cdot m_{e}}{{r_{g,ge}}^{2}}

law of gravitation(ground, earth)

E_{g,mg} = \frac{G \cdot m_{g}}{{r_{g,m}}^{2}}

law of gravitation(moon, ground)

E_{g,me} = \frac{G \cdot m_{e}}{{r_{g,me}}^{2}}

law of gravitation(moon, earth)

E_{g,eg} = \frac{G \cdot m_{g}}{{r_{g,e}}^{2}}

law of gravitation(earth, ground)

E_{g,em} = \frac{G \cdot m_{m}}{{r_{g,em}}^{2}}

law of gravitation(earth, moon)

m_{e} = M_{o}

the Moon's revolution