Physics Book - User contributions [en]

String Theory

2016-11-26T02:52:27Z

Mdickerson30:

Claimed by Madeleine Dickerson

==Introduction to String Theory==
String theory is a theoretical framework aimed at addressing the short-comings of Einstein's theory of general relativity and quantum physics - two primary tools for understanding modern physics. String theory is often referred to as the ''theory of everything'' - a just identification given its ability to bridge the gap between gravity and quantum physics. The theory of general relativity and quantum mechanics are the primary, established explanations for the way the universe works on a macroscopic and microscopic scale, respectively. However, these two physical explanations address different ends of the spectrum of matter, in terms of relativistic effects, fundamental forces, and physical properties. String theory provides explanations where modern physics fails. These fields of interest include the early universe, black holes, and atomic nuclei. In order to understand the purpose of the ''theory of everything'' it is vital to have a sound understanding of general relativity's purpose as well as the role of relativistic quantum mechanics - that is, in terms of gravitational effects on matter and the fundamental physics of the universe. General relativity serves an important purpose in determining the gravitational affects on planetary-sized objects and observable particles, for example. However, the "gap" in Einstein's theory lies within its inability to address gravity's effect on elementary particles. General relativity successfully explains macro-scaled particles, given it's ability to incorporate gravitational effects in its theoretical and mathematical foundation. However, the same cannot be said for elementary particles. Accordingly, these elementary particles require relativistic quantum mechanics to explain its physical properties, with the underlying assumption that gravity's affect on this scale is negligible. This is the disconnect between modern physics' primary schools of thought. As a result, the purpose of the string theory is realized in its ability to successfully incorporate gravitational affects, independent of the scale of particles. Thus, the ''theory of everything'' emerged.

''See reference 3.''

===Relevant Mathematical Equations (Bekenstein-Hawking Formula)===

:<math>S= \frac{c^3kA}{4\hbar G}</math>

The Bekenstein-Hawking formula for entropy of a black hole, shown above, provides a theoretical value for the entropy of a black hole. While this equation provides an expected value that correlates to the entropy of a black hole, according to macroscopic features resulting from its microstates, the formula lacks a derivation for a black hole's entropy based on counting microstates on a quantum scale with consideration of gravitational aspects. In 1996, Andrew Strominger and Cumrun Vafa provided a derivation for a black hole's entropy with respect to the number of microstates of a black hole, and in terms of the string theory. The results obtained from these calculations provided solutions matching Berkenstein and Hawking's original formula. Thus, string theory was validated as a mode of quantum gravity.

See ''reference 1''.

===Theoretical Foundation===

String theory is founded on the basis that elementary particles are components of strings that are microscopically small - to a degree that technology is not available to visualize. A valid and common depiction of strings lined with elementary particles is a visualization of these same particles laying on a violin string. String theory advocates propose that elementary particles are believed to be in "excited" states due to vibrations in these strings. This is a significant given the bending of spacetime, existence of black holes and wormholes - solutions to Einstein's equations which are not explained entirely by quantum physics or general relativity in their own right. On a theoretical basis, string theory provides solutions based on gravity acting on elementary particles (theories of quantum gravity), a unification of quantum mechanics and Einstein's theory of general relativity.

[[File:String Theory.jpg]]

''See reference 2''.

==Potential Discoveries and Applications==

===Wormholes===
A potential usage of string theory is to provide the quantum gravitational solutions that Einstein's theory of general relativity fail to recognize at the center of black holes or worm holes. This inability is due to its lack of consideration of quantum forces alongside gravity. String theory is integral to discovering wormholes' (derived solutions to Einstein's equations) level of stability. Specifically, string theory's quantum forces and gravity enables the determination of the radiative affects and stability of these unexplained astrophysical phenomenon. In conclusion, these recently gained understandings of string theory may ultimately yield answers to questions such as...

1. Given Kerr wormholes and their potential to connect distant points in the universe, is it theoretically feasible to use these solutions to travel vast lengths through the universe?

''See reference 2''.

===Wormhole Diagram===

[[File:w-003.jpg]]

==Connectedness==
#How is this topic connected to something that you are interested in? String Theory is a topic that is particularly intriguing due to far-reaching applications that today are widely considered science fiction. (i.e. interstellar space travel utilizing wormholes).
#How is it connected to your major? While string theory does not have any direct application in Mechanical Engineering, a general understanding of the basics yield a profound understanding o the way in which the physical world operates - a vital component in the engineering field.
#Is there an interesting industrial application? Industrial applications for string theory may not be realized in the near future, but in the long-run an understanding of string theory could have a profound, yet unforeseen, impact on the world - in a similar revolutionary and unpredicted manner as that of quantum mechanics and the development of modern communication and flow of information.

