Thursday, July 5, 2012

Two days ago, the CERN team announced that they had found a new particle whose properties are consistent with the long sought-after Higgs Boson’s. Whether or not it is the elusive boson however, is still to be determined by further research. To read more about the event, follow this link to the new BBC article.

If you have no clue what this is about, the above video is a quick and nice introduction to the Higgs Boson submitted by one of our followers, the lovely oh-yeah-and-what. Thanks for the awesome submission!
SIWS loves feedback from followers and we’ll do our best to respond. If you have any questions, ideas, or concerns, feel free to drop us a message, email us at sayitwithscience@gmail.com or like and post on our Facebook page. You can even make a submission post and we might publish it and credit you, like we did with this one!
Take care and happy science-ing! 

Wednesday, October 12, 2011
Hypercubes
What is a hypercube (also referred to as a tesseract) you say! Well, let’s start with what you know already. We know what a cube is, it’s a box! But how else could you describe a cube? A cube is 3 dimensional. Its 2 dimensional cousin is a square. 
A hypercube is just to a cube what a cube is to a square. A hypercube is 4 dimensional! (Actually— to clarify, hypercubes can refer to cubes of all dimensions. “Normal” cubes are 3 dimensional, squares are 2 dimensional “cubes, etc. This is because a hypercube is an n-dimensional figure whose edges are aligned in each of the space’s dimensions, perpendicular to each other and of the same length. A tesseract is specifically a 4-d cube). 

[source]
Another way to think about this can be found here:

Start with a point. Make a copy of the point, and move it some distance away. Connect these points. We now have a segment. Make a copy of the segment, and move it away from the first segment in a new (orthogonal) direction. Connect corresponding points. We now have an ordinary square. Make a copy of the square, and move it in a new (orthogonal) direction. Connect corresponding points. We now have a cube. Make a copy and move it in a new (orthogonal, fourth) direction. Connect corresponding points. This is the tesseract.

If a tesseract were to enter our world, we would only see it in our three dimensions, meaning we would see forms of a cube doing funny things and spinning on its axes. This would be referred to as a cross-section of the tesseract. Similarly, if we as 3-dimensional bodies were to enter a 2-dimensional world, its 2-dimension citizens would “observe” us as 2-dimensional cross objects as well! It would only be possible for them to see cross-sections of us.
Why would this be significant? Generally, in math, we work with multiple dimensions very often. While it may seem as though a mathematican must then work with 3 dimensions often, it is not necessarily true. The mathematician deals with these dimensions only mathematically. These dimensions do not have a value because they do not correspond to anything in reality; 3 dimensions are nothing ordinary nor special. 
Yet, through modern mathematics and physics, researchers consider the existence of other (spatial) dimensions.  What might be an example of such a theory? String theory is a model of the universe which supposes there may be many more than the usual 4 spacetime dimensions (3 for space, 1 for time). Perhaps understanding these dimensions, though seemingly impossible to visualize, will come in hand. 
Carl Sagan also explains what a tesseract is. 
Image: Peter Forakis, Hyper-Cube, 1967, Walker Art Center, Minneapolis

Hypercubes

What is a hypercube (also referred to as a tesseract) you say! Well, let’s start with what you know already. We know what a cube is, it’s a box! But how else could you describe a cube? A cube is 3 dimensional. Its 2 dimensional cousin is a square. 

A hypercube is just to a cube what a cube is to a square. A hypercube is 4 dimensional! (Actually— to clarify, hypercubes can refer to cubes of all dimensions. “Normal” cubes are 3 dimensional, squares are 2 dimensional “cubes, etc. This is because a hypercube is an n-dimensional figure whose edges are aligned in each of the space’s dimensions, perpendicular to each other and of the same length. A tesseract is specifically a 4-d cube). 

[source]

Another way to think about this can be found here:

Start with a point. Make a copy of the point, and move it some distance away. Connect these points. We now have a segment. Make a copy of the segment, and move it away from the first segment in a new (orthogonal) direction. Connect corresponding points. We now have an ordinary square. Make a copy of the square, and move it in a new (orthogonal) direction. Connect corresponding points. We now have a cube. Make a copy and move it in a new (orthogonal, fourth) direction. Connect corresponding points. This is the tesseract.

If a tesseract were to enter our world, we would only see it in our three dimensions, meaning we would see forms of a cube doing funny things and spinning on its axes. This would be referred to as a cross-section of the tesseract. Similarly, if we as 3-dimensional bodies were to enter a 2-dimensional world, its 2-dimension citizens would “observe” us as 2-dimensional cross objects as well! It would only be possible for them to see cross-sections of us.

