Math as a Muse (Part I)

I have had an affinity for math for as long as I can remember.  Even in elementary school, when working with numbers or shapes, it always seemed like magic to me.  Now, as an adult, I find this magic hiding in unexpected places, like the relationships between notes in music, and in the geometry of architecture.  It was the latter that called out to me last week.

I was standing in a place of worship.  Admittedly, I do not spend a great deal of time in such places.  On this particular occasion, in a church, my mind was wandering, and I began examining all of the geometry around me: the slopes in the roof, the shapes of the stained glass, and the angle in which the sunlight came through them.  When my gaze returned forward, I began to carefully examine all of the equally spaced columns (known as pews) between me and the front of the church, where the Reverend stood.  Almost immediately, a fun math problem presented itself to me, and I spent the next twenty minutes analyzing it in my mind.

As shown in the figure below, I can see less and less of each column the further they are from me, as each column is obstructed by the one that precedes it.  But, what governs how much of a given column I can see?  I called the column directly in front of me n = 0, and then came 1, 2, and so on.  My aim was to find a function that described the height of each column that I could see for each column, h(n).  I decided that it depended on three other parameters: the column spacing (s), as well as the vertical and horizontal distance from my eyes to the zeroth column immediately in front of me (H and L).

Searching for visible height h as function of pew number n (mad paint skills, I know...)

Richard Feynman Comes Alive in Unorthodox Autobiography

I finished reading Richard Feynman's "Surely You're Joking, Mr. Feynman!" last week, and still find myself laughing about it today.  What could have been a conventional autobiography of the Nobel Prize winner for physics is instead a collection of quirky stories, through which one really gets to know the man.  To give you a sense of the tone of the book, Feynman mentions the Nobel Prize he won about halfway through it, as a sort of after-thought - the focus is rather on what he is truly proud of, like, for example, his ability to break into safes that contained top-secret information about the Manhattan project during the second World War.

"Slow Mo Guys" = Great Teaching Tool

"Sir, you're going too fast!" - it is a complaint I hear in my physics classes every so often.  Whether it is the case or not, it is true that there is an ideal speed for progressing through science content in a classroom setting.

Similarly, there is an ideal speed for the viewing of the countless science phenomena that occur in nature.  Often times, events like chemical reactions or travelling waves elapse over too short a time interval to be properly grasped.  This is actually the reason why many scientific phenomena that are now well understood went misunderstood for so long (and why others still go misunderstood).

Take something simple, like an apple falling from a tree.  Five centuries ago, people believed that the fall was at a constant speed, which was governed by the apple's mass (heavy apples would fall faster than light ones).  Of course, this assessment is wrong on many levels, but one can easily appreciate why such a faulty conclusion could be arrived at.  The entire fall of an apple might take one second, which is an insufficient amount of time for a person to gauge an event.

Had mankind invented the video camera a few centuries earlier than it did, enabling it to see the world in slow motion, early science would have evolved more rapidly than it did.  The apple could then be seen to displace more and more with each passing frame, invalidating the constant speed theory.

Some people today may not see how valuable adjusting the frame rate of an event is outside sports and action movies.  Fortunately, a few young people certainly do, and they are responsible for my current favourite YouTube channel: "Slow Mo Guys".

Life is Like a Non-conservative Force

"Mama always said, life was like a box of chocolates..."  Had Mama been a physicist, she may have instead used non-conservative forces as an analogy to depict life's winding roads.  Let us first explore the meaning of a non-conservative force, and then attempt to draw parallels between it and life.

In physics, it is important to understand the distinction between conservative and non-conservative forces.  For one thing, it comes in handy when trying to solve problems using a work/energy approach, which countless mechanics students are no doubt busily doing as I write.

The conservation of energy principle is merely a statement of the first law of thermodynamics, which, for mechanics, translates to: "The change in the total mechanical energy of a system between states 1 and 2 is equal to the total work done on the system by non-conservative forces between states 1 and 2."  The term 'state' refers to a particular position and velocity of the system's components (time does elapse in between states, but the particular amount is not significant for the analysis).