== Scientific Development Timeline==

'''1970''': String theory is proposed to understand the quantum mechanics of oscillating strings.

'''1976''': Supergravity is proposed as a means of explaining the interdependence of gravity and sub-atomic particles' spectrum of excitations - an integral component in string theory.

'''1980''': Initially perceived to discredit string theory as a unification of quantum mechanics, classical physics, and particle physics, the inconsistencies of the theory were found to nullify each other when considered as special cases. This is the year that string theory becomes widely accepted as a potential unifying explanation in the scientific community of the time.

'''1991-1995''': String theory exploration, in terms of black holes, results in development towards the understanding of the different forms of string theory, in terms of their relationship.

'''1996''': String theory provides microscopic understanding of black hole entropy and the nature of black hole quantum physics.

''See reference 3''.

== Additional Resources ==

===Further reading===

[http://www.physicsbook.gatech.edu/Elementary_Particles_and_Particle_Physics_Theory Elementary Particles and Particle Physics Theory]

[http://www.physicsbook.gatech.edu/Quantum_Theory Quantum Theory]

[http://www.physicsbook.gatech.edu/Big_Bang_Theory Big Bang Theory]

===External links===

[http://www.superstringtheory.com/index.html Superstring Theory]

[http://mkaku.org/home/articles/blackholes-wormholes-and-the-tenth-dimension/ Wormholes and Blackholes]

[http://web.physics.ucsb.edu/~strings/superstrings/bholes.htm String Theory]

[https://www.youtube.com/watch?v=YtdE662eY_M Brian Greene: Making Sense of String Theory (YouTube)]

==References==

'''Reference 1''': "SUPERSTRINGS! Black Holes ." <i>Web.physics.ucsb.edu</i>. UCSB, n.d. Web. 03 Dec. 2015. <http://web.physics.ucsb.edu/~strings/superstrings/bholes.htm>.

'''Reference 2''': Kaku, Micho, Dr. "Blackholes, Wormholes and the Tenth Dimension." <i>Mkaku.org</i>. N.p., n.d. Web. 03 Dec. 2015. <http://mkaku.org/home/articles/blackholes-wormholes-and-the-tenth-dimension/>.

'''Reference 3''': Shwarz, Patricia, Dr. "The Official String Theory Web Site." <i>Superstringtheory.com</i>. N.p., n.d. Web. 03 Dec. 2015. <http://www.superstringtheory.com/index.html>.

[[Category:Theory]]

Escape Velocity

2016-10-31T20:45:56Z

Mdickerson30:

[[File:Escape-velocity-diagram.gif|400px|thumb|right| A diagram showing several orbits around Earth. Orbit <math>p</math> is a parabolic orbit, with the object travelling at escape velocity. Orbit <math>H</math> is an example of the hyperbolic orbit of an object travelling faster than the escape velocity.]]
CLAIMED BY MADELEINE DICKERSON

Escape velocity is defined as the minimum velocity required for an object to escape the gravitational force of a large object. The sum of an object's kinetic energy and its gravitational potential energy is equal to zero. The gravitational potential energy is negative due to the fact that kinetic energy is always positive. The velocity of the object will be be zero at infinite distance from the center of gravity. There is no net force on an object as it escapes and zero acceleration is perceived.

==The Main Idea==

The formula for escape velocity at a givendistance from a body is calculated by the formula
:<math>v_e = \sqrt{\frac{2GM}{r}},</math>

where <math>G</math> is the universal [[gravitational constant]] (<math>G = 6.7\times 10^{-11} \text{N}\cdot\text{m}^2 / \text{kg}^2</math>), <math>M</math> is the mass of the large body to be escaped, and <math>r</math> the distance from the [[center of mass]] of the mass <math>M</math> to the object. This equation assumes there are no other forces acting on either body. As a side note, the escape velocity stated here could really be called escape speed due to the fact that the quantity is independent of direction. Notice that the equation does not include the mass of the orbiting body.

===A Mathematical Model===

When we model escape velocity, we consider the situation when an object's velocity takes it to a point an infinite distance away. The lowest possible escape velocity has a final speed of zero, and any speed higher results in a nonzero final speed. To derive the formula for the escape velocity, the energy principle is used, and we assume that the only two objects in our system are the orbiting body and the planet.