Why would this be significant? Generally, in math, we work with multiple dimensions very often. While it may seem as though a mathematican must then work with 3 dimensions often, it is not necessarily true. The mathematician deals with these dimensions only mathematically. These dimensions do not have a value because they do not correspond to anything in reality; 3 dimensions are nothing ordinary nor special. 

Yet, through modern mathematics and physics, researchers consider the existence of other (spatial) dimensions.  What might be an example of such a theory? String theory is a model of the universe which supposes there may be many more than the usual 4 spacetime dimensions (3 for space, 1 for time). Perhaps understanding these dimensions, though seemingly impossible to visualize, will come in hand. 

Carl Sagan also explains what a tesseract is

Image: Peter Forakis, Hyper-Cube, 1967, Walker Art Center, Minneapolis

Friday, August 26, 2011
[Image source]
What exactly is “redshift”? 
Redshift is defined as: 

a shift toward longer wavelengths of the spectral lines emitted by a celestial object  that is caused by the object moving away from the earth.

If you can understand that, great! But for those of us who cannot, consider the celestial bodies which make up our night sky. Did you think they were still, adamant, everlasting constants? They may seem to stick around forever, but…
Boy, you were wrong. I’ll have you know that stars are born and, at some point, they die. They move, they change. Have you heard about variable stars? Stars undergo changes, sometimes in their luminosity. (We are, indeed, made of the same stuff as stars).
So, stars move. All celestial bodies do, actually. You might have heard about some mysterious, elusive thing called dark energy. Dark energy is thought to be the force that causes the universe to expand at a growing rate. If it is proven to exist, dark energy will be able to explain why redshift occurs.
Maybe you can understand redshift by studying a visual:

[Image source]
These are spectral lines from an object. What do you notice is different in the unshifted, “normal” emission lines from the redshifted and blueshifted lines?
The redshifted line is observed as if everything is “shifted” a bit to the right— towards the red end of the spectrum; whereas the blueshifted line is moved to the left towards the bluer end of the spectrum.
Imagine if you were standing here on earth and some many lightyears away, a hypothetical “alien” was standing on their planet. With this image in mind, consider a galaxy in between the two of you that is moving towards the alien. You would then observe redshift (stretched out wavelength) and the alien would observe blueshift (shortened wavelength). 
Here, Symmetry Magazine explains redshift in their “Explain it in 60 seconds” series. 
A simple, everyday example of this concept can be observed if you stand in front of a road. As a car (one without a silencer) drives by, the pitch you observe changes. This is known as the Doppler effect. Watch this quick youtube video titled “Example of Dopper Shift using car horn”: 
(You may not be able to view it from the dashboard, only by opening this post on the actual blog page. You can watch the video by clicking this link). 

Notice how as the car drives past the camera man, the sound changes drastically.
Understanding redshift is important to scientists, especially astronomers and astrophysicists. They must account for this observable difference to make the right conclusions. Redshift is one the concepts which helped scientists determine that celestial bodies are actually moving further away from us at an accelerating rate.

[Image source]

What exactly is “redshift”? 

Redshift is defined as: 

a shift toward longer wavelengths of the spectral lines emitted by a celestial object  that is caused by the object moving away from the earth.

If you can understand that, great! But for those of us who cannot, consider the celestial bodies which make up our night sky. Did you think they were still, adamant, everlasting constants? They may seem to stick around forever, but…

Boy, you were wrong. I’ll have you know that stars are born and, at some point, they die. They move, they change. Have you heard about variable stars? Stars undergo changes, sometimes in their luminosity. (We are, indeed, made of the same stuff as stars).

So, stars move. All celestial bodies do, actually. You might have heard about some mysterious, elusive thing called dark energy. Dark energy is thought to be the force that causes the universe to expand at a growing rate. If it is proven to exist, dark energy will be able to explain why redshift occurs.

Maybe you can understand redshift by studying a visual:

[Image source]

These are spectral lines from an object. What do you notice is different in the unshifted, “normal” emission lines from the redshifted and blueshifted lines?

The redshifted line is observed as if everything is “shifted” a bit to the right— towards the red end of the spectrum; whereas the blueshifted line is moved to the left towards the bluer end of the spectrum.

Imagine if you were standing here on earth and some many lightyears away, a hypothetical “alien” was standing on their planet. With this image in mind, consider a galaxy in between the two of you that is moving towards the alien. You would then observe redshift (stretched out wavelength) and the alien would observe blueshift (shortened wavelength). 

