Physics Book - User contributions [en]

3 or More Body Interactions

2019-07-25T19:00:50Z

Lduffy8: /* Examples */

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.
==Main Idea==
Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

'''Restricted 3-Body Problem'''

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

[[File:Three body problem figure-8 orbit animation.gif|250px|thumb|right|An example of a stable solution to the 3-body problem.]]
'''Other Solutions'''

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

==Examples==
This is a simple example that corresponds to the restricted three body problem. More complex examples require computer simulations and integrators in order to solve.

''The moon has a mass of <math>m_{moon}</math> kg. If it is located a distance <math>d_1</math> m from the Earth (mass <math>m_E</math> kg) and a distance <math>d_2</math> m from the Sun (mass <math>m_{Sun}</math> kg). What are the magnitudes of the forces of the Sun on the Moon and the Earth on the Moon?''

Recall that the following equation applies to all bodies in this problem:
::*<math>F_g=\frac{GM_1M_2}{r^2}</math>
Here, we can replace the relevant distances and masses into the equation relatively easily to come up with the following force magnitudes:
::*<math>F_{Sun,Moon}=\frac{Gm_{moon}m_{Sun}}{d_2^2}</math> for the Sun on the Moon
::*<math>F_{Earth,Moon} = \frac{Gm_{moon}m_E}{d_1^2}</math> for the Earth on the Moon

For a fun extra reference, the Sun's mass is about 300,000 times greater than Earth's mass, but the Sun-Moon distance is about 389 times greater than the Earth-Moon distance. Via a simple fraction, as shown below, the magnitude of the force that the Earth exerts on the Moon is thus much lesser than the magnitude of the force the Sun exerts on the Moon, despite the fact that the Moon is orbiting around the Earth.
::*<math>\frac{F_{Earth}}{F_{Sun}}=\frac{m_E}{300000m_E}\frac{(389d_E)^2}{d_E^2}\approx{}\frac{150000}{300000}=0.5</math>

==Connectedness==
My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

==History==
The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when ''Principia'' was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of [https://en.wikipedia.org/wiki/Chaos_theory chaos theory].

==See Also==

Electric Force

2019-07-25T18:45:32Z

Lduffy8: /* A Mathematical Model */

'''CLAIMED BY Luis Pimentel Spring 2018'''

==The Main Idea==

There are two kinds of electric charge: positive and negative. Particles with like charges will repel each other, and particles with unlike charges attract each other. We generally approximate charges as 'point particles', or objects with radii and masses so small in comparison to other distances and masses used in problems that they are inconsequential. The object is then treated as if all its charge and mass were concentrated at a single point. In many of the problems we will encounter, protons and electrons are treated as point charges.

[[File:Attract_repel_1.png|250px|thumb|right|This figure depicts how different charges either attract or repel each other.]]
Two charges will exert a force on each other that depends on the sign of the charges. This force will be attractive when the two charges are opposite, and repulsive when the two charges are the same. The strength of this electrical interaction is a vector quantity that has magnitude and direction. Each individual charge will additionally create an [[Electric Field]] around it, whose magnitude depends on the sign and magnitude of the charge, and which decreases in magnitude radially from the source charge. If the electric field at a particular location is known, then this field can be used to calculate the electric force of the particle being acted upon. The electric force is directly proportional to the amount of charge within each particle being acted upon by the other's electric field. Moreover, the magnitude of the force is inversely proportional to the square distance between the two interacting particles. It is important to remember that a particle cannot have an electric force on itself; there must be at least two interacting, charged components.

===A Mathematical Model===
'''The Coulomb Force Law'''

The formula for the magnitude of the electric force between two point charges is:

::*<math>F=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>
Where '''<math>{q}_{1}</math>''' and '''<math>{q}_{2}</math>''' are the magnitudes of electric charge of point 1 and point 2, and '''<math>r</math>''' is the distance between the two point charges. The units for electric force are the same as the units for all forces: Newtons. The expression <math>\frac{1}{4 \pi \epsilon_0 }</math> is known as the electric constant and carries the value 9e9. Epsilon naught defines the electric permittivity of free space (the permittivity of air - this constant has different values if the force is acting in different materials!).

Interestingly enough, one can see a relationship between this formula and the formula for gravitational force (<math>F={G} \frac{|{m}_{1}{m}_{2}|}{r^2} </math>). From this relationship, one can conclude that the interactions of two objects as a result of their charges or masses follow similar fundamental laws of physics. Namely, these two forces are both examples of the [https://en.wikipedia.org/wiki/Inverse-square_law inverse square law].

'''Derivations of Electric Force'''

The electric force on a particle can also be written as:

::*<math>\vec F=q\vec E </math>
Where '''<math>q</math>''' is the charge of the particle and '''<math>\vec E </math>''' is the external electric field. It is interesting to note that the electric field from a charge(<math>q_{1}</math>) takes the form shown below:

::*<math>|\vec E|=\frac{1}{4 \pi \epsilon_0 } \frac{|q_1|}{r^2} </math>

Therefore, the magnitude of electric force between point charge 1 and point charge 2 can be written as:

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=|{q}_{2}|\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}|}{r^2}=|{q}_{2}||\vec{E}_{1}| </math>

The units of charge are in Coulombs and the units for electric field are in Newton/Coulombs, so this derivation is correct in its dimensions since multiplying the two units gives just Newtons.

===A Computational Model===

A computational representation of the Electric force can be created using VPython.
The code below shows how we can find the net force, momentum and final position between two charged particles in VPython. This example uses two positively charged protons.

<code>
''''' #CONSTANTS '''''
E= 9e9 ''''' # Electric Force constant '''''
q1 = +1.6e-19 ''''' # Charge of proton 1'''''
q2 = +1.6e-19 ''''' # Charge of proton 2 '''''

while t < 200 :
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r = proton.pos - proton2.pos
rmag = mag(r)
rhat = r/rmag
Fnet = E*((q1*q2)/(mag2(r)))*rhat
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r2 = proton2.pos - proton.pos
rmag2 = mag(r2)
rhat2 = r2/rmag2
Fnet2 = E*((q1*q2)/(mag2(r2)))*rhat2
''''' # Update positions of BOTH protons '''''
pproton = pproton + Fnet*deltat
vproton = pproton/mproton
proton.pos = proton.pos + vproton*deltat
pproton2 = pproton2 + Fnet2*deltat
vproton2 = pproton2/mproton
proton2.pos = proton2.pos + vproton2*deltat
</code>

Notice that this iterative process is very similar to that working with [http://www.physicsbook.gatech.edu/Gravitational_Force Gravitational Forces].

Notice, however, that the direction of motion of each particle is not determined by relative position, but by the charges of the particles. The product of these charges ultimately determine whether the two particles will attract each other, or repel. This is demonstrated in the following simulations:

The trinket model linked demonstrates the Electric force interaction of a proton and an electron. [https://trinket.io/glowscript/c1cdb42527 Proton-Electron Electric Force Interaction]

The trinket model linked demonstrates the Electric force interaction between two protons. [https://trinket.io/glowscript/d5f7516fc2 Proton-Proton Electric Force Interaction]

==Examples==

===Simple===
'''Problem: '''Find the electric force of a -3 C particle in a region with an electric field of <math><7, 5, 0></math>N/C.

'''Step 1: '''Substitute values into the correct formula.

<math>\vec F=q\vec E </math>

<math>\vec F=(-3 C)<7, 5, 0></math>N/C

<math>\vec F=<-21, -15, 0></math>N

The electric force vector for this particle is <math><-21, -15, 0></math>N.

===Midding===
'''Problem: '''Find the magnitude of electric force on two charged particles located at <math> <0, 0, 0></math>m and <math> <0, 10, 0></math>m. The first particle has a charge of +5 nC and the second particle has a charge of -10 nC. Is the force attractive or repulsive?

'''Step 1: '''Find the distance between the two point charges.

<math>d=\sqrt{(0 m-0 m)^2+(0 m-10 m)^2+(0 m-0 m)^2}=\sqrt{100 m}=10 </math>m.

The distance between the two points is 10 m.

'''Step 2: '''Substitute values into the correct formula.

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=\frac{1}{4 \pi \epsilon_0 } \frac{|(5 nC)(-10 nC)|}{(10m)^2} </math>

<math>|\vec F|=4.5e-9 </math> N

The magnitude of electric force is <math>|\vec F|=4.5e-9 </math> N.