In our system , the energy principle states that

:<math>(K + U_g)_i = (K + U_g)_f \,</math>

When finding the minimum escape velocity, <math>K-f = 0 </math> because we take the final velocity to be zero, and <math>U_{gf} = 0 </math> because its final distance is expressed as infinity, therefore

:<math>(K + U_g)_i = \frac{1}{2}mv_e^2 + \frac{-GMm}{r} = 0</math>

Solving for <math>v_e</math> yields:

:<math>v_e = \sqrt{\frac{2GM}{r}}</math>

Note that because both the kinetic and potential energy terms contain a common factor <math>m</math>, the final escape velocity is independent of the mass of the orbiting body.

===Bound versus unbound systems===

When an object is orbiting a massive body, it can be in one of two states: bound and unbound. If the object is in a bound state, we see an elliptical trajectory, in which the orbiting body never escapes the gravitational influence of the more massive body. In an unbound state, however, we observe a parabolic or hyperbolic trajectory, in which the object is able to escape the gravitational influence of the orbiting body and escape to infinity. The diagram to the left shows an unbound system, in which the sum of the kinetic and potential energy of the orbiting body is greater then 0. As distance goes to infinity in this system, gravitational potential energy approaches zero, but the object retains a positive kinetic energy, and therefore a positive velocity. The image on the right shows a bound system, in which the sum of the kinetic and potential energy is negative. In this system, kinetic energy reaches zero at a specific maximum distance, at which point the object begins to fall back towards the massive body, never to escape.

[[File:Ediagram_MSPaint_UnboundSystemWithExtraEnergy.png|500px|thumb|left|The energy diagram of an unbound system, in which the object has excess kinetic energy. The vertical axis represents energy, while the horizontal axis represents distance. At exactly the escape velocity, the sum of <math>K</math> and <math>U</math> is exactly 0.]]
[[File:EDiagram_MSPaint_BoundSystem.png|500px|thumb|right|The energy diagram of a bound system, in which the object has insufficient kinetic energy to escape. The vertical axis represents energy, while the horizontal axis represents distance. ]]

<br clear=all>

==Examples==

===Escaping Jupiter===
The radius of Jupiter is <math>71.5\times 10^6 \text{m}</math>, and its mass is <math>1900\times 10^{24} \text{kg}</math>. What is the escape speed of an object launched straight up from just above the atmosphere of Jupiter?

<div style="">
System = Jupiter + object
<math>

\Delta E = 0\\
v_i = ?\\
v_f = 0 \text{m/s}\\
r_i = 71.5\times 10^6 \text{m}\\
r_f = \infty\\
m = m_{Object}\\
M = m_{Jupiter} = 1900\times 10^{24} \text{kg} \\

</math>

Starting from the Energy Principle:

<math>

\Delta E = W + Q\\
\Delta E = 0 + 0 = 0\\
\Delta K + \Delta U = 0\\
\frac{1}{2}m(v_f^2-v_i^2) + (\frac{-GMm}{r_f} - \frac{-GMm}{r_i}) = 0\\
\frac{1}{2}m(0-v_i^2) + (0 - \frac{-GMm}{r_i}) = 0\\
-\frac{1}{2}mv_i^2 + \frac{GMm}{r_i} = 0\\
\frac{GMm}{r_i} = \frac{1}{2}mv_i^2\\
\frac{GM}{r_i} = \frac{1}{2}v_i^2\\
v_i = \sqrt{\frac{2GM}{r_i}}\\
= \sqrt{\frac{2(6.7\times 10^{-11} \text{N}\cdot\text{m}^2 / \text{kg}^2)(1900\times 10^{24} \text{kg})}{(71.5\times 10^6 \text{m})}}\\
= 5.97 \times 10^4 \text{m/s}
</math>

===Escaping Earth===
Compute the escape velocity for Earth if its mass is <math>5.98 \times 10^{24} \text{kg}</math> and its radius is <math>6.37 \times 10^{6} \text{m}</math>.
<math>
v_e = \sqrt{\frac{2GM}{r}} \\
= \sqrt{\frac{2(6.7\times 10^{-11} \text{N}\cdot\text{m}^2 / \text{kg}^2)(5.98\times 10^{24} \text{kg})}{(6.37\times 10^6 \text{m})}}\\
= 1.12 \times 10^4 \text{m/s}
</math>

</div>

==Connectedness==

In real life, escape velocity is not as easy to calculate as in the above examples. The primary complicating factor is the fact that there are more than two bodies in the universe, so in systems with multiple massive attracting bodies, escape velocity can become more complicated. One example is the escape velocity of an object from Earth: an object that achieves the escape velocity of Earth could theoretically escape Earth's influence, but it would remain in orbit around the sun unless its speed were much greater. The concept of escape velocity is widely used in orbital mechanics and rocketry and is critical for the planning of space missions.