Here, Symmetry Magazine explains redshift in their “Explain it in 60 seconds” series. 

A simple, everyday example of this concept can be observed if you stand in front of a road. As a car (one without a silencer) drives by, the pitch you observe changes. This is known as the Doppler effect. Watch this quick youtube video titled “Example of Dopper Shift using car horn”: 

(You may not be able to view it from the dashboard, only by opening this post on the actual blog page. You can watch the video by clicking this link). 

Notice how as the car drives past the camera man, the sound changes drastically.

Understanding redshift is important to scientists, especially astronomers and astrophysicists. They must account for this observable difference to make the right conclusions. Redshift is one the concepts which helped scientists determine that celestial bodies are actually moving further away from us at an accelerating rate.

Thursday, August 11, 2011
Fractal Geometry: An Artistic Side of Infinity
Fractal Geometry is beautiful. Clothes are designed from it and you can find fractal calendars for your house. There’s just something about that infinitely endless pattern that intrigues the eye— and brain. 
Fractals are “geometric shapes which can be split into parts which are a reduced-size copy of the whole” (source: wikipedia). They demonstrate a property called self-similarity, in which parts of the figure are similar to the greater picture. Theoretically, each fractal can be magnified and should be infinitely self-similar. 
One simple fractal which can easily display self-similarity is the Sierpinski Triangle. You can look at the creation of such a fractal:

What do you notice? Each triangle is self similar— they are all equilateral triangles. The side length is half of the original triangle. And what about the area? The area is a quarter of the original triangle. This pattern repeats again, and again. 
Two other famous fractals are the Koch Snowflake and the Mandelbrot Set. 
The Koch Snowflake looks like: 
 (source: wikipedia)
It is constructed by going in 1/3 of the of the side of an equilateral triangle and creating another equilateral triangle. You can determine the area of a Koch Snowflake by following this link.
The Mandelbrot set…

… is:

the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. (source: wikipedia)

It is a popular fractal named after Benoît Mandelbrot. More on creating a Mandelbrot set is found here, as well as additional information. 
You can create your own fractals with this fractal generator. 
But what makes fractals extraordinary?
Fractals are not simply theoretical creations. They exist as patterns in nature! Forests can model them, so can clouds and interstellar gas! 
Artists are fascinated by them, as well. Consider The Great Wave off Kanagawa by Katsushika Hokusai: 

Even graphic artists use fractals to create mountains or ocean waves. You can watch Nova’s episode of Hunting the Hidden Dimension for more information. 

Fractal Geometry: An Artistic Side of Infinity

Fractal Geometry is beautiful. Clothes are designed from it and you can find fractal calendars for your house. There’s just something about that infinitely endless pattern that intrigues the eye— and brain. 

Fractals are “geometric shapes which can be split into parts which are a reduced-size copy of the whole” (source: wikipedia). They demonstrate a property called self-similarity, in which parts of the figure are similar to the greater picture. Theoretically, each fractal can be magnified and should be infinitely self-similar. 

One simple fractal which can easily display self-similarity is the Sierpinski Triangle. You can look at the creation of such a fractal:

What do you notice? Each triangle is self similar— they are all equilateral triangles. The side length is half of the original triangle. And what about the area? The area is a quarter of the original triangle. This pattern repeats again, and again. 

Two other famous fractals are the Koch Snowflake and the Mandelbrot Set

The Koch Snowflake looks like: 

 (source: wikipedia)

It is constructed by going in 1/3 of the of the side of an equilateral triangle and creating another equilateral triangle. You can determine the area of a Koch Snowflake by following this link.

The Mandelbrot set…

… is:

the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. (source: wikipedia)

It is a popular fractal named after Benoît Mandelbrot. More on creating a Mandelbrot set is found here, as well as additional information. 

You can create your own fractals with this fractal generator. 

But what makes fractals extraordinary?

Fractals are not simply theoretical creations. They exist as patterns in nature! Forests can model them, so can clouds and interstellar gas! 

Artists are fascinated by them, as well. Consider The Great Wave off Kanagawa by Katsushika Hokusai: 

Even graphic artists use fractals to create mountains or ocean waves. You can watch Nova’s episode of Hunting the Hidden Dimension for more information. 

Thursday, August 4, 2011
Have you ever looked out on a starry night and wondered what else is out there? Perhaps, who else? And if there were to be someone, something there— would they be looking out for you, too?
Don’t worry, you’re not alone. Others have theorized about it: Frank Drake (an American Radio astronomer who wrote the famous Arecibo message) made an entire equation. Behold, The Drake Equation.
N = R* × fp × ne × fl × fi × fc × L
The Drake Equation is an equation for  predicting the number of civilizations in the Milky Way Galaxy capable of  interstellar communication.
Short descriptions of what the variables of the equation represent can be found here.

