What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.
A Computational Model
A computational representation of Gravitational force can be created using VPython.
The code below show how we can find the net force, momentum and final position in Python.
while t < 235920:
rate(5000)
#Calculate the Gravitational Force acting on the craft due to Earth and Moon
Gravitational Interaction between the space craft, Earth and the Moon. The red arrow represents the momentum of the spacecraft at the moment
#Craft to Earth Calculations
r = craft.pos - Earth.pos
rmag = sqrt(r.x**2 + r.y**2 + r.z**2)
fgravmag = (G * mEarth * mcraft) / (rmag**2
rhat = r / rmag
fgrav = -1 * fgravmag * rhat
#Craft to Moon Calculations
rmc = craft.pos - Moon.pos
rmcmag = sqrt(rmc.x**2 + rmc.y**2 + rmc.z**2)
fgravmoonmag = (G * mMoon * mcraft) / (rmcmag**2)
rmchat = rmc / rmcmag
fgravmoon = -1 * fgravmoonmag * rmchat
#Fnet is sum of these two forces
fnet = fgrav + fgravmoon
#Update the position of the spacecraft by calculating the new momentum and the new velocity
pcraft = pcraft + (fnet*deltat)
vcraft = pcraft / mcraft
craft.pos = craft.pos + (vcraft*deltat)
Units
In Si Unit, Gravitational Force F is measured in Newtons (N), the two masses, m1 and m2 are measures in kilograms (Kg), the distance is measured in meters (m), and the gravitational constant G is measured in N m2/ kg−2 and has a value of 6.674×10−11 N m2/ kg−2. The value of constant G appeared in Newton's law of universal gravitation, but it was not measured until seventy two years after Newton's death by Henry Cavendish with his Cavendish experiment in 1798. The value of gravitational constant was also the first test of Newton's law between two masses in Laboratory.
Examples
Problems taken from Textbook and old Test on T-square
Simple
Determine the force of gravitational attraction between the earth (m = [math]\displaystyle{ 6 x 10^{24} kg }[/math]) and a 70 kg physics student if the student is in an airplane at 40,000 feet above earth's surface. This would place the student a distance of [math]\displaystyle{ 6.38 x 10^6 m }[/math] from earth's center.
The solution of the problem involves substituting known values of G ([math]\displaystyle{ 6.673 x 10^{-11} }[/math] N m2/kg2), m1 ([math]\displaystyle{ 6 x 10^{24} }[/math] kg), m2 (70 kg) and d ([math]\displaystyle{ 6.38 x 10^6 m }[/math]) into the universal gravitation equation and solving for Fgrav. The solution is as follows:
[math]\displaystyle{ |\mathbf{F_{g}}| = G \frac{m_E m_M}{r^2} }[/math]
[math]\displaystyle{ |\mathbf{F_{g}}| = 6.7x10^{-11} \frac{6e24 * 70}{6.38 x 10^6} }[/math]
[math]\displaystyle{ |\mathbf{F_{g}}| =686 N }[/math]
Middling
The mass of the Earth is [math]\displaystyle{ 6 x 10^ {24} }[/math] kg, and the mass of the Moon is [math]\displaystyle{ 7 x 10^ {22} }[/math] kg. At a particular instance the moon is at location [math]\displaystyle{ \lt 2.8 x 10^8,0,-2.8 x 10^8\gt }[/math] m, in a coordinate system whose origin is at the center of the earth. (a) What is [math]\displaystyle{ \vec{\mathbf{r}} }[/math], the relative position vector from the Earth to the Moon? (b) What is [math]\displaystyle{ |\vec{\mathbf{r}}| }[/math]? (c) What is the unit vector [math]\displaystyle{ \vec{\mathbf{r}} }[/math]? (d) What is the gravitation force exerted by the Earth on the Moon? Your answer should be in vector.
The solution of the problem involves substituting known values of G ([math]\displaystyle{ 6.673 x 10^{-11} }[/math]N m2/kg2), m1[math]\displaystyle{ 6 x 10^ {24} }[/math] kg, m2[math]\displaystyle{ 7 x 10^ {22} }[/math] kg and d [math]\displaystyle{ \lt 2.8 x 10^8,0,-2.8 x 10^8\gt }[/math] m m into the universal gravitation equation and solving for Fgrav. The solution is as follows:
Gravitational Force Between Moon and Earth
(a) The position vector of Moon relative to Earth is,
[math]\displaystyle{ \vec{\mathbf{r}} }[/math] = [math]\displaystyle{ \lt 2.8 x 10^8,0,-2.8 x 10^8\gt - \lt 0,0,0\gt }[/math]
[math]\displaystyle{ \vec{\mathbf{r}} }[/math] = [math]\displaystyle{ \lt 2.8 x 10^8,0,-2.8 x 10^8\gt }[/math] m
(b) The magnitude of position vector of Moon relative to Earth is,
[math]\displaystyle{ |\vec{\mathbf{r}}| }[/math] = [math]\displaystyle{ \sqrt{(2.8 x 10^8)^2+0^2+(-2.8 x 10^8)^2} }[/math]
[math]\displaystyle{ |\vec{\mathbf{r}}| }[/math] = [math]\displaystyle{ 4.0 x 10^8 }[/math] m
[math]\displaystyle{ \vec{\mathbf{F}}_{g} }[/math] = [math]\displaystyle{ -1.76x10^{20}*\lt 0.7,0,-0.7\gt N }[/math]
[math]\displaystyle{ \vec{\mathbf{F}}_{g} }[/math] = [math]\displaystyle{ \lt -1.232x10^{20},0,1.232x10^{20}\gt N }[/math]
Difficult
In the following problems you will be asked to calculate the net gravitational force acting on the Moon.To do so, please use the following variables:
Mass
mS - Mass of the Sun
mE - Mass of the Earth
mM - Mass of the Moon
Initial Positions
[math]\displaystyle{ \vec{\mathbf{r_{S}}} = \lt 0,0,0\gt m }[/math]- Position of the Sun
[math]\displaystyle{ \vec{\mathbf{r_{E}}} }[/math] = <[math]\displaystyle{ L,0,0\gt }[/math] m- Position of the Earth
[math]\displaystyle{ \vec{\mathbf{r_{M}}} }[/math] = <[math]\displaystyle{ L,h,0\gt }[/math] m - Position of the Moon
(a)Calculate the gravitational force on the Moon due to the Earth(b)Calculate the gravitational force on the Moon due to the Sun(c)Calculate the net gravitational force on the Moon
(a) The gravitational force on the Moon due to the Earth is,