'''Step 3: '''Determine if force is attractive or repulsive.

Since the first particle is positively charged and the second is negatively charged, the force is attractive. The particles are attracted to each other.

===Difficult===
'''Problem:'''

[[File:SpringTest1Prob1.png]]

Using the graphic above, find a) the net force acting on particle -q' and b) the direction of the net force on this charge.

'''Step 1: '''Calculate the net force.

[[File:Force1.png]]
[[File:Force2.png]]

'''Step 2: '''Evaluate the direction of the force.

[[File:ForceDir.png]]

Problems involving electric force exclusively will not be more complicated than the above. However, the the electric force can be used in calculation of a net force acting on a particle in combination with non-Coulomb electric force and magnetic force.

==Connectedness==
Electric force is ubiquitous in everyday life, although it is not always evident. One interesting example of the electric force is the attraction of clothes to one another after being washed. The charges caused by the machine-drying process create opposite, attractive charges on different pieces of clothing which cause them to stick together. Another example can be seen in refrigerator magnets due to the electromagnetic forces that enable magnetism.

I am a physics major, and as such I must consider the electric force in a lot of different situations, and not just obvious ones. The electric and magnetic fields around ultra-dense objects such as neutron stars and black holes cause electrons to accelerate, and create radiation, for example. Knowing how to treat the electromagnetic force is extremely helpful in understanding various phenomena in Space.

Thinking more complexly, the electric force is also prevalent in almost all forms of modern technology involving electricity. One particular example is the process of charging a smartphone: the electric force allows a current to be generated which transfers charge from outlets to the internal battery of these devices. A final, slightly more complicated example of the electric force is seen in the production of abrasive paper, whereby positively charged-smoothing particles are attracted to a negatively charged, smooth surface to create papers like sandpaper.

==History==

French physicist Charles-Augustin de Coulomb discovered in 1785 that the magnitude of electric force between two charged particles is directly proportional to the product of the absolute value of the two charges and inversely proportional to the distance squared between the two particles. He experimented with a torsion balance which consisted of an insulated bar suspended in the air by a silk thread. Coulomb attached a metal ball with a known charge to one end of the insulated bar. He then brought another ball with the same charge near the first ball. This distance between the two balls was recorded. The balls repelled each other, causing the silk thread to twist. The angle of the twist was measured and by knowing how much force was required for the thread to twist through the recorded angle, Coulomb was able to calculate the force between the two balls and derive the formula for electric force.

This [https://www.youtube.com/watch?v=FYSTGX-F1GM| video] explains Coulomb's experiment and the corresponding derivation of his law.

It wasn't until much later, however, that the fundamental charge (the charge on an electron) was discovered when Millikan ran his [https://physics.aps.org/articles/v5/9 oil drop experiment] in 1913 that this value was determined.

== See Also ==

Electric Field: [http://www.physicsbook.gatech.edu/Electric_Field]

Net Force: [http://www.physicsbook.gatech.edu/Lorentz_Force]

==References==

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4th Edition

https://en.wikipedia.org/wiki/Coulomb's_law

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html

http://www.jfinternational.com/ph/coulomb-law.html

[[Category:Forces]]

Introduction to Magnetic Force

2019-07-25T18:42:20Z

Lduffy8: /* See Also */

This page provides a short introduction to the concept of the magnetic force.
==Main Idea==
The magnetic force acts upon moving charges (or currents). This means that the magnitude of the magnetic force has a direct dependence on the velocity of the charge it is acting upon. Furthermore, the magnetic force acts via the magnetic field. This magnetic force acts in a direction perpendicular to both the velocity of the charge and the direction of the magnetic field.

The magnetic field is a vector field (meaning that it extends in all three dimensions). Magnetic fields are provided most simply either by moving charges or by dipoles. When the magnetic field strength is known, it makes finding the magnetic force quite simple.

Magnetic fields and electric fields are, in fact, related, and both together comprise the electromagnetic force. This force is the second strongest of the four fundamental forces, behind the nuclear strong force, and ahead of the weak force and the gravitational force.
===A Mathematical Model===
Magnetic force can be represented as follows, where ''q'' is the charge on a particle, ''v'' is that particles velocity, and ''B'' is the magnetic field. Here, <math>\theta</math> represents the angle between the velocity and magnetic field vectors. The direction of the magnetic field can be found using the [http://physicsed.buffalostate.edu/SeatExpts/resource/rhr/rhr.htm right hand rule].
::*<math>\vec{\mathbf{F_B}}=q\vec{\mathbf{v}}\times{}\vec{\mathbf{B}}</math>
::*<math>|\vec{\mathbf{F_B}}|=qvBsin(\theta{})</math>

The above means that any particle at rest will NOT experience any magnetic force.

The magnetic force also applies to a current carrying wire. Because a current is essentially made of moving charges, this means that the current carrying wire will experience a force. This force is directly proportional to the length of the wire and the current in that wire, as is displayed below. Here ''I'' is the magnitude of the current, which is in units amperes, or Coulombs/second, ''l'' is in units of length, and is in the same direction as the current (for the cross product).
::*<math>\vec{\mathbf{F_B}}=I\vec{\mathbf{l}}\times{}\vec{\mathbf{B}}</math>
::*<math>|\vec{\mathbf{F_B}}|=IlBsin(\theta{})</math>

==Examples==
===Simple===
''Say you have a 1 C charge moving at a rate of 3 m/s. This velocity makes an angle of 60 degrees with respect to a magnetic field with strength 2 Tesla. What is the magnitude of the force the charge experiences?''

The units given in the problem statement are all proper SI units, and so the numbers can just be plugged in to the following equation:
::<math> F_B=qvBsin(\theta{})</math>
::<math>F_B=(1 C)(3 m/s)(2 T)(sin(60))=5.196 N</math>
Thus the answer to this problem is, straightforwardly, 5.196 N.

===Middling===
[[File:IntroBimage.png|300px|thumb|right|A diagram depicting the problem for the middling section. Here, the magnetic field is going out of the page.]]
''Suppose you have a wire with a current <math>I=6</math> amps. This wire is sitting in a constant magnetic field of magnitude 1 T. This wire has length 6 cm. What is the magnitude and the direction of the force acting upon the wire? The magnetic field is pointing out of the page (reference the diagram at right).''

The easiest first step would be to determine the direction of the magnetic force. By using the [http://physicsed.buffalostate.edu/SeatExpts/resource/rhr/rhr.htm right hand rule], we can determine that when the current length, which is going up, is crossed with a vector pointing out of the page, the resulting vector would point towards the right, in the so-called positive x-direction.

Next, we can determine the magnitude of the resultant force via the following relation. Because <math>\theta{}</math> here is equal to 90 degrees, <math>sin(\theta{})</math> is equal simply to one.
::*<math>F_B=IlBsin(\theta{})</math>
::*<math>F_B=(6A)(0.06m)(1T)=0.36N</math>
Finally, as stated before, we know that this is solely in the positive x-direction due to the right hand rule.

===Hard===
[http://physicsbook.gatech.edu/Magnetic_Force#Middling See this problem under the Magnetic Force section on the Physics 2 page]

==Connectedness==
The magnetic force contributes heavily to specific phenomenon in astrophysics. Specifically, neutron stars have large magnetic fields, causing electrons to be accelerated and emit a type of radiation known as synchrotron radiation. Indeed, there are certain types of neutron stars known as [https://en.wikipedia.org/wiki/Magnetar magnetars] that have unusually high magnetic field strengths that then power the emission of X-rays and gamma rays (the highest energy type of electromagnetic radiation). Furthermore, there are also strong magnetic fields around black holes.

As a physics major, understanding how magnetic fields affect the motion of particles and their behavior is fundamental.

The magnetic force is connected directly to MRI's (magnetic resonance imaging). Using MRI's, doctors can learn a great deal about the current condition of the human body, including tracking cancerous tumors and diagnosing different injuries and diseases.

==History==
Initially, electricity and magnetism were thought to be separate forces. When Maxwell published his ''A Treatise on Electricity and Magnetism'', however, it was realized that the two forces were interrelated. Later, in 1820, a Danish physicist realized that the magnetic field from a circuit could be turned on and off if the circuit was turned on and off. This was when it was realized that magnetic fields radiate from all sides of a wire carrying an electric current, providing proof that electricity and magnetism were definitely interrelated.