The variables represent the average rate of star formation per year in our galaxy, the fraction of those stars which have planets, the average number of planets that can potentially support life per star which has planets, the fraction of those which actually go on to develop life in the future, the fraction of those which go on to develop intelligent life, the fraction of those which can release detectable signals of their existence, and (finally) the length of time for which these civilizations release signals.

That all seems like a mess, but you get the idea.
According to Drake’s parameters:

50% of new stars develop planets
0.4 planets will be habitable
90% of habitable planets develop life
10% of new instances of life develop intelligence
10% of such life develops interstellar communications
These civilizations, might, on average, last 10,000 years.

To be fair, we are not sure on the actual figures. Drake’s values gives an answer of 10, meaning that 10 of these theoretical civilizations would be able to communicate.
But the importance of Drake’s equations is not necessarily the numerical value. It lies in all the questions that the equation led him to. Who knows exactly how many stars there are and what not? These figures are yet to be discovered.
So next time you look above, remember to always question. You’re not alone in questioning and you don’t know where these questions can lead you. Like Drake, you might be led to discover companions from different worlds.

Have you ever looked out on a starry night and wondered what else is out there? Perhaps, who else? And if there were to be someone, something there— would they be looking out for you, too?

Don’t worry, you’re not alone. Others have theorized about it: Frank Drake (an American Radio astronomer who wrote the famous Arecibo message) made an entire equation. Behold, The Drake Equation.

N = R* × fp × ne × fl × fi × fc × L

The Drake Equation is an equation for predicting the number of civilizations in the Milky Way Galaxy capable of interstellar communication.

Short descriptions of what the variables of the equation represent can be found here.

The variables represent the average rate of star formation per year in our galaxy, the fraction of those stars which have planets, the average number of planets that can potentially support life per star which has planets, the fraction of those which actually go on to develop life in the future, the fraction of those which go on to develop intelligent life, the fraction of those which can release detectable signals of their existence, and (finally) the length of time for which these civilizations release signals.

That all seems like a mess, but you get the idea.

According to Drake’s parameters:

  • 50% of new stars develop planets
  • 0.4 planets will be habitable
  • 90% of habitable planets develop life
  • 10% of new instances of life develop intelligence
  • 10% of such life develops interstellar communications
  • These civilizations, might, on average, last 10,000 years.

To be fair, we are not sure on the actual figures. Drake’s values gives an answer of 10, meaning that 10 of these theoretical civilizations would be able to communicate.

But the importance of Drake’s equations is not necessarily the numerical value. It lies in all the questions that the equation led him to. Who knows exactly how many stars there are and what not? These figures are yet to be discovered.

So next time you look above, remember to always question. You’re not alone in questioning and you don’t know where these questions can lead you. Like Drake, you might be led to discover companions from different worlds.

