Weighing the Earth by the Moon

Q. Weighing the Earth: the distance to the Moon is 3.84 × 10⁸ m and the time period of the Moon's revolution is 27.3 d. What is the mass of the Earth, in kg? expects 6.02 × 10²⁴ kg

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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?

The playground

Inputs

distance

r_{o}

384000000.00 m

period

T

27.3 d

Outputs

mass

m_{e}

The solution

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 Ruby

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

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

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

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