These findings kickstarted a period of intensive research into attempting to find a mathematical representation of the relationship between electricity and magnetism, and the result would be one of the defining accomplishments of the 19th century in physics.

==See Also==
::*[https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnets-magnetic/a/what-is-magnetic-force An introductory article from Khan Academy]
::*[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html Hyperphysics magnetic force page]
::*[https://en.wikipedia.org/wiki/Magnetic_resonance_imaging MRI wikipedia page; if you're curious on how they work]
::*[https://phys.org/news/2018-11-magnetic-fields-deep-space-wiggle.html A fun article on magnetic fields in space!]

Introduction to Magnetic Force

2019-07-25T18:41:24Z

Lduffy8: /* See Also */

Reciprocity

2019-07-25T18:38:48Z

Lduffy8:

claimed by Nicole Romer (Spring 2017)

edited by Laurentino Castro (Spring 2018)

This section covers the top of reciprocity which explains why forces, such as gravitational and electric forces, act upon each other with equal magnitudes.

[[File:BookForce.JPG|thumb|This is an example of reciprocity. The book is exerting a contact force on the table and the table is exerting a contact force on the book.]]

==The Main Idea==

Reciprocity is the idea that the force object 1 exerts on object 2 is the same as the force object 2 exerts on object 1. This idea comes from Newton's Third Law of Motion, which states that every action has an equal and opposite reaction. For example, if I put my hand on a table, I am exerting a contact force on the table. Because my hand doesn't go straight through the table but instead rests on top of it, the table must also be exerting a force on me that is equivalent to the force I am placing upon it. Though it seems like I am putting in more effort, the forces are the same.

Newton's 3rd law implies that all forces come in pairs. The two forces are called "action" and "reaction" pairs. When forces are in these pairs, the magnitude of the two forces equal each other. However, in vector form, the two forces would be in opposite directions of each other, so one force would have a negative sign on it. This applies to both the gravitational and electric forces. Contact forces that we observe (such as the normal force) are actually due to electrical forces at the atomic level.

Because these forces are equal, so are the changes in momentum of the objects the forces are acting upon (remember, <math>|\vec{\mathbf{F}}|=|\frac{d\vec{\mathbf{p}}}{dt}| </math>). Two planets exerting attractive forces on each other will have the same change in momentum over the same amount of time. However if one planet is much larger it will have a much smaller change in velocity then the smaller planet. This is because p = mv and thus <math> {\Delta}v={\Delta}p/m</math>. So even though the forces will be the same, they may appear unequal due to a smaller change in velocity in a larger object.

The following image showcases the principle of reciprocity as the two people pulling exert an equal and opposite force on each other through tension. Also the object laying motionless on a surface has a net force of zero because the normal force of the table is equal to that of gravity on the object.

[[File:pulling_updated.jpg]]

===A Mathematical Model===
The force due to gravity is:
::*<math>|\vec{\mathbf{F_g}}|=\frac{Gm_{1}m_{2}}{r^2}</math>

Although the force of gravity is dependent on the mass of both objects, both will experience the same magnitude of gravitational force. This can be clearly seen from the above equation.

The force due to electric interactions is:
::*<math>|\vec{\mathbf{F_{elec}}}|=\frac{1}{4{\pi}{\epsilon}_0}\frac{q_{1}q_{2}}{r^2}</math>

[[File:Attract_repel_1.png|250px|thumb|right|This figure depicts how different charges either attract or repel each other. The force on each charge is equal and opposite to that on the other charge in the pairing.]]
Electric charges that are of different sign will attract each other with the same magnitude of force. Furthermore, two like charges will repel each other, and the same magnitude of repelling force will act on each of the charges.

A more formulaic representation of the principle of reciprocity is as follows:
::*<math>\vec{\mathbf{F_{2 on 1}}}=\vec{\mathbf{F_{1 on 2}}}</math>

This can be represented in vector form as follows:
::* <math>\vec{\mathbf{F_{2on1}}}</math> = ''<F,0,0>''
::* <math>\vec{\mathbf{F_{1on2}}}</math> = ''<-F,0,0>''

===A Computational Model===

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===

''If you exert 20 N on the table, what would be the normal force of the table on you?''

Since you are exerting 20 Newtons, due to reciprocity the table will be exerting a normal force of 20 Newtons.

''If you push a box to the right with a force of 15 N, what is the magnitude and direction of the reciprocal force from the box on you?''

The box exerts a force of 15 N on you towards the left.

===Middling===

''A 60 kilogram man stands on the surface of the Earth. What is the force Earth exerts on the man? What is the force the man exerts on the Earth?''
Recall, due to reciprocity, the force that the Earth exerts on the man should be the same as the force that the man exerts on the Earth. We know how much the Man exerts on the Earth because he is standing on the Earth with a force equal to his weight.
::<math> \mathbf{F_g}=mg=(60kg)(9.8\frac{m}{s^2})=588N</math>
Thus 588 N is the force that the man exerts on the Earth in the downwards direction. Because of reciprocity, the Earth must also be exerting 588 N on the man, but in the upwards direction.

''A box is sitting on Earth's surface. The Earth exerts a force of 196 N on the box. What is the force on the Earth? What is the box's mass?''
Here, we have the force. Because of reciprocity, the force of the box on the Earth must also be equal to 196 N. Then, finding the mass of the box is relatively straightforward, as is seen below:
::<math> m = F/g = 196N/9.8 = 20 kg</math>

===Difficult===
[[File:reciprocity_difficult_1.png|250px|thumb|right|Problem set-up]]

''Two blocks of mass <math>m_1</math> (under rod) and <math>m_3</math> (above rod) are connected by a rod of mass <math>m_2</math>. A constant unknown force F pulls upward on the top block while both blocks and the rod move upward at a constant velocity v near the surface of the Earth. The direction of the gravitational force on each block points down. Find <math>F_{1on2}</math>, the magnitude of the force exerted by the bottom block on the rod.''

See the diagram to the side for a depiction of the situation. Because the velocity is constant, the net force on the system, and thus on block 1, is equal to zero.
::<math>F_{net}=0=F_{2on1}+F_g </math>
::<math>\vec{\mathbf{F_{2on1}}}-m_{1}g\hat{\mathbf{y}}=0</math>
::<math>\vec{\mathbf{F_{2on1}}}=m_{1}g\hat{\mathbf{y}}</math>

Because <math>\vec{\mathbf{F_{1on2}}}=-\vec{\mathbf{F_{2on1}}}</math> it stands that:
::<math>\vec{\mathbf{F_{1on2}}}=-m_{1}g\hat{\mathbf{y}}</math>

''You are pulling a rope with a force of 15 N. The rope is connected to a box on a frictionless surface. The rope and box are moving at a constant rate. Find the force of the rope of the block, the block on the rope, and the rope on you.''

Because the velocity of the system is constant, the net force on the system is equal to zero (<math>F_{net} = 0</math>).
::The Force of the rope on you is -15 N (equal and opposite to the force you are exerting on the rope)
::The Force of the rope on the box is 15 N (this must be true in order to make the net force on the rope equal to zero)
::The Force of the box on the rope is -15 N

==Connectedness==
# The first physics I ever learned was Newtons laws. Before heading into any science class, I always thought, every reaction gets an equal and opposite reaction. I didnt really understand it. That is a fundamental principle that we use in almost all physics problems. It has been test questions and homework questions. The thing that intrigues me the most is how an ant can be pushing against a rhino and though the rhino is so much bigger, they are still exerting the same force.
#I am an industrial engineering major and though there is very minimal use of physics in that field, I do believe it is something that will help us go about our days knowing that force isn't how much effort you put in but about the action reaction pairs.
#Forces are something we deal with everyday. Everything we touch, me typing this page right now is all the result of forces. An important industry that deals with this is the automobile industry. If we understand the forces of the wheels on the road, we will know how to make wheels that best suit an automobile.
#This topics is one of the fundamental underlying principles of physics. It applies to pretty much everything in physics.
#Physics applies to almost any major. As an Environmental Engineer reciprocity is important to remember reciprocity, specifically in waste water management. Water pressure (force from water on pipe and force from pipe on water) is important when designing water infrastructure.
#Reciprocity is important to remember in buildings. When creating a structure that can sustain a storm, you have to think about how high winds will change how parts of the structure interact with each other.
#I remember back in elementary school the example that my teacher used was: when you kick a can, which experiences a greater force, the can or your foot? I was astonished to find out that both experienced a equal force due to reciprocity. I spent weeks kicking cans trying to understand this phenomenon until I learned for myself the forces were experienced relative to mass. This was my first real exposure to seeking to understand the world through the lens of physics.