Tuesday, August 2, 2011
Number Harmony
It is easy to recognize octaves because the frequency of an octave above a certain pitch is exactly twice the frequency of that pitch. Octaves harmonize so well that they almost sound identical, so we call these notes by the same name: an octave above or below middle C is  another C; an octave above or below concert A, 440 Hz, is another A (880  or 220 Hz). Mathematically, if a certain note H has frequency f then a note with frequency 2nf, where n is an integer, is n octaves above H (if n is negative, it is a positive power of 1/2 and represents |n| octaves below H).
Not alone in their ability to harmonize well, octaves are joined by all the intervals that make up a major or minor scale (in the Western music system), notably including perfect fifths (fifth note of a scale, 3/2 times the frequency of the starting note) and major or minor thirds (third note of a scale, respectively 5/4 or 6/5 times the frequency of the starting note). All these frequencies are ratios of relatively small whole numbers - this contributes to the harmony of the notes, just like the ratio 2/1 does for octaves. The simpler the frequency ratio, the higher the quality of harmony achieved by an interval when played out loud. The only requirement is for  the ratio to be a (positive) rational number, able to be written with whole numbers for the numerator and denominator.
However, suppose you tuned a piano perfectly according to one of the scales. Then you can play that scale and it would be perfectly in tune - but the harmony of all the other scales get thrown off! For example, E is both the third note of a C major scale and the second note of a D major scale. By tuning the piano to the C major scale, you guarantee that an E has frequency 5/4 times the frequency of a C (C to E is a major third). In a perfect C scale, D has frequency 9/8 times that of a C. Call these frequencies fC, fD, and fE.
fE = (5/4)fCfD = (9/8)fC⇒ fE = (5/4)(8/9)fD = (10/9)fD
This is still a relatively simple rational number ratio, but it’s the wrong ratio. In a perfect D major scale, E has frequency 9/8 that of D. The relative error when tuning to C is
|10/9 - 9/8| = |80-81|/72 = 1/72.
In the first days of the harpsichord and piano (keyboard instruments), tuners chose one scale to tune to, sacrificing the harmony of the other scales. Interestingly, some of the music from that era took that into account; on one hand some scales were considered “sweeter” than others based on common tuning practices, and on the other some songs were purposely written in one of the sour-sounding scales for their dissonant harmonies.
Today’s most common tuning, or temperament, is called equal temperament. Each scale sounds equally good (or equally bad, depending on your tolerance of imperfection), and the only interval which is perfectly preserved is octaves. Since, in the Western music system, there are 12 semitones from octave to octave (12 white and black keys from a note to an octave above the note), each of those keys is assigned the frequency of exactly the twelfth root of 2 times the key preceding it. What’s great about that, of course, is that this is a completely egalitarian system: no scale is sweeter or sourer sounding that any other. Yet the cost is the complete destruction of the rational number harmonies: the twelfth root of 2 is as irrational as they come, and could never in any number theorist’s wildest dreams be written as a ratio of whole numbers.
—
Further reading:”Why you’ve never really heard the Moonlight Sonata,” Jan Swafford, Slate Magazine

Number Harmony

It is easy to recognize octaves because the frequency of an octave above a certain pitch is exactly twice the frequency of that pitch. Octaves harmonize so well that they almost sound identical, so we call these notes by the same name: an octave above or below middle C is another C; an octave above or below concert A, 440 Hz, is another A (880 or 220 Hz). Mathematically, if a certain note H has frequency f then a note with frequency 2nf, where n is an integer, is n octaves above H (if n is negative, it is a positive power of 1/2 and represents |n| octaves below H).

Not alone in their ability to harmonize well, octaves are joined by all the intervals that make up a major or minor scale (in the Western music system), notably including perfect fifths (fifth note of a scale, 3/2 times the frequency of the starting note) and major or minor thirds (third note of a scale, respectively 5/4 or 6/5 times the frequency of the starting note). All these frequencies are ratios of relatively small whole numbers - this contributes to the harmony of the notes, just like the ratio 2/1 does for octaves. The simpler the frequency ratio, the higher the quality of harmony achieved by an interval when played out loud. The only requirement is for  the ratio to be a (positive) rational number, able to be written with whole numbers for the numerator and denominator.

However, suppose you tuned a piano perfectly according to one of the scales. Then you can play that scale and it would be perfectly in tune - but the harmony of all the other scales get thrown off! For example, E is both the third note of a C major scale and the second note of a D major scale. By tuning the piano to the C major scale, you guarantee that an E has frequency 5/4 times the frequency of a C (C to E is a major third). In a perfect C scale, D has frequency 9/8 times that of a C. Call these frequencies fC, fD, and fE.

fE = (5/4)fC
fD = (9/8)fC
fE = (5/4)(8/9)fD = (10/9)fD

This is still a relatively simple rational number ratio, but it’s the wrong ratio. In a perfect D major scale, E has frequency 9/8 that of D. The relative error when tuning to C is

|10/9 - 9/8| = |80-81|/72 = 1/72.

In the first days of the harpsichord and piano (keyboard instruments), tuners chose one scale to tune to, sacrificing the harmony of the other scales. Interestingly, some of the music from that era took that into account; on one hand some scales were considered “sweeter” than others based on common tuning practices, and on the other some songs were purposely written in one of the sour-sounding scales for their dissonant harmonies.

Today’s most common tuning, or temperament, is called equal temperament. Each scale sounds equally good (or equally bad, depending on your tolerance of imperfection), and the only interval which is perfectly preserved is octaves. Since, in the Western music system, there are 12 semitones from octave to octave (12 white and black keys from a note to an octave above the note), each of those keys is assigned the frequency of exactly the twelfth root of 2 times the key preceding it. What’s great about that, of course, is that this is a completely egalitarian system: no scale is sweeter or sourer sounding that any other. Yet the cost is the complete destruction of the rational number harmonies: the twelfth root of 2 is as irrational as they come, and could never in any number theorist’s wildest dreams be written as a ratio of whole numbers.