In equation form, these words look like this:

Mechanical Analysis of Baumgartner's Dive (Part II)

(This is the second and final article of the Felix Baumgartner dive saga - click here for part 1)

By now you have no doubt heard that Felix Baumgartner has shattered several records with his successful sky dive on October 14, 2012.  Fearless Felix stepped off of his perch, fell freely for 4 min 18 sec, and then pulled his parachute, coasting safely to the surface about five minutes later.

The lead up to the historic event was similar to that of a rocket launch, complete with weather delays.  This jump was originally set for October 8, but on several occasions, it got bumped.  You know you are involved in something risky when a little too much wind is cause for serious worry.

Imagine you are Felix, and you wake up on October 8, having probably not slept much the night before, ready for the leap of your life.  You down a few red bulls, get your adrenalin up, and then some guy in a lab coat gives you the news that the jump must be postponed.  Repeat this a few more times, and you just might go mad.  I do not know this for certain, but I would imagine that a psychologist was on site with Baumgartner to help him maintain his mental well-being through this go/no-go roller coaster that lasted more than a week.

Many videos of the dive have circulated on YouTube, though most have been yanked by the sponsor (Red Bull).  Here is their 90 second summary of the event.

One can only imagine what it must have been like to look down from 128,000 ft (8,000 ft more than originally planned), and to behold the planet.  From that altitude, one can begin to get a sense of the Earth's curvature.  With a final salute (to his family, and mankind I suppose), Baumgartner stepped off from his pod and quickly vanished from view.

Based on some of the information given in the video, as well as some educated guesses, I have constructed approximate graphs of Baumgartner's speed and altitude as a function of time for the free-fall portion of his descent.

(Note that it is possible to generate theoretical results by solving the governing equation numerically, but as I do not have access to the particular parameters associated with his specially designed space suit, such as mass and drag coefficient, I elected to plot these 'experimentally')

Mechanical Analysis of Baumgartner's Dive (Part I)

120,000 feet... 

For a normal person, it represents the distance travelled during a fairly long commute to work.  For Felix Baumgartner, the Austrian daredevil, it represents the altitude from which he plans on free-falling towards the Earth this coming Monday, October 8, 2012.

For someone like myself, any height is too high to jump from with nothing but a parachute to save me from death.  However, even sky divers, who are themselves barely sane, see Baumgartner's jump as nothing short of lunacy.

You see, 120,000 feet is 36,576 m - that's more than 36 kilometers!  To put this into perspective, his descent will begin at an altitude that is three times that at which typical commercial airplanes fly.  It is above the troposphere, in the middle of the stratosphere.  So, "How will he get there?" you ask.  Why, he will wait inside a man-sized pod that is lifted by a large balloon, of course.  When the balloon reaches the correct altitude, the pod will open, and down he will fall.

There are literally countless risks associated with this particular sky dive that aims to crush the previous altitude record of 102,000 feet. 

To begin with, the air way up there is extremely cold.  Should Baumgartner's special suit fail even a little, the convection associated with the high speed sub-zero air flowing by him will freeze him almost instantly.

Not only is it much colder up there, but the air pressure is just 1% of that on the surface.  For this reason, the daredevil will have an oxygen tank strapped to him.  And, with this pressure change, comes a change to the most important environmental factor when it comes to aerodynamics: fluid density.

The density of air on the surface of the Earth is about 1.2 kg/m³.  In the middle of the stratosphere, it is more like 0.01 kg/m³.  A quick application of Newton's second law shows that this has a dramatic effect on the terminal velocity of the dive...

Terminal velocity

A sky diver reaches his or her terminal velocity when his or her body ceases to accelerate.  When this inertial term vanishes, we are left with a simple force balance: Drag force = Gravitational force.  The force of gravity can be approximated as mg (even at an altitude of 36 km, using the gravitational acceleration one experiences on the surface of the Earth, 9.8 m/s², introduces very little error).  The drag force is a bit more complex, and is given by:

Drag force = (1/2)ρC_DAv²

In this expression, ρ is the density (kg/m³) of the fluid, C_D is the drag coefficient (unitless) of the falling body, which is essentially a measure of how aerodynamic it is (it is greater for objects that are not streamlined), A is the projected surface area (m²) of the body, and v is the relative velocity (m/s) of the body with respect to the fluid.  It is clear that drag is largest when large objects move within dense fluids at high speeds.  This is why we can often ignore drag for, say, a ball that is tossed through the air by a child. 