==History==

[[File:Issac_Newton_New1.jpg]]

Isaac Newton was born in Woolsthorpe, England. When he was a child, one day he was resting under an apple tree when suddenly an apple fell on his head. He thought about why things fall down and not fall back up. He spent years figuring out the phenomenon. After all this, he came up with three laws of motion. This is when he discovered gravitation as a force. Where Newton's Law comes into play is that the Earth is exerting a force on us to stay with it since closer objects exert stronger forces on each other. We are also exerting a force on Earth so that we stay on the ground and don't go flying off. The date of this story is not known, and some even believe it to be a myth. However William Stukeley, author of ''Memoirs of Sir Isaac Newton's Life'' noted that he had a conversation with Newton and Newton talked about why an apple falls to the ground due to gravitational interaction.

== See also ==

===Further reading===

:'''1:'''http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law

:'''2:'''Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4

===External links===

https://www.youtube.com/watch?v=NfuKfbpkIrQ

==References==

:'''1:''' http://www.mainlesson.com/display.php?author=baldwin&book=thirty&story=newton

:'''2:''' https://www.newscientist.com/blogs/culturelab/2010/01/newtons-apple-the-real-story.html

:'''3:''' http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law

:'''4:''' http://www.schoolphysics.co.uk/age16-19/Mechanics/Statics/text/Vectors_in_equilibrium/index.html

:'''5:''' Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4

[[Category:Which Category did you place this in?]]

Introduction to Magnetic Force

2019-07-25T18:36:41Z

Lduffy8: /* Hard */

Introduction to Magnetic Force

2019-07-25T18:34:42Z

Lduffy8: /* History */

This page provides a short introduction to the concept of the magnetic force.
==Main Idea==
The magnetic force acts upon moving charges (or currents). This means that the magnitude of the magnetic force has a direct dependence on the velocity of the charge it is acting upon. Furthermore, the magnetic force acts via the magnetic field. This magnetic force acts in a direction perpendicular to both the velocity of the charge and the direction of the magnetic field.

The magnetic field is a vector field (meaning that it extends in all three dimensions). Magnetic fields are provided most simply either by moving charges or by dipoles. When the magnetic field strength is known, it makes finding the magnetic force quite simple.

Magnetic fields and electric fields are, in fact, related, and both together comprise the electromagnetic force. This force is the second strongest of the four fundamental forces, behind the nuclear strong force, and ahead of the weak force and the gravitational force.
===A Mathematical Model===
Magnetic force can be represented as follows, where ''q'' is the charge on a particle, ''v'' is that particles velocity, and ''B'' is the magnetic field. Here, <math>\theta</math> represents the angle between the velocity and magnetic field vectors. The direction of the magnetic field can be found using the [http://physicsed.buffalostate.edu/SeatExpts/resource/rhr/rhr.htm right hand rule].
::*<math>\vec{\mathbf{F_B}}=q\vec{\mathbf{v}}\times{}\vec{\mathbf{B}}</math>
::*<math>|\vec{\mathbf{F_B}}|=qvBsin(\theta{})</math>

The above means that any particle at rest will NOT experience any magnetic force.

The magnetic force also applies to a current carrying wire. Because a current is essentially made of moving charges, this means that the current carrying wire will experience a force. This force is directly proportional to the length of the wire and the current in that wire, as is displayed below. Here ''I'' is the magnitude of the current, which is in units amperes, or Coulombs/second, ''l'' is in units of length, and is in the same direction as the current (for the cross product).
::*<math>\vec{\mathbf{F_B}}=I\vec{\mathbf{l}}\times{}\vec{\mathbf{B}}</math>
::*<math>|\vec{\mathbf{F_B}}|=IlBsin(\theta{})</math>

==Examples==
===Simple===
''Say you have a 1 C charge moving at a rate of 3 m/s. This velocity makes an angle of 60 degrees with respect to a magnetic field with strength 2 Tesla. What is the magnitude of the force the charge experiences?''

The units given in the problem statement are all proper SI units, and so the numbers can just be plugged in to the following equation:
::<math> F_B=qvBsin(\theta{})</math>
::<math>F_B=(1 C)(3 m/s)(2 T)(sin(60))=5.196 N</math>
Thus the answer to this problem is, straightforwardly, 5.196 N.

===Middling===
[[File:IntroBimage.png|300px|thumb|right|A diagram depicting the problem for the middling section. Here, the magnetic field is going out of the page.]]
''Suppose you have a wire with a current <math>I=6</math> amps. This wire is sitting in a constant magnetic field of magnitude 1 T. This wire has length 6 cm. What is the magnitude and the direction of the force acting upon the wire? The magnetic field is pointing out of the page (reference the diagram at right).''

The easiest first step would be to determine the direction of the magnetic force. By using the [http://physicsed.buffalostate.edu/SeatExpts/resource/rhr/rhr.htm right hand rule], we can determine that when the current length, which is going up, is crossed with a vector pointing out of the page, the resulting vector would point towards the right, in the so-called positive x-direction.

Next, we can determine the magnitude of the resultant force via the following relation. Because <math>\theta{}</math> here is equal to 90 degrees, <math>sin(\theta{})</math> is equal simply to one.
::*<math>F_B=IlBsin(\theta{})</math>
::*<math>F_B=(6A)(0.06m)(1T)=0.36N</math>
Finally, as stated before, we know that this is solely in the positive x-direction due to the right hand rule.

===Hard===

==Connectedness==
The magnetic force contributes heavily to specific phenomenon in astrophysics. Specifically, neutron stars have large magnetic fields, causing electrons to be accelerated and emit a type of radiation known as synchrotron radiation. Indeed, there are certain types of neutron stars known as [https://en.wikipedia.org/wiki/Magnetar magnetars] that have unusually high magnetic field strengths that then power the emission of X-rays and gamma rays (the highest energy type of electromagnetic radiation). Furthermore, there are also strong magnetic fields around black holes.

As a physics major, understanding how magnetic fields affect the motion of particles and their behavior is fundamental.

The magnetic force is connected directly to MRI's (magnetic resonance imaging). Using MRI's, doctors can learn a great deal about the current condition of the human body, including tracking cancerous tumors and diagnosing different injuries and diseases.

==History==
Initially, electricity and magnetism were thought to be separate forces. When Maxwell published his ''A Treatise on Electricity and Magnetism'', however, it was realized that the two forces were interrelated. Later, in 1820, a Danish physicist realized that the magnetic field from a circuit could be turned on and off if the circuit was turned on and off. This was when it was realized that magnetic fields radiate from all sides of a wire carrying an electric current, providing proof that electricity and magnetism were definitely interrelated.

These findings kickstarted a period of intensive research into attempting to find a mathematical representation of the relationship between electricity and magnetism, and the result would be one of the defining accomplishments of the 19th century in physics.

==See Also==

Introduction to Magnetic Force

2019-07-25T18:29:40Z

Lduffy8: /* Connectedness */

This page provides a short introduction to the concept of the magnetic force.
==Main Idea==
The magnetic force acts upon moving charges (or currents). This means that the magnitude of the magnetic force has a direct dependence on the velocity of the charge it is acting upon. Furthermore, the magnetic force acts via the magnetic field. This magnetic force acts in a direction perpendicular to both the velocity of the charge and the direction of the magnetic field.

The magnetic field is a vector field (meaning that it extends in all three dimensions). Magnetic fields are provided most simply either by moving charges or by dipoles. When the magnetic field strength is known, it makes finding the magnetic force quite simple.