Further reading:
Why you’ve never really heard the Moonlight Sonata,” Jan Swafford, Slate Magazine

Monday, August 1, 2011
SOH-CAH-TOA
 
This simple mnemonic (which bears a resemblance to a Native American name) is used to remember how to compute the sine, cosine and tangent of a right-angled triangle and the relationships between them.
When labeling a triangle, it is imperative to take note of the following parts:
Hypotenuse (H) - the longest side of the triangle
Angle (θ) - theta is often used to name the acute angle of a triangle one’s trying to find the sine/cosine/tangent of, however this is not always the case; any consistent labeling is fine
Adjacent (A) - the side of the triangle next to the selected angle
Opposite (O) - the side of the triangle opposite the selected angle 
Note: Do not think that the way the pictured triangle above is the only way you can label a triangle. The opposite and adjacent can easily switch places if the other angle is selected!
              Opposite

Sin θ = -----------------

              Adjacent


              Adjacent

Cos θ = -----------------

             Hypotenuse


              Opposite

Tan θ = -----------------

             Hypotenuse

SOH-CAH-TOA

This simple mnemonic (which bears a resemblance to a Native American name) is used to remember how to compute the sine, cosine and tangent of a right-angled triangle and the relationships between them.

When labeling a triangle, it is imperative to take note of the following parts:

  • Hypotenuse (H) - the longest side of the triangle
  • Angle (θ) - theta is often used to name the acute angle of a triangle one’s trying to find the sine/cosine/tangent of, however this is not always the case; any consistent labeling is fine
  • Adjacent (A) - the side of the triangle next to the selected angle
  • Opposite (O) - the side of the triangle opposite the selected angle 

Note: Do not think that the way the pictured triangle above is the only way you can label a triangle. The opposite and adjacent can easily switch places if the other angle is selected!

              Opposite

Sin θ = -----------------

              Adjacent


              Adjacent

Cos θ = -----------------

             Hypotenuse


              Opposite

Tan θ = -----------------

             Hypotenuse
Friday, July 29, 2011
The Neutron Star
When a star runs out of fuel it eventually goes through a gravitational collapse. There are several possible outcomes, and three come about by simply taking into account only the mass of the star that has just collapsed. If the star is less than 1.5 solar masses, then a white dwarf is formed. Yet if its mass is larger than 5 solar masses, it will create a black hole. What about if the star’s mass was in between those two points? Well, that’s when a neutron star is formed. Due to the inward collapse of such a star, electrons combine with protons to form neutrons – thus giving the resultant celestial body the name “neutron star”.
Neutron stars characteristically have extremely high densities. Anything falling into a neutron star is super-accelerated by gravity (which is 100 billion times stronger than what we experience on earth). “If you dropped a marshmallow onto a neutron star, it would have the energy of an atomic bomb,” says Chip Meegan from the Marshal Space Flight Center of NASA. Neutron stars also have insanely strong magnetic fields, approximately 2 x 1011 times those of Earth. These stars are usually very hot. The degeneracy pressure due to the Pauli exclusion principle (no two neutrons or any other fermions can occupy the same place and quantum state simultaneously) ensures the neutron star’s stability and prevents it from collapse. (The only situation in which a neutron star would collapse into a black hole is if it is gradually absorbing matter from an accompanying binary star.)
Pulsars!
Very simply, pulsars are rotating neutron stars. Because they conserve the angular momentum of the stars from which they were formed, pulsars can spin at rates over 700 times revolutions per second. They stream jets of highly energetic particles (which have speeds close to that of light) out from their magnetic poles, producing extremely powerful beams of radiation. This combined with their rotational movement allows them to appear as if they are pulsing when they are observed. The rotational and magnetic axes of these stars are misaligned, which causes the beam of light from the jets to “sweep” around as the pulsar rotates, giving rise to the lighthouse effect.

The Neutron Star

When a star runs out of fuel it eventually goes through a gravitational collapse. There are several possible outcomes, and three come about by simply taking into account only the mass of the star that has just collapsed. If the star is less than 1.5 solar masses, then a white dwarf is formed. Yet if its mass is larger than 5 solar masses, it will create a black hole. What about if the star’s mass was in between those two points? Well, that’s when a neutron star is formed. Due to the inward collapse of such a star, electrons combine with protons to form neutrons – thus giving the resultant celestial body the name “neutron star”.