===External links===
http://www.scientificamerican.com/article/bring-science-home-reaction-time/

https://www.youtube.com/watch?v=7w56rwAtUZU

==References==
"Escape Velocity | Physics." Encyclopedia Britannica Online. Encyclopedia Britannica, n.d. Web. 05 Dec. 2015.
[http://www.britannica.com/science/escape-velocity]

Giancoli, Douglas C. "Physics for Scientists and Engineers with Modern Physics." Google Books. Google, n.d. Web. 05 Dec. 2015.
[https://books.google.com/books?id=xz-UEdtRmzkC&pg=PA199&dq=escape+velocity+gravitational+potential+energy&hl=en&sa=X&ved=0CC0Q6AEwA2oVChMI8_PO4_PBxwIVBJmICh3T6gGl#v=onepage&q=escape%20velocity%20gravitational%20potential%20energy&f=false]

"Escape Velocity." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.
[https://en.wikipedia.org/wiki/Escape_velocity]

Velocity, Escape, and ©200. ESCAPE VELOCITY EXAMPLES (n.d.): n. pag. 13 June 2003. Web. 5 Dec. 2015.
[http://www.beaconlearningcenter.com/documents/1483_01.pdf]

Escape Velocity

2016-10-31T20:45:40Z

Mdickerson30:

[[File:Escape-velocity-diagram.gif|400px|thumb|right| A diagram showing several orbits around Earth. Orbit <math>p</math> is a parabolic orbit, with the object travelling at escape velocity. Orbit <math>H</math> is an example of the hyperbolic orbit of an object travelling faster than the escape velocity.]]
CLAIMED BY MADELEINE DICKERSON
Escape velocity is defined as the minimum velocity required for an object to escape the gravitational force of a large object. The sum of an object's kinetic energy and its gravitational potential energy is equal to zero. The gravitational potential energy is negative due to the fact that kinetic energy is always positive. The velocity of the object will be be zero at infinite distance from the center of gravity. There is no net force on an object as it escapes and zero acceleration is perceived.

==The Main Idea==

The formula for escape velocity at a givendistance from a body is calculated by the formula
:<math>v_e = \sqrt{\frac{2GM}{r}},</math>

where <math>G</math> is the universal [[gravitational constant]] (<math>G = 6.7\times 10^{-11} \text{N}\cdot\text{m}^2 / \text{kg}^2</math>), <math>M</math> is the mass of the large body to be escaped, and <math>r</math> the distance from the [[center of mass]] of the mass <math>M</math> to the object. This equation assumes there are no other forces acting on either body. As a side note, the escape velocity stated here could really be called escape speed due to the fact that the quantity is independent of direction. Notice that the equation does not include the mass of the orbiting body.

===A Mathematical Model===

When we model escape velocity, we consider the situation when an object's velocity takes it to a point an infinite distance away. The lowest possible escape velocity has a final speed of zero, and any speed higher results in a nonzero final speed. To derive the formula for the escape velocity, the energy principle is used, and we assume that the only two objects in our system are the orbiting body and the planet.

In our system , the energy principle states that

:<math>(K + U_g)_i = (K + U_g)_f \,</math>

When finding the minimum escape velocity, <math>K-f = 0 </math> because we take the final velocity to be zero, and <math>U_{gf} = 0 </math> because its final distance is expressed as infinity, therefore

:<math>(K + U_g)_i = \frac{1}{2}mv_e^2 + \frac{-GMm}{r} = 0</math>

Solving for <math>v_e</math> yields:

:<math>v_e = \sqrt{\frac{2GM}{r}}</math>

Note that because both the kinetic and potential energy terms contain a common factor <math>m</math>, the final escape velocity is independent of the mass of the orbiting body.

===Bound versus unbound systems===

When an object is orbiting a massive body, it can be in one of two states: bound and unbound. If the object is in a bound state, we see an elliptical trajectory, in which the orbiting body never escapes the gravitational influence of the more massive body. In an unbound state, however, we observe a parabolic or hyperbolic trajectory, in which the object is able to escape the gravitational influence of the orbiting body and escape to infinity. The diagram to the left shows an unbound system, in which the sum of the kinetic and potential energy of the orbiting body is greater then 0. As distance goes to infinity in this system, gravitational potential energy approaches zero, but the object retains a positive kinetic energy, and therefore a positive velocity. The image on the right shows a bound system, in which the sum of the kinetic and potential energy is negative. In this system, kinetic energy reaches zero at a specific maximum distance, at which point the object begins to fall back towards the massive body, never to escape.