Magnetic fields and electric fields are, in fact, related, and both together comprise the electromagnetic force. This force is the second strongest of the four fundamental forces, behind the nuclear strong force, and ahead of the weak force and the gravitational force.
===A Mathematical Model===
Magnetic force can be represented as follows, where ''q'' is the charge on a particle, ''v'' is that particles velocity, and ''B'' is the magnetic field. Here, <math>\theta</math> represents the angle between the velocity and magnetic field vectors. The direction of the magnetic field can be found using the [http://physicsed.buffalostate.edu/SeatExpts/resource/rhr/rhr.htm right hand rule].
::*<math>\vec{\mathbf{F_B}}=q\vec{\mathbf{v}}\times{}\vec{\mathbf{B}}</math>
::*<math>|\vec{\mathbf{F_B}}|=qvBsin(\theta{})</math>

The above means that any particle at rest will NOT experience any magnetic force.

The magnetic force also applies to a current carrying wire. Because a current is essentially made of moving charges, this means that the current carrying wire will experience a force. This force is directly proportional to the length of the wire and the current in that wire, as is displayed below. Here ''I'' is the magnitude of the current, which is in units amperes, or Coulombs/second, ''l'' is in units of length, and is in the same direction as the current (for the cross product).
::*<math>\vec{\mathbf{F_B}}=I\vec{\mathbf{l}}\times{}\vec{\mathbf{B}}</math>
::*<math>|\vec{\mathbf{F_B}}|=IlBsin(\theta{})</math>

==Examples==
===Simple===
''Say you have a 1 C charge moving at a rate of 3 m/s. This velocity makes an angle of 60 degrees with respect to a magnetic field with strength 2 Tesla. What is the magnitude of the force the charge experiences?''

The units given in the problem statement are all proper SI units, and so the numbers can just be plugged in to the following equation:
::<math> F_B=qvBsin(\theta{})</math>
::<math>F_B=(1 C)(3 m/s)(2 T)(sin(60))=5.196 N</math>
Thus the answer to this problem is, straightforwardly, 5.196 N.

===Middling===
[[File:IntroBimage.png|300px|thumb|right|A diagram depicting the problem for the middling section. Here, the magnetic field is going out of the page.]]
''Suppose you have a wire with a current <math>I=6</math> amps. This wire is sitting in a constant magnetic field of magnitude 1 T. This wire has length 6 cm. What is the magnitude and the direction of the force acting upon the wire? The magnetic field is pointing out of the page (reference the diagram at right).''

The easiest first step would be to determine the direction of the magnetic force. By using the [http://physicsed.buffalostate.edu/SeatExpts/resource/rhr/rhr.htm right hand rule], we can determine that when the current length, which is going up, is crossed with a vector pointing out of the page, the resulting vector would point towards the right, in the so-called positive x-direction.

Next, we can determine the magnitude of the resultant force via the following relation. Because <math>\theta{}</math> here is equal to 90 degrees, <math>sin(\theta{})</math> is equal simply to one.
::*<math>F_B=IlBsin(\theta{})</math>
::*<math>F_B=(6A)(0.06m)(1T)=0.36N</math>
Finally, as stated before, we know that this is solely in the positive x-direction due to the right hand rule.

===Hard===

==Connectedness==
The magnetic force contributes heavily to specific phenomenon in astrophysics. Specifically, neutron stars have large magnetic fields, causing electrons to be accelerated and emit a type of radiation known as synchrotron radiation. Indeed, there are certain types of neutron stars known as [https://en.wikipedia.org/wiki/Magnetar magnetars] that have unusually high magnetic field strengths that then power the emission of X-rays and gamma rays (the highest energy type of electromagnetic radiation). Furthermore, there are also strong magnetic fields around black holes.

As a physics major, understanding how magnetic fields affect the motion of particles and their behavior is fundamental.

The magnetic force is connected directly to MRI's (magnetic resonance imaging). Using MRI's, doctors can learn a great deal about the current condition of the human body, including tracking cancerous tumors and diagnosing different injuries and diseases.

==History==
==See Also==

3 or More Body Interactions

2019-07-25T18:13:04Z

Lduffy8:

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.
==Main Idea==
Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

'''Restricted 3-Body Problem'''

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

[[File:Three body problem figure-8 orbit animation.gif|250px|thumb|right|An example of a stable solution to the 3-body problem.]]
'''Other Solutions'''

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

==Examples==
''The moon has a mass of <math>m_{moon}</math> kg. If it is located a distance <math>d_1</math> m from the Earth (mass <math>m_E</math> kg) and a distance <math>d_2</math> m from the Sun (mass <math>m_{Sun}</math> kg). What are the magnitudes of the forces of the Sun on the Moon and the Earth on the Moon?''

Recall that the following equation applies to all bodies in this problem:
::*<math>F_g=\frac{GM_1M_2}{r^2}</math>
Here, we can replace the relevant distances and masses into the equation relatively easily to come up with the following force magnitudes:
::*<math>F_{Sun,Moon}=\frac{Gm_{moon}m_{Sun}}{d_2^2}</math> for the Sun on the Moon
::*<math>F_{Earth,Moon} = \frac{Gm_{moon}m_E}{d_1^2}</math> for the Earth on the Moon

For a fun extra reference, the Sun's mass is about 300,000 times greater than Earth's mass, but the Sun-Moon distance is about 389 times greater than the Earth-Moon distance. Via a simple fraction, as shown below, the magnitude of the force that the Earth exerts on the Moon is thus much lesser than the magnitude of the force the Sun exerts on the Moon, despite the fact that the Moon is orbiting around the Earth.
::*<math>\frac{F_{Earth}}{F_{Sun}}=\frac{m_E}{300000m_E}\frac{(389d_E)^2}{d_E^2}\approx{}\frac{150000}{300000}=0.5</math>

==Connectedness==
My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

==History==
The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when ''Principia'' was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of [https://en.wikipedia.org/wiki/Chaos_theory chaos theory].

==See Also==

Introduction to Magnetic Force

2019-07-01T22:02:31Z

Lduffy8: /* Middling */

Introduction to Magnetic Force

2019-07-01T22:01:51Z

Lduffy8: /* Middling */

Introduction to Magnetic Force

2019-07-01T22:01:22Z

Lduffy8: /* Middling */

Introduction to Magnetic Force

2019-07-01T21:56:34Z

Lduffy8: /* Middling */

File:IntroBimage.png

2019-07-01T21:55:57Z

Lduffy8: A problem diagram for the intro to B-field page.

A problem diagram for the intro to B-field page.

Introduction to Magnetic Force

2019-07-01T21:49:11Z

Lduffy8: /* Examples */

Main Page

2019-07-01T21:47:08Z

Lduffy8: /* Fundamental Interactions */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
*[[About]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Kinematics]]
*[[Projectile Motion]]
*[[Relativistic Momentum]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Analytical Prediction]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Heat Capacity]]
*[[Calorific Value(Heat of combustion)]]
*[[Specific Heat Capacity]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Angular Velocity]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Momentum Compared to Linear Momentum]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[Angular Momentum of Multiparticle Systems]]
*[[The Moments of Inertia]]
*[[Moment of Inertia for a cylinder]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Insulators====
<div class="mw-collapsible-content">
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Conductors====
<div class="mw-collapsible-content">
*[[Conductivity]]
*[[Charge Transfer]]
*[[Resistivity]]
*[[Polarization of a conductor]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

3 or More Body Interactions

2019-07-01T21:44:59Z

Lduffy8: /* Simple */

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.
==Main Idea==
Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

'''Restricted 3-Body Problem'''

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

[[File:Three body problem figure-8 orbit animation.gif|250px|thumb|right|An example of a stable solution to the 3-body problem.]]
'''Other Solutions'''

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

==Examples==
===Simple===
''The moon has a mass of <math>m_{moon}</math> kg. If it is located a distance <math>d_1</math> m from the Earth (mass <math>m_E</math> kg) and a distance <math>d_2</math> m from the Sun (mass <math>m_{Sun}</math> kg). What are the magnitudes of the forces of the Sun on the Moon and the Earth on the Moon?''