Neutron stars characteristically have extremely high densities. Anything falling into a neutron star is super-accelerated by gravity (which is 100 billion times stronger than what we experience on earth). “If you dropped a marshmallow onto a neutron star, it would have the energy of an atomic bomb,” says Chip Meegan from the Marshal Space Flight Center of NASA. Neutron stars also have insanely strong magnetic fields, approximately 2 x 1011 times those of Earth. These stars are usually very hot. The degeneracy pressure due to the Pauli exclusion principle (no two neutrons or any other fermions can occupy the same place and quantum state simultaneously) ensures the neutron star’s stability and prevents it from collapse. (The only situation in which a neutron star would collapse into a black hole is if it is gradually absorbing matter from an accompanying binary star.)

Pulsars!

Very simply, pulsars are rotating neutron stars. Because they conserve the angular momentum of the stars from which they were formed, pulsars can spin at rates over 700 times revolutions per second. They stream jets of highly energetic particles (which have speeds close to that of light) out from their magnetic poles, producing extremely powerful beams of radiation. This combined with their rotational movement allows them to appear as if they are pulsing when they are observed. The rotational and magnetic axes of these stars are misaligned, which causes the beam of light from the jets to “sweep” around as the pulsar rotates, giving rise to the lighthouse effect.

Monday, July 25, 2011
Originally, in the Newtonian formulation of classical mechanics, equations of motion were determined by summing up vector forces (à la free body diagrams). Is there a different way to find the equations of motion?



In place of drawing a free body diagram, we can represent a system more rigorously by describing its configuration space. The configuration space (often denoted Q) of a system is a mathematical space (a differential manifold) where every point in the space represents a particular state or configuration of the system. A curve drawn through a configuration space, then, represents the evolution of a system through a sequence of configurations.



Consider a rod along which a pendulum can slide. We need two numbers to describe the state of this system: the angle of the swinging pendulum and the position of the pendulum’s base along the rod. These two numbers are generalized coordinates for our system. Just like a traditional, linear vector space has a coordinate basis (like x, y, and z), our configuration space can use our generalized coordinates as a basis; let’s choose to name the position on the rod x and the angle of the pendulum φ. Since x can take any real value and φ can take any value from 0 to 2π (or 0 to 360o, if you like), the x dimension can be represented by a line (R1) and the φ dimension by a circle (or a one-sphere, S1). When we combine these dimensions, our new space — the configuration space of this system — is shaped like an infinite cylinder, R1 x S1. Just imagine connecting a circle to every point on a line… or, conversely, a line to every point on a circle.



The general process of examining a system and the constraints on its movement is a standard first step for solving mechanics problems analytically. After accounting for the constraints on a system, the ways a system can vary are called the degrees of freedom. Their count is often represented by the variable s. Notice: s = dim(Q).



Now that we’ve represented the configuration of our system, we need to talk about the forces present. There are several different ways that we can set up scalar fields on our configuration manifold that represent quantities related to the energy of the system. The simplest to deal with is often the Lagrangian, L = T - V = (Total kinetic energy) - (Total potential energy). Some fancy mathematics (a.k.a. calculus of variations) shows that when we define the Lagrangian in terms of our coordinates and their time derivatives, we can easily derive the equations of motion using the Euler-Lagrange equation.



For more complicated systems, configurations spaces may look different. A double pendulum (a pendulum on a pendulum) would have the topology S1 x S1 = T2, the torus (as pictured). Many systems will have higher dimensions that prevent them from being easily visualized.



Exercise left to the reader: the Lagrangian explicitly takes the time derivatives of the coordinates as arguments; information about the velocities of the system is needed to derive the equations of motion. But this information isn’t included in Q, so Lagrangian dynamics actually happens on TQ, the tangent bundle to Q. This new manifold includes information about how the system changes from every given configuration; since it needs to include a velocity coordinate for each configuration coordinate, dim(TQ) = 2s. TQ is also called Γv, the velocity phase space. T*Q, the cotangent bundle to Q, is the dual of TQ, and is traditionally just called the phase space, Γ; this is where Hamiltonian mechanics takes place.

Originally, in the Newtonian formulation of classical mechanics, equations of motion were determined by summing up vector forces (à la free body diagrams). Is there a different way to find the equations of motion?

In place of drawing a free body diagram, we can represent a system more rigorously by describing its configuration space. The configuration space (often denoted Q) of a system is a mathematical space (a differential manifold) where every point in the space represents a particular state or configuration of the system. A curve drawn through a configuration space, then, represents the evolution of a system through a sequence of configurations.

