The Moon's Month

Q. Kepler's constant for orbits round the Earth is k = 1 × 10⁻¹³ s²/m³, so that T² = kR³. The Moon is at a distance of 3.84 × 10⁵ km from the Earth. What is its time period of revolution, in days? expects 27.5 d

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Q. Kepler's constant for orbits round the Earth is k = 1 × 10⁻¹³ s²/m³ $k_{K,o} = 1 \times 10^{−13}$ s²/m³, so that T² = kR³. The Moon is at a distance of 3.84 × 10⁵ km $r_{o} = 3.84 \times 10^{5}$ km from the Earth. What is its time period of revolution, in days?

The playground

Inputs

constant

k_{K,o}

1e-13 s²/m³

distance

r_{o}

3.84e+05 km

Outputs

period

T

The solution

Solved with EarthSatellites

Given

$k_{K,o} = 1 \times 10^{−13}$ s²/m³
$r_{o} = 3.84 \times 10^{5}$ km

Find $T$ .

Working

$r_{o} = 3.84 \times 10^{5}$ km $= 3.84 \times 10^{8}$ m
law of periods(round) $T^{2} = k_{K,o} \cdot {r_{o}}^{3}$
$T^{2} = 1 \times 10^{−13} \cdot {(3.84 \times 10^{8})}^{3}$
$T = 27.54$ d
$T = 2.37956097 \times 10^{6}$ s $= 27.5412149$ d
$T \approx 27.5$ dto 3 significant figures

Conclusion

Every orbit round the Earth shares one constant, so the period follows from the distance alone, as the square root of its cube times that constant.

The Ruby

1require_relative "../physics"23question :the_moon_s_month do4  kind :numerical5  describes "Kepler's constant for orbits round the Earth is k = %{constant}, so that T² = kR³. The Moon is at a " \6            "distance of %{distance} from the Earth. %{period}, in days?"7  given constant: "1e-13 s2/m3"8  given distance: "3.84e5 km"9  asks period: "What is its time period of revolution"10end

1require_relative "../questions/the_moon_s_month"2require_relative "../science/gravitation/earth_satellites"34solve :the_moon_s_month, using: EarthSatellites do5  objects :moon6  paths :orbit7  moon.goes(:round, on: orbit)89  take constant: orbit.keplers_constant10  take distance: orbit.radius1112  period = round.period13  answer period:, in: "d", figures: 31415  conclusion "Every orbit round the Earth shares one constant, so the period follows from the distance " \16             "alone, as the square root of its cube times that constant."17end

1require_relative "../solutions/the_moon_s_month"23play_with :the_moon_s_month do4  title "The Moon's Month"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	—
position of moon at end of round	$x_{1}$	m	—
position of moon at start of round	$x_{0}$	m	—
average velocity of moon during round	$v_{av}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of moon during round	$c_{av}$	m/s	—
path length of moon during round	$d$	m	—
speed of moon during round	$v$	m/s	0 … ∞
angular speed of moon during round	$ω$	rad/s	—
radius of orbit	$r_{o}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of moon during round	$ν$	Hz	0 … ∞
revolutions of moon during round	$n_{rev}$	—	—
centripetal acceleration of moon during round	$a_{c}$	m/s²	—
against gravity of moon during round	$q$	—	—
centripetal force on moon during round	$F_{c}$	N	—
mass of moon	$m$	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	$E_{b}$	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/s²	—
radius of the body of moon	$R$	m	0 … ∞
gravitational intensity of ground by moon	$E_{g,g}$	m/s²	—
separation of ground from moon	$r_{g,g}$	m	0 … ∞
gravitational intensity of moon by ground	$E_{g,g}$	m/s²	—
separation of moon from ground	$r_{g}$	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 = x_{1} - x_{0}

change of position(round)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(round)

c_{av} = \frac{d}{Δ t}

path a time(round)

v = ω \cdot r_{o}

speed round the circle(round)

ω = 2 \cdot \frac{π}{T}

once round a period(round)

ν = \frac{1}{T}

rounds a second(round)

ν = \frac{n_{rev}}{Δ t}

rounds over the time(round)

a_{c} = \frac{v^{2}}{r_{o}}

towards the centre(round)

q = \frac{a_{c}}{g}

measured in gs(round)

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

mass times the turning(round)

v^{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} = \frac{G \cdot M_{o} \cdot 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} = \frac{G \cdot m}{R^{2}}

gravity at the surface(moon)

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

law of gravitation(ground, moon)

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

law of gravitation(moon, ground)