Recall that the following equation applies to all bodies in this problem:
::*<math>F_g=\frac{GM_1M_2}{r^2}</math>
Here, we can replace the relevant distances and masses into the equation relatively easily to come up with the following force magnitudes:
::*<math>F_{Sun,Moon}=\frac{Gm_{moon}m_{Sun}}{d_2^2}</math> for the Sun on the Moon
::*<math>F_{Earth,Moon} = \frac{Gm_{moon}m_E}{d_1^2}</math> for the Earth on the Moon

For a fun extra reference, the Sun's mass is about 300,000 times greater than Earth's mass, but the Sun-Moon distance is about 389 times greater than the Earth-Moon distance. Via a simple fraction, as shown below, the magnitude of the force that the Earth exerts on the Moon is thus much lesser than the magnitude of the force the Sun exerts on the Moon, despite the fact that the Moon is orbiting around the Earth.
::*<math>\frac{F_{Earth}}{F_{Sun}}=\frac{m_E}{300000m_E}\frac{(389d_E)^2}{d_E^2}\approx{}\frac{150000}{300000}=0.5</math>

==Connectedness==
My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

==History==
The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when ''Principia'' was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of [https://en.wikipedia.org/wiki/Chaos_theory chaos theory].

==See Also==

3 or More Body Interactions

2019-07-01T21:44:25Z

Lduffy8: /* Simple */

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.
==Main Idea==
Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

'''Restricted 3-Body Problem'''

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

[[File:Three body problem figure-8 orbit animation.gif|250px|thumb|right|An example of a stable solution to the 3-body problem.]]
'''Other Solutions'''

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

==Examples==
===Simple===
''The moon has a mass of <math>m_{moon}</math> kg. If it is located a distance <math>d_1</math> m from the Earth (mass <math>m_E</math> kg) and a distance <math>d_2</math> m from the Sun (mass <math>m_{Sun}</math> kg). What are the magnitudes of the forces of the Sun on the Moon and the Earth on the Moon?''

Recall that the following equation applies to all bodies in this problem:
::*<math>F_g=\frac{GM_1M_2}{r^2}</math>
Here, we can replace the relevant distances and masses into the equation relatively easily to come up with the following force magnitudes:
::*<math>F_{Sun,Moon}=\frac{Gm_{moon}m_{Sun}}{d_2^2}</math> for the Sun on the Moon
::*<math>F_{Earth,Moon} = \frac{Gm_{moon}m_E}{d_1^2}</math> for the Earth on the Moon

For a fun extra reference, the Sun's mass is about 300,000 times greater than Earth's mass, but the Sun-Moon distance is about 389 times greater than the Earth-Moon distance. Via a simple fraction, as shown below, the magnitude of the force that the Earth exerts on the Moon is thus much lesser than the magnitude of the force the Sun exerts on the Moon, despite the fact that the Moon is orbiting around the Earth.
::*<math>\frac{F_{Earth}}{F_{Sun}}=\frac{m_E}{300000m_E}\frac{(389d_E)^2}{d_E^2}\approx{}0.5</math>

==Connectedness==
My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

==History==
The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when ''Principia'' was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of [https://en.wikipedia.org/wiki/Chaos_theory chaos theory].

==See Also==

3 or More Body Interactions

2019-07-01T21:41:59Z

Lduffy8: /* Simple */

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.
==Main Idea==
Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

'''Restricted 3-Body Problem'''

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

[[File:Three body problem figure-8 orbit animation.gif|250px|thumb|right|An example of a stable solution to the 3-body problem.]]
'''Other Solutions'''

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

==Examples==
===Simple===
''The moon has a mass of <math>m_{moon}</math> kg. If it is located a distance <math>d_1</math> m from the Earth (mass <math>m_E</math> kg) and a distance <math>d_2</math> m from the Sun (mass <math>m_{Sun}</math> kg). What are the magnitudes of the forces of the Sun on the Moon and the Earth on the Moon?''

Recall that the following equation applies to all bodies in this problem:
::*<math>F_g=\frac{GM_1M_2}{r^2}</math>
Here, we can replace the relevant distances and masses into the equation relatively easily to come up with the following force magnitudes:
::*<math>F_{Sun,Moon}=\frac{Gm_{moon}m_{Sun}}{d_2^2}</math> for the Sun on the Moon
::*<math>F_{Earth,Moon} = \frac{Gm_{moon}m_E}{d_1^2}</math> for the Earth on the Moon

For reference, the Sun's mass is about 300,000 times greater than Earth's mass, but the Sun-Moon distance is about 389 times greater than the Earth-Moon distance. Via a simple fraction, as shown below, the magnitude of the force that the Earth exerts on the Moon is thus greater than the magnitude of the force the Sun exerts on the Moon.
::*<math>\frac{F_{Earth}}{F_{Sun}}=\frac{m_E}{300000m_E}\frac{(389d_E)^2}{d_E^2}=</math>

==Connectedness==
My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

==History==
The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when ''Principia'' was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of [https://en.wikipedia.org/wiki/Chaos_theory chaos theory].

==See Also==

3 or More Body Interactions

2019-07-01T21:41:30Z

Lduffy8: /* Examples */

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.
==Main Idea==
Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

'''Restricted 3-Body Problem'''

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

[[File:Three body problem figure-8 orbit animation.gif|250px|thumb|right|An example of a stable solution to the 3-body problem.]]
'''Other Solutions'''

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

==Examples==
===Simple===
''The moon has a mass of <math>m_{moon}</math> kg. If it is located a distance <math>d_1</math> m from the Earth (mass <math>m_E</math> kg) and a distance <math>d_2</math> m from the Sun (mass <math>m_{Sun}</math> kg). What are the magnitudes of the forces of the Sun on the Moon and the Earth on the Moon?''

Recall that the following equation applies to all bodies in this problem:
::*<math>F_g=\frac{GM_1M_2}{r^2}</math>
Here, we can replace the relevant distances and masses into the equation relatively easily to come up with the following force magnitudes:
::*<math>F_{Sun,Moon}=\frac{Gm_{moon}m_{Sun}}{d_2^2}</math> for the Sun on the Moon
::*<math>F_{Earth,Moon} = \frac{Gm_{moon}m_E}{d_1^2}</math> for the Earth on the Moon

For reference, the Sun's mass is about 300,000 times greater than Earth's mass, but the Sun-Moon distance is about 389 times greater than the Earth-Moon distance. Via a simple fraction, as shown below, the magnitude of the force that the Earth exerts on the Moon is thus greater than the magnitude of the force the Sun exerts on the Moon.
::<math>*\frac{F_{Earth}}{F_{Sun}}=\frac{m_E}{300000m_E}{(389d_E)^2}{d_E^2}=</math>

==Connectedness==
My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

==History==
The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when ''Principia'' was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of [https://en.wikipedia.org/wiki/Chaos_theory chaos theory].

==See Also==

3 or More Body Interactions

2019-07-01T21:31:01Z

Lduffy8: /* History */

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.
==Main Idea==
Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

'''Restricted 3-Body Problem'''

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

[[File:Three body problem figure-8 orbit animation.gif|250px|thumb|right|An example of a stable solution to the 3-body problem.]]
'''Other Solutions'''

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

==Examples==

==Connectedness==
My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

==History==
The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when ''Principia'' was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of [https://en.wikipedia.org/wiki/Chaos_theory chaos theory].

==See Also==

Gravitational Force in Space and Other Applications

2019-07-01T21:23:26Z

Lduffy8: /* Hard */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===
[http://physicsbook.gatech.edu/Gravitational_Force#Difficult Please see this problem, listed generally under the gravitational force page.]

==Connectedness==
Space has always fascinated me, starting from my time as a little kid. Looking up at the stars and wondering how they came to be and how they move has always been a part of my life. Knowing how gravity works at a massive level helps to further my understanding and quench my thirst for knowledge with regards to the stars.

I am a physics major, specifically concentrating in astrophysics, and thus this knowledge is extremely applicable to me. I do research in classical dynamics in space, and thus gravitational interactions are incredibly important to consider in my research. More broadly, however, the knowledge of how the planets move and their gravity can be used when conceptualizing how to send spacecraft in the most efficient way to other places in the solar system. Lots of launches make use of so-called [https://en.wikipedia.org/wiki/Gravity_assist gravity assist] maneuvers in order to get to the planet or body in question in the quickest possible manner without utilizing too much fuel.

[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html GPS devices] also make use of the notion of relativity in order to locate their targets. Without considering general relativity (which takes into account gravity in space), GPS devices would not function as planned and would be impossible to use.