Consider a rod along which a pendulum can slide. We need two numbers to describe the state of this system: the angle of the swinging pendulum and the position of the pendulum’s base along the rod. These two numbers are generalized coordinates for our system. Just like a traditional, linear vector space has a coordinate basis (like x, y, and z), our configuration space can use our generalized coordinates as a basis; let’s choose to name the position on the rod x and the angle of the pendulum φ. Since x can take any real value and φ can take any value from 0 to 2π (or 0 to 360o, if you like), the x dimension can be represented by a line (R1) and the φ dimension by a circle (or a one-sphere, S1). When we combine these dimensions, our new space — the configuration space of this system — is shaped like an infinite cylinder, R1 x S1. Just imagine connecting a circle to every point on a line… or, conversely, a line to every point on a circle.

The general process of examining a system and the constraints on its movement is a standard first step for solving mechanics problems analytically. After accounting for the constraints on a system, the ways a system can vary are called the degrees of freedom. Their count is often represented by the variable s. Notice: s = dim(Q).

Now that we’ve represented the configuration of our system, we need to talk about the forces present. There are several different ways that we can set up scalar fields on our configuration manifold that represent quantities related to the energy of the system. The simplest to deal with is often the Lagrangian, L = T - V = (Total kinetic energy) - (Total potential energy). Some fancy mathematics (a.k.a. calculus of variations) shows that when we define the Lagrangian in terms of our coordinates and their time derivatives, we can easily derive the equations of motion using the Euler-Lagrange equation.

For more complicated systems, configurations spaces may look different. A double pendulum (a pendulum on a pendulum) would have the topology S1 x S1 = T2, the torus (as pictured). Many systems will have higher dimensions that prevent them from being easily visualized.

Exercise left to the reader: the Lagrangian explicitly takes the time derivatives of the coordinates as arguments; information about the velocities of the system is needed to derive the equations of motion. But this information isn’t included in Q, so Lagrangian dynamics actually happens on TQ, the tangent bundle to Q. This new manifold includes information about how the system changes from every given configuration; since it needs to include a velocity coordinate for each configuration coordinate, dim(TQ) = 2s. TQ is also called Γv, the velocity phase space. T*Q, the cotangent bundle to Q, is the dual of TQ, and is traditionally just called the phase space, Γ; this is where Hamiltonian mechanics takes place.

Sunday, July 10, 2011
 
Galileo Galilei was interested in ordinary water pumps and how they function. At that time, people used suction pumps to draw water from wells. Galileo once noticed that a well functioning pump failed to lift water more than 34 ft above the free water level in the well.
Evangelista Torricelli*, who invented the vacuum and the barometer, was able to explain the failure of the pump. He took a glass tube about 3 ft long that was sealed at one end and filled it with mercury. He put his finger on the open end, and turned the tube over so it stood in a vessel containing mercury. Torricelli removed his finger carefully, so that no air got in the tube, and noticed that the mercury in the tube sinks and comes to rest about 30 in. above the mercury in the open vessel.
Since there is a vacuum in the space above the mercury in the tube, no atmospheric pressure can be acting on the top of the mercury column. However, there is atmospheric pressure acting on the mercury in the vessel, and that is equivalent to the weight of the mercury in the tube. 
It should be noted that mercury is around 13.6 times heavier than water. So:
13.6*30 inches=408 inches
408 inches is 34 feet of water, and this explains the failure of a suction pump to lift water above this height. 
*The SI unit for pressure is in atmospheres. 1 Atmosphere=760. Millimeters of Mercury=760 Torr. The unit Torr is named after Evangelista Torricelli. 

Galileo Galilei was interested in ordinary water pumps and how they function. At that time, people used suction pumps to draw water from wells. Galileo once noticed that a well functioning pump failed to lift water more than 34 ft above the free water level in the well.

Evangelista Torricelli*, who invented the vacuum and the barometer, was able to explain the failure of the pump. He took a glass tube about 3 ft long that was sealed at one end and filled it with mercury. He put his finger on the open end, and turned the tube over so it stood in a vessel containing mercury. Torricelli removed his finger carefully, so that no air got in the tube, and noticed that the mercury in the tube sinks and comes to rest about 30 in. above the mercury in the open vessel.

Since there is a vacuum in the space above the mercury in the tube, no atmospheric pressure can be acting on the top of the mercury column. However, there is atmospheric pressure acting on the mercury in the vessel, and that is equivalent to the weight of the mercury in the tube. 

It should be noted that mercury is around 13.6 times heavier than water. So:

13.6*30 inches=408 inches

408 inches is 34 feet of water, and this explains the failure of a suction pump to lift water above this height. 

*The SI unit for pressure is in atmospheres. 1 Atmosphere=760. Millimeters of Mercury=760 Torr. The unit Torr is named after Evangelista Torricelli.