Substituting these parameters into the force balance and solving for v, we get the terminal velocity equation as follows:

Ascent Towards a Type I Civilization

As a species that is in the midst of more than a century of massive technological evolution, homo sapiens, with all of their blinking gadgets and other paraphernalia, rarely fancy themselves as primitive.  It comes then as a surprise to most to find out that we have still yet to attain a civilization status of Type I.  That's right!  Take that, fragile collective ego of mankind.  Despite all that you may feel your species has accomplished, you presently belong to a Type Zero civilization, as did your cave-dwelling ancestors.

While we do indeed have access to vastly superior technology than homo sapiens have had in their long history, we do not yet qualify as a Type I.  We are, however, well on our way.

So, what is this civilization classification system, and who established it?

The system measures the technological advancement of a civilization by assessing the amount of space it takes up, and the extent to which it utilizes the energy resources within that space.  It is known as the Kardashev scale, and can be expressed in terms of the order of magnitude of power that a civilization extracts for its personal use.  This very forward-thinking and pragmatic scale was proposed in 1964 by Soviet astronomer Nikolai Kardashev.  A civilization that has attained a certain level of technological development is described as follows:

Universal Gravitation (Journey to the Center of the Earth)

Having seen parts of the original film and the previews for what must have been a horrible remake, I can assert with confidence that the physics behind a Journey to the Center of the Earth are of much greater interest than any film that goes by that name (from the preview of the latter, it appears that dinosaurs currently reside somewhere within our planet, where they are safe from all the pesky breathable air up here).

While we usually think of the gravity due to planets when we reside on their surface or orbit around them, it is intriguing to consider the role that gravitation plays inside a large body.

Newton's law of universal gravitation states that all bodies having mass emit a gravitational field and thereby attract all other masses to them.  The magnitude of the attractive force (which, by Newton's third law, acts on both bodies) is governed by the mass of each body, as well as their proximity to one another.  The force is stronger for more massive bodies, and increases as the gap between them reduces.  In equation form, the magnitude of the gravitational force, Fg (in Newtons), that acts on both bodies is given by

Fg = GM₁M₂/r^{2    (refer to figure below)}

Here, G is known as the universal gravitational constant (6.673 × 10^-11 m³/kg s²), M₁ and M₂ are the masses of each of the bodies (kg), and r is the distance that separates them (m).

Extremes in Engineering and Politics

As I watch the election season unfold in America, I am constantly stunned at how far to one side each party and their supporters are and seemingly must be.  It is, by and large, Republicans to the right, Democrats to the left, and no middle ground in sight.  When we categorize ourselves as one of these two extremes, we may enjoy the apparent sense of community that comes from it.  After some time, we begin to identify with one extreme.  However, as a whole, such categorization has the net effect of polarizing a nation.

In the twenty-first century, it seems that so many of us are unimpressed by an optimized solution.  There is nothing sexy about a calculated compromise.  On YouTube, it is the biggest this and the fastest that which tend to garner millions of views.  We simply are not interested in anything average.

I think that politicians ought to replace their segragating rhetoric with a middle of the road approach.  But the fault lies not only with them.  The voting population must recognize the value of a moderate approach.  Why do we tend not to award those who are steady and balanced?  Is it related to our fascination with outlandish personalities like those of the Kardashians?

Vive le Québec Intolérant

Today, for a change, I write of nothing scientific.  You may find it to be educational nonetheless, particularly if you do not reside in Canada, as it concerns the provincial election of its most diverse and bizarre province: Quebec.  As the majority of my readers reside in the United States, where an election with global impact is on the horizon, I will continue on.

Let's begin with some geography.  Canada has ten provinces and three territories.  Quebec is one of the bigger provinces in terms of both land mass and population, and is located towards the East.  It is rich in terms of water, and, as a result, is one of the only North American states whose energy production is primarily sustainable (hydro power).  The population of Quebec is concentrated along the St. Laurence river, which flows from West to East.  The city of Montreal, the second most populous in all of Canada, is an island along the St. Laurence towards the western side of the province; it happens to be my home.