==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

==See Also==
::*[https://www.science.org.au/curious/space-time/gravity This link] provides a general timeline of gravity, and dives into some basics of relativity with helpful diagrams.
::*[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html Here, one can find] a helpful article detailing how GPS Systems are, in fact, proof of the concept of general relativity.
::*[https://en.wikipedia.org/wiki/History_of_gravitational_theory A general overview] of the history of gravitational theory.
::*[http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html#grav Hyperphysics] page describing a general overview of gravity.

Gravitational Force in Space and Other Applications

2019-07-01T21:18:50Z

Lduffy8: /* See Also */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
Space has always fascinated me, starting from my time as a little kid. Looking up at the stars and wondering how they came to be and how they move has always been a part of my life. Knowing how gravity works at a massive level helps to further my understanding and quench my thirst for knowledge with regards to the stars.

I am a physics major, specifically concentrating in astrophysics, and thus this knowledge is extremely applicable to me. I do research in classical dynamics in space, and thus gravitational interactions are incredibly important to consider in my research. More broadly, however, the knowledge of how the planets move and their gravity can be used when conceptualizing how to send spacecraft in the most efficient way to other places in the solar system. Lots of launches make use of so-called [https://en.wikipedia.org/wiki/Gravity_assist gravity assist] maneuvers in order to get to the planet or body in question in the quickest possible manner without utilizing too much fuel.

[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html GPS devices] also make use of the notion of relativity in order to locate their targets. Without considering general relativity (which takes into account gravity in space), GPS devices would not function as planned and would be impossible to use.

==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

==See Also==
::*[https://www.science.org.au/curious/space-time/gravity This link] provides a general timeline of gravity, and dives into some basics of relativity with helpful diagrams.
::*[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html Here, one can find] a helpful article detailing how GPS Systems are, in fact, proof of the concept of general relativity.
::*[https://en.wikipedia.org/wiki/History_of_gravitational_theory A general overview] of the history of gravitational theory.
::*[http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html#grav Hyperphysics] page describing a general overview of gravity.

Gravitational Force in Space and Other Applications

2019-07-01T21:16:04Z

Lduffy8: /* See Also */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
Space has always fascinated me, starting from my time as a little kid. Looking up at the stars and wondering how they came to be and how they move has always been a part of my life. Knowing how gravity works at a massive level helps to further my understanding and quench my thirst for knowledge with regards to the stars.

I am a physics major, specifically concentrating in astrophysics, and thus this knowledge is extremely applicable to me. I do research in classical dynamics in space, and thus gravitational interactions are incredibly important to consider in my research. More broadly, however, the knowledge of how the planets move and their gravity can be used when conceptualizing how to send spacecraft in the most efficient way to other places in the solar system. Lots of launches make use of so-called [https://en.wikipedia.org/wiki/Gravity_assist gravity assist] maneuvers in order to get to the planet or body in question in the quickest possible manner without utilizing too much fuel.

[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html GPS devices] also make use of the notion of relativity in order to locate their targets. Without considering general relativity (which takes into account gravity in space), GPS devices would not function as planned and would be impossible to use.

==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

==See Also==
::*[https://www.science.org.au/curious/space-time/gravity This link] provides a general timeline of gravity, and dives into some basics of relativity with helpful diagrams.
::*[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html Here, one can find] a helpful article detailing how GPS Systems are, in fact, proof of the concept of general relativity.

Gravitational Force in Space and Other Applications

2019-07-01T21:13:40Z

Lduffy8: /* Connectedness */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
Space has always fascinated me, starting from my time as a little kid. Looking up at the stars and wondering how they came to be and how they move has always been a part of my life. Knowing how gravity works at a massive level helps to further my understanding and quench my thirst for knowledge with regards to the stars.

I am a physics major, specifically concentrating in astrophysics, and thus this knowledge is extremely applicable to me. I do research in classical dynamics in space, and thus gravitational interactions are incredibly important to consider in my research. More broadly, however, the knowledge of how the planets move and their gravity can be used when conceptualizing how to send spacecraft in the most efficient way to other places in the solar system. Lots of launches make use of so-called [https://en.wikipedia.org/wiki/Gravity_assist gravity assist] maneuvers in order to get to the planet or body in question in the quickest possible manner without utilizing too much fuel.

[http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html GPS devices] also make use of the notion of relativity in order to locate their targets. Without considering general relativity (which takes into account gravity in space), GPS devices would not function as planned and would be impossible to use.

==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

==See Also==

Gravitational Force in Space and Other Applications

2019-07-01T21:13:12Z

Lduffy8: /* Connectedness */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
Space has always fascinated me, starting from my time as a little kid. Looking up at the stars and wondering how they came to be and how they move has always been a part of my life. Knowing how gravity works at a massive level helps to further my understanding and quench my thirst for knowledge with regards to the stars.

I am a physics major, specifically concentrating in astrophysics, and thus this knowledge is extremely applicable to me. I do research in classical dynamics in space, and thus gravitational interactions are incredibly important to consider in my research. More broadly, however, the knowledge of how the planets move and their gravity can be used when conceptualizing how to send spacecraft in the most efficient way to other places in the solar system. Lots of launches make use of so-called [https://en.wikipedia.org/wiki/Gravity_assist gravity assist] maneuvers in order to get to the planet or body in question in the quickest possible manner without utilizing too much fuel.

GPS devices also make use of the notion of relativity in order to locate their targets. Without considering general relativity (which takes into account gravity in space), GPS devices would not function as planned and would be impossible to use.

==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

==See Also==

Gravitational Force in Space and Other Applications

2019-07-01T21:09:38Z

Lduffy8: /* Connectedness */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
Space has always fascinated me, starting from my time as a little kid. Looking up at the stars and wondering how they came to be and how they move has always been a part of my life. Knowing how gravity works at a massive level helps to further my understanding and quench my thirst for knowledge with regards to the stars.

==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

==See Also==

Gravitational Force in Space and Other Applications

2019-07-01T21:07:33Z

Lduffy8:

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

==See Also==

Gravitational Force in Space and Other Applications

2019-07-01T21:07:06Z

Lduffy8: /* History */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|400px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

Gravitational Force in Space and Other Applications

2019-07-01T21:06:53Z

Lduffy8: /* History */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

[[File:grtest1.png|200px|thumb|right|An image depicting how light is deflected around massive bodies, causing us to view objects at an 'apparent' position that differs from their actual positon.]]
As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

File:Grtest1.png

2019-07-01T21:06:10Z

Lduffy8: One of the proofs of relativity, the deflection of light around massive bodies can be seen drawn out here. Compare the apparent position with the actual position of the star.

One of the proofs of relativity, the deflection of light around massive bodies can be seen drawn out here. Compare the apparent position with the actual position of the star.

Gravitational Force in Space and Other Applications

2019-07-01T21:04:57Z

Lduffy8: /* History */

This page looks into the gravitational force in space, and other applications of the gravitational force. Kepler's Laws are also explained here.

==Main Idea==
The gravitational force, as it acts in space, can be used to explain the motions of celestial bodies, and thus is of extreme interest when predicting future events in our Solar System.

Johannes Kepler, prior to Newton, had created laws to explain the motion of celestial bodies. His laws, explained below, were further proven by Newton's idea of gravity, which can explain why these laws are true.

[[File:Kepler_1.png|200px|thumb|right|A diagram depicting Kepler's 1st Law: Namely, that planets revolve around the central body in an elliptical orbit, with the central body at one of the foci. The eccentricity of the ellipse has been greatly exaggerated here.]]
::* Kepler's first law states that all planets move about the Sun in an elliptical orbit, with the Sun at one of the foci of the ellipse. This can be further elaborated to say that every body orbits around the center of mass in its system on an elliptical orbit. For example, in the Earth-Moon system, the Moon orbits around the Earth in an elliptical orbit, with the Earth at one of the foci of the ellipse. An ellipse, in this case, can be a circle, oval, parabola, or hyperbola, depending on the type of motion of the orbiting body. The gravity of the Sun keeps the object bound on the orbit, with the force of gravity always pointing radially inwards towards the Sun. This constantly changing direction of gravitational force is what keeps the planet bound.

