Where the Sun and the Earth Pull Alike

solved UniversalGravitation

Q. A rocket is fired from the earth towards the sun $r_{g,rs} = r_{g,es} - r_{g,re}$ . At what distance from the earth's centre is the gravitational force on the rocket $E_{g,rs} = E_{g,re}$ zero? Mass of the sun = 2 × 10³⁰ kg $m_{s} = 2 \times 10^{30}$ kg, mass of the earth = 6 × 10²⁴ kg $m_{e} = 6 \times 10^{24}$ kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 10¹¹ m $r_{g,es} = 1.5 \times 10^{11}$ m). Answer in metres.

Inputs

sun

m_{s}

2e+30 kg

earth

m_{e}

6e+24 kg

orbit

r_{g,es}

150000000000.00 m

Outputs

distance

r_{g,re}

How it's solved

Solved with UniversalGravitation

Given

$m_{s} = 2 \times 10^{30}$ kg
$m_{e} = 6 \times 10^{24}$ kg
$r_{g,es} = 1.5 \times 10^{11}$ m
$E_{g,rs} = E_{g,re}$ from “gravitational force on the rocket”
$r_{g,rs} = r_{g,es} - r_{g,re}$ from “from the earth towards the sun”

Find $r_{g,re}$ .

Working

law of gravitation(rocket, sun) $E_{g,rs} = \frac{G \cdot m_{s}}{{r_{g,rs}}^{2}}$
gravitational force on the rocket $E_{g,rs} = E_{g,re}$
law of gravitation(rocket, earth) $E_{g,re} = \frac{G \cdot m_{e}}{{r_{g,re}}^{2}}$
from the earth towards the sun $r_{g,rs} = r_{g,es} - r_{g,re}$
$r_{g,re} = 2.594 \times 10^{8}$ m
$r_{g,re} \approx 2.6 \times 10^{8}$ mto 2 significant figures

Conclusion

The sun is far heavier than the earth, so its pull falls to match the earth's only very near the earth, where the two pulls on the rocket cancel.

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 :where_the_sun_and_the_earth_pull_alike do4  kind :numerical5  describes "A rocket is fired from the earth towards the sun. %{distance} is the gravitational force on the rocket " \6            "zero? Mass of the sun = %{sun}, mass of the earth = %{earth}. Neglect the effect of other planets etc. " \7            "(orbital radius = %{orbit}). Answer in metres."8  given sun: "2e30 kg"9  given earth: "6e24 kg"10  given orbit: "1.5e11 m"11  asks distance: "At what distance from the earth's centre"12end

solution.rb

1require_relative "../questions/where_the_sun_and_the_earth_pull_alike"2require_relative "../science/gravitation/universal_gravitation"34solve :where_the_sun_and_the_earth_pull_alike, using: UniversalGravitation do5  objects :rocket, :sun, :earth67  take sun: sun.mass8  take earth: earth.mass9  take orbit: earth.separation(from: sun)1011  deduce_from "gravitational force on the rocket", rocket.gravitational_intensity(by: sun) =>12                                                   rocket.gravitational_intensity(by: earth)13  deduce_from "from the earth towards the sun", rocket.separation(from: sun) =>14                                                earth.separation(from: sun) - rocket.separation(from: earth)1516  distance = rocket.separation(from: earth)17  answer distance:, in: "m", figures: 21819  conclusion "The sun is far heavier than the earth, so its pull falls to match the earth's only very near the " \20             "earth, where the two pulls on the rocket cancel."21end

playground.rb

1require_relative "../solutions/where_the_sun_and_the_earth_pull_alike"23play_with :where_the_sun_and_the_earth_pull_alike do4  title "Where the Sun and the Earth Pull Alike"5end

The laws it drew on

quantity	written	units	within	from
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
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 rocket	$g_{s,r}$	m/s²	—
mass of rocket	$m_{r}$	kg	0 … ∞
radius of the body of rocket	$R_{r}$	m	0 … ∞
surface gravity of sun	$g_{s,s}$	m/s²	—
mass of sun	$m_{s}$	kg	0 … ∞
radius of the body of sun	$R_{s}$	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 rocket	$E_{g,gr}$	m/s²	—
separation of ground from rocket	$r_{g,gr}$	m	0 … ∞
gravitational intensity of ground by sun	$E_{g,gs}$	m/s²	—
separation of ground from sun	$r_{g,gs}$	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 rocket by ground	$E_{g,rg}$	m/s²	—
separation of rocket from ground	$r_{g,r}$	m	0 … ∞
gravitational intensity of rocket by sun	$E_{g,rs}$	m/s²	—
separation of rocket from sun	$r_{g,rs}$	m	0 … ∞
gravitational intensity of rocket by earth	$E_{g,re}$	m/s²	—
separation of rocket from earth	$r_{g,re}$	m	0 … ∞
gravitational intensity of sun by ground	$E_{g,sg}$	m/s²	—
separation of sun from ground	$r_{g,s}$	m	0 … ∞
gravitational intensity of sun by rocket	$E_{g,sr}$	m/s²	—
separation of sun from rocket	$r_{g,sr}$	m	0 … ∞
gravitational intensity of sun by earth	$E_{g,se}$	m/s²	—
separation of sun from earth	$r_{g,se}$	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 rocket	$E_{g,er}$	m/s²	—
separation of earth from rocket	$r_{g,er}$	m	0 … ∞
gravitational intensity of earth by sun	$E_{g,es}$	m/s²	—
separation of earth from sun	$r_{g,es}$	m	0 … ∞

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

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

gravity at the surface(ground)

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

gravity at the surface(rocket)

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

gravity at the surface(sun)

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

gravity at the surface(earth)

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

law of gravitation(ground, rocket)

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

law of gravitation(ground, sun)

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

law of gravitation(ground, earth)

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

law of gravitation(rocket, ground)

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

law of gravitation(rocket, sun)

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

law of gravitation(rocket, earth)

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

law of gravitation(sun, ground)

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

law of gravitation(sun, rocket)

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

law of gravitation(sun, earth)

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

law of gravitation(earth, ground)

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

law of gravitation(earth, rocket)

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

law of gravitation(earth, sun)

E_{g,rs} = E_{g,re}

gravitational force on the rocket

r_{g,rs} = r_{g,es} - r_{g,re}

from the earth towards the sun