[[File:Ediagram_MSPaint_UnboundSystemWithExtraEnergy.png|500px|thumb|left|The energy diagram of an unbound system, in which the object has excess kinetic energy. The vertical axis represents energy, while the horizontal axis represents distance. At exactly the escape velocity, the sum of <math>K</math> and <math>U</math> is exactly 0.]]
[[File:EDiagram_MSPaint_BoundSystem.png|500px|thumb|right|The energy diagram of a bound system, in which the object has insufficient kinetic energy to escape. The vertical axis represents energy, while the horizontal axis represents distance. ]]

<br clear=all>

==Examples==

===Escaping Jupiter===
The radius of Jupiter is <math>71.5\times 10^6 \text{m}</math>, and its mass is <math>1900\times 10^{24} \text{kg}</math>. What is the escape speed of an object launched straight up from just above the atmosphere of Jupiter?

<div style="">
System = Jupiter + object
<math>

\Delta E = 0\\
v_i = ?\\
v_f = 0 \text{m/s}\\
r_i = 71.5\times 10^6 \text{m}\\
r_f = \infty\\
m = m_{Object}\\
M = m_{Jupiter} = 1900\times 10^{24} \text{kg} \\

</math>

Starting from the Energy Principle:

<math>

\Delta E = W + Q\\
\Delta E = 0 + 0 = 0\\
\Delta K + \Delta U = 0\\
\frac{1}{2}m(v_f^2-v_i^2) + (\frac{-GMm}{r_f} - \frac{-GMm}{r_i}) = 0\\
\frac{1}{2}m(0-v_i^2) + (0 - \frac{-GMm}{r_i}) = 0\\
-\frac{1}{2}mv_i^2 + \frac{GMm}{r_i} = 0\\
\frac{GMm}{r_i} = \frac{1}{2}mv_i^2\\
\frac{GM}{r_i} = \frac{1}{2}v_i^2\\
v_i = \sqrt{\frac{2GM}{r_i}}\\
= \sqrt{\frac{2(6.7\times 10^{-11} \text{N}\cdot\text{m}^2 / \text{kg}^2)(1900\times 10^{24} \text{kg})}{(71.5\times 10^6 \text{m})}}\\
= 5.97 \times 10^4 \text{m/s}
</math>

===Escaping Earth===
Compute the escape velocity for Earth if its mass is <math>5.98 \times 10^{24} \text{kg}</math> and its radius is <math>6.37 \times 10^{6} \text{m}</math>.
<math>
v_e = \sqrt{\frac{2GM}{r}} \\
= \sqrt{\frac{2(6.7\times 10^{-11} \text{N}\cdot\text{m}^2 / \text{kg}^2)(5.98\times 10^{24} \text{kg})}{(6.37\times 10^6 \text{m})}}\\
= 1.12 \times 10^4 \text{m/s}
</math>

</div>

==Connectedness==

In real life, escape velocity is not as easy to calculate as in the above examples. The primary complicating factor is the fact that there are more than two bodies in the universe, so in systems with multiple massive attracting bodies, escape velocity can become more complicated. One example is the escape velocity of an object from Earth: an object that achieves the escape velocity of Earth could theoretically escape Earth's influence, but it would remain in orbit around the sun unless its speed were much greater. The concept of escape velocity is widely used in orbital mechanics and rocketry and is critical for the planning of space missions.

===External links===
http://www.scientificamerican.com/article/bring-science-home-reaction-time/

https://www.youtube.com/watch?v=7w56rwAtUZU

==References==
"Escape Velocity | Physics." Encyclopedia Britannica Online. Encyclopedia Britannica, n.d. Web. 05 Dec. 2015.
[http://www.britannica.com/science/escape-velocity]

Giancoli, Douglas C. "Physics for Scientists and Engineers with Modern Physics." Google Books. Google, n.d. Web. 05 Dec. 2015.
[https://books.google.com/books?id=xz-UEdtRmzkC&pg=PA199&dq=escape+velocity+gravitational+potential+energy&hl=en&sa=X&ved=0CC0Q6AEwA2oVChMI8_PO4_PBxwIVBJmICh3T6gGl#v=onepage&q=escape%20velocity%20gravitational%20potential%20energy&f=false]

"Escape Velocity." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.
[https://en.wikipedia.org/wiki/Escape_velocity]

Velocity, Escape, and ©200. ESCAPE VELOCITY EXAMPLES (n.d.): n. pag. 13 June 2003. Web. 5 Dec. 2015.
[http://www.beaconlearningcenter.com/documents/1483_01.pdf]