[[File:Kepler_2.png|200px|thumb|right|A diagram depicting Kepler's 2nd law. The pink areas are the same, and the time it took the planet to sweep out the areas was also the same. The black dot is the central body (for us, the Sun).]]
::* Kepler's second law, the law of equal areas, states that a planetary body sweeps out an equal area in its orbit in equal time. This is also shown in a diagram. Basically, if you draw a line from the planet to the Sun, the area that is created in some time segment by the motion of the planet will be the same as the area created in an equivalent time segment at a different point in the objects orbit. This can also be explained by the gravitational force, and the conservation of angular momentum, which will be covered later in the course.

[[File:Semimajor_explanation.png|200px|thumb|left|A diagram showing where the semi-major and semi-minor axes of an ellipse are located.]]
::* Kepler's third law states that the square of the orbital period of the planet (the time it takes to go around the Sun) is directly proportional to the cube of its semi-major axis (which is half of the length of the longer axis of the ellipse, and is shown in a diagram). This law is incredibly useful, especially in determining the semi-major axis of exoplanets when we know their period from transits (it can be detected when the planet passes in front of the star. By measuring the timing of these transits, the period of the planet can be determined).

Newton's law of gravitation also functions in space much the same as it does almost everywhere else classically, and expressions of this force can be used to determine whether or not planets or other celestial bodies are bound to a star, or if they are just floating through space.
===A Mathematical Model===
The easiest of Kepler's three laws to formulate mathematically is Kepler's 3rd Law. By setting the net force equal to the gravitational force, we can derive a simple equation for the period of the planet.

:: <math> F_{net}=ma=\frac{Gm_{1}m_{2}}{r^{2}} </math>

In a circular orbit, the acceleration of planet is equal to the square of its velocity over the radius of the orbit. This will be further explained later, when the centripetal force is covered, but for this purpose, we can accept this relationship and replace ''ma'' in our equation.

::<math> F_{net}=\frac{mv^2}{r}= \frac{Gm_{1}m_{2}}{r^{2}} </math>

Through simple manipulation of this net force, we can determine that the period (which is equal to <math>\frac{2{\pi}r}{v}</math>) squared is proportional to the radius cubed. While this is just a simple derivation for a circular orbit, the same holds true for elliptical orbits, and, in fact, when you use years for the units of period, and astronomical units (1 AU = the distance from the Earth to the Sun) for the semi-major axis (''a''), the following equation holds true for the solar system:
::* <math>P^2=a^3</math>
A more rigorous approach (with mass in solar masses, and <math>M_{total}</math> representing the total mass of the two bodies) provides the following equation:
::*<math>(M_{total})P^2=a^3</math>

==Examples==
===Simple===
''If a planet is revolving around a star with the same mass as the Sun with a period of 2 years, what is the semimajor axis of that planet?''

Recall, we know the formula to find the semi-major axis if period is known, as is shown below:
::<math> P^2 = a^3 </math>
Here, P = 2 years, which can then be replaced into the formula to find the final answer.
::<math> a^3 = 2^2 = 4 </math>
::<math> a = (4)^{1/3} = 1.587 AU</math>

===Middling===
''Imagine a stellar binary. In this system, there is a 1.1 solar mass star and a 3 solar mass star orbiting each other with a period of 30 days. What is the semi-major axis of this orbit?''

First, the period of this system must be converted from days to years.
::<math> years = days/365.25=30/365.25= 0.0821 years</math>
This value for years can then be put into the following equations:
::<math>(M_1+M_2)P^2=a^3</math>
::<math>a=((M_1+M_2)P^2)^{1/3}=((1.1+3)(0.0821)^2))^{1/3}</math>
::<math>a=0.3023 AU</math>
Thus, the semimajor axis of the system is 0.3023 AU.

===Hard===

==Connectedness==
==History==
Isaac Newton made the connection between what keeps the planets rotating around the Sun and what keeps us standing on the Earth. In 1687, when the ''Principia'' was published, Newton noted that the force that held the planets on their orbits, and the force that kept the moon on its orbit around the Sun, must be the same.

Through Newton's theory, the existence of Neptune was predicted and subsequently confirmed.

As we know now, however, there are some slight discrepancies between observations and Newton's theory that couldn't be explained prior to the work of Albert Einstein. These discrepancies could, primarily, be noted in Space. One of these, for example, was an issue that arose in the way that Mercury's orbit changed over time. Simply put, there were some perturbations in the orbit that simply could not be explained by Newton's theory of gravity. This issue was first noted in 1859 by Le Verrier (who had also used gravity to predict the existence of Neptune, in fact). These precessions would not be fully explained until the early twentieth century (1915ish) when Albert Einstein first proposed the theory of relativity. Furthermore, the deflection of light around massive bodies such as the Sun, as can be seen in the diagram to the right, although somewhat predicted by Newton's theories, occurred to a much greater degree than Newton had predicted. These discrepancies, however, were perfectly in accordance with general relativity. It was in 1919 that Eddington measured these deflections.

Gravitational Force in Space and Other Applications

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Lduffy8: /* Middling */

Introduction to Magnetic Force

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Lduffy8: /* Main Idea */

Introduction to Magnetic Force

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Lduffy8: /* Simple */

Introduction to Magnetic Force

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Lduffy8: /* A Mathematical Model */

Introduction to Magnetic Force

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Lduffy8: /* A Mathematical Model */

Introduction to Magnetic Force

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Lduffy8: /* A Mathematical Model */

Introduction to Magnetic Force

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Lduffy8: /* A Mathematical Model */

Introduction to Magnetic Force

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Lduffy8: /* A Mathematical Model */

Introduction to Magnetic Force

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Lduffy8: /* A Mathematical Model */

Introduction to Magnetic Force

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Lduffy8: /* Main Idea */

Introduction to Magnetic Force

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Lduffy8: /* A Mathematical Model */

This page provides a short introduction to the concept of the magnetic force.
==Main Idea==
===A Mathematical Model===
Magnetic force can be represented as follows, where ''q'' is the charge on a particle, ''v' is that particles velocity, and ''B'' is the magnetic field.
::*<math>\vec{\mathbf{F_B}}=q\vec{\mathbf{v}}\times{}\vec{\mathbf{B}}</math>

==Examples==
===Simple===
===Middling===
===Hard===
==Connectedness==
==History==
==See Also==

Introduction to Magnetic Force

2019-06-27T03:03:43Z

Lduffy8: Created page with "This page provides a short introduction to the concept of the magnetic force. ==Main Idea== ===A Mathematical Model=== Magnetic force can be represented as follows, where ''q'..."

This page provides a short introduction to the concept of the magnetic force.
==Main Idea==
===A Mathematical Model===
Magnetic force can be represented as follows, where ''q'' is the charge on a particle, ''v' is that particles velocity, and ''B'' is the magnetic field.
::*<math>\mathbf{F_B}=q\mtahbf{v}\times{}\mathbf{B}</math>
==Examples==
===Simple===
===Middling===
===Hard===
==Connectedness==
==History==
==See Also==

Main Page

2019-06-27T02:59:29Z

Lduffy8: /* Fundamental Interactions */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Help with VPython====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Kinematics]]
*[[Projectile Motion]]
*[[Relativistic Momentum]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Analytical Prediction]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]
*[[Fluid Mechanics]]
*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Momentum with respect to external Forces]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Heat Capacity]]
*[[Calorific Value(Heat of combustion)]]
*[[Specific Heat Capacity]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Models of Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Angular Velocity]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Momentum Compared to Linear Momentum]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[Angular Momentum of Multiparticle Systems]]
*[[The Moments of Inertia]]
*[[Moment of Inertia for a cylinder]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Insulators====
<div class="mw-collapsible-content">
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Conductors====
<div class="mw-collapsible-content">
*[[Conductivity]]
*[[Charge Transfer]]
*[[Resistivity]]
*[[Polarization of a conductor]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Gravitational Force in Space and Other Applications

2019-06-27T02:57:45Z

Lduffy8: /* Middling */

Gravitational Force in Space and Other Applications

2019-06-27T02:57:15Z

Lduffy8: /* Middling */

Gravitational Force in Space and Other Applications

2019-06-27T02:56:53Z

Lduffy8: /* Middling */

Gravitational Force in Space and Other Applications

2019-06-27T02:52:09Z

Lduffy8: /* A Mathematical Model */

Gravitational Force in Space and Other Applications

2019-06-27T02:50:59Z

Lduffy8: /* Hard */