Sunday, January 24, 2021

Enforced Rotation of Tarzan Rope (Problem)

I had so much fun with the rope problem I posted to start off 2021 (which was subsequently solved by Anthony Attia - see his elegant solution here), that I want to continue to explore this theme.  While that problem seemed tough (seeking the steady state profile of a uniform rope pinned at its top end and suspended vertically in a uniform horizontal wind), it turned out to be fairly simple.  It was almost disappointing.  To remedy the situation, consider an even more intriguing problem...

Imagine a Tarzan rope (bulk density 'p') that you suspend vertically in uniform surface gravity 'g'.  You then take the top end of the rope with length 'L' and move it with uniform circular motion in the horizontal plane (radius 'R' and angular frequency 'w').  Ignoring aerodynamic effects (because that would cause a 3D problem and have no clean analytical solution), what profile will the rope assume?  That is, if we froze the video at any given instant, what lateral deflection function, y(x), describes the rope's shape?  Treat the rope like a string (cannot support shear loads).

I spent some time on the problem, and it turns out to be even more interesting that I expected.  I will not give any hints this time.  I am curious to see if anyone will post a solution.  If you do, please provide a description of how you did it.

I am excited to share my solution, but I will be patient, and see what, if anything, gets submitted here.

Saturday, January 23, 2021

A Tarzan Rope in the Wind (Solution)

This is very exciting: a former student of mine, Anthony Attia, has submitted a solution to the Tarzan rope problem I posted some weeks ago.  Anthony was in my Mechanics class at Vanier College in 2016.  He is now pursuing undergraduate studies in mechanical engineering and simultaneously doing a stage at my former employer, MDA Space.

As is the case with some students, Anthony and I have stayed in touch since he graduated from college.  This post, however, is the first one in more than ten years of this blog's existence that someone other than me has written; it is about time.  Watch as Anthony analyzes a uniform rope, pinned at the top and vertically suspended, subjected to a horizontal uniform wind.

The following text appears here with Anthony Attia's consent:


When faced with a complex physical phenomenon, it is quite common to simplify the problem to a point where an analytical solution can be formulated. The simplification is done by stating assumptions throughout the approach. The more assumptions we take, the more likely our approximated answer will diverge from the true value. As students of science, it is our duty to ensure that we are equipped with enough knowledge to apply the proper assumptions.

Tarzan’s rope problem can be as complex as we want it to be. We can treat the rope as either flexible or rigid, we can treat the wind force as a function of time or a constant, we can consider the effects of cold temperature on the characteristic properties of air or we can neglect them. For the sake of maintaining my sanity and that of the reader’s, we shall treat the rope as a pinned rigid body who is subjected to a constant drag force that is acting in the horizontal direction. An important fact about assumptions is that there cannot be an incorrect one per say, however, every single one of them must be justified.

In my preliminary analysis, I will assume the rope to be rigid, effectively assuming that the profile of the rope will be linear when displaced.  Generally, this assumption would not be valid with a rope, but I will make it anyway and check the extent to which it was good later.

With that in mind, we can begin trying to find the velocity of the wind, by relating the drag force FD and the weight W.  Consider the model below, which depicts the scenario:

Given that the net drag force is acting on the center of gravity in the horizontal direction and the weight is acting in the vertical direction, the ratio of these forces, FD/W, ends up being equal to tan(θ).  We can take the sum of all torques about the pin and put them equal to zero.  Then, using the following definitions, we may express the wind speed as a function of the other parameters.

Surface gravity: g
Air density: p
Wind speed: V
Rope angle: θ
Rope mass: m
Rope diameter: d
Rope length: L
Rope shape (cylinder) coefficient of drag: Cd

The wind speed is then given by: 

Knowing this, we may begin computations to determine the wind speed that causes a specific rope deflection.  Assuming some reasonable 'Tarzan rope' values, it takes a 15 m/s wind to rotate the rope by 30°.  This seems reasonable.  But, we can only feel so much confidence in this result, as it is based on an assumption that may not be justifiable.

Say, however, that we now want to treat the rope as a flexible body; how would we proceed? Before answering that question, we must properly understand the behavior of weight and drag. In the previous figure, the drag force was lumped into a single vector whose line of action passes through the center of mass of the rope.  Let us do a quick thought experiment, if we were walking headwind, would our entire body feel pushed by the drag force or just a single point? The answer is the former, so why did we draw a single vector? That vector is actually the resultant or net drag force acting on the rope. If we were to properly illustrate the aerodynamic force that the body is subjected to, we would have to draw many smaller vectors that are acting on the entire exposed surface. These types of forces are called distributed load: though they act on every point of the body, we may sometimes use a single vector to represent the resulting effect (note that gravity is similarly distributed and then a resultant is used). Every segment of the rope has a mass equal to dm and the sum of all segment masses will yield the total mass m. Now, to solve the flexible body problem, we must assess a differential segment dm that is exposed to a differential drag of dFd by drawing its free body diagram.

Newton’s second law in x and in y yields:

These equations simplify to:

Equalizing the two equation we get:

It is evident that the equation obtained for the flexible body problem is the same as the rigid body problem, however, it is in a differential form. To remove the differentials, we must apply an integrating operator to the equation.  If we do so, the same expression linking the angle to the wind speed is obtained.

We conclude that both approaches lead to the same answer, but one requires an understanding of calculus, whilst the other requires only an understanding of mechanics. As one of my professors used to say, the simplest solution is often the best solution!


It gives me much pride to see a former student of mine express himself as he does here.  I get the same result on my end as that which he found.  The reason that the rigid body assumption works is because, due to the symmetry of the scenario, the uniformity of the fields and rope, the rope's profile must be linear.

In this problem, we have the weight force and the drag force.  They act vertically, and horizontally, respectively, onto each element.  While weight acts on dm elements, and drag acts on dA elements, both are uniform: we may think of each as a uniform field.  Effectively, they combine to form a uniform net field, and the rope simply aligns itself with it.  Though I initially thought the rope would have some curvature, it does not.  I am almost disappointed that the result is so simple.  I will try to pose a problem that has a stranger result in my next post.

Still, what I really want to emphasize here, is something greater than the problem itself.  I am thrilled that The Engineer's Pulse just had its first guest writer; he happens to be a fine engineer in the making.

Saturday, December 26, 2020

A Tarzan Rope in the Wind (Problem)

With all of my grades entered, my mind can turn off for two weeks.  In my case, that means exploring my curiosity.  Today, that resulted in a fascinating mechanics problem.

My kids have a Tarzan rope in the backyard - a rope suspended vertically and hanging freely.  I noticed this morning that it had been displaced significantly by the wind: it was now draped over a swing that hangs nearby.  "That must have been some wind," I thought.  Rope has a small ratio of surface area to mass, which means it should not be overly affected by aerodynamic drag forces.  With some physics, I should be able to estimate the minimum speed of last night's wind.

To make the exercise worthwhile, I have no intention of simply solving a numerical problem: boring.  Instead, I will solve a generalized problem before specifying any parameters.  Before doing so, I will make some assumptions that will hopefully render the problem to one that can be solved without numerical software.

I will assume that the wind is lateral and constant.  This may lead to an overestimate of the wind, because it is possible that some sort of driving frequency was present in the wind, causing the fundamental mode of the rope to resonate somewhat.  Still, it is probably a fair assumption.  Also, the fact that the wind force is not time variant will reduce the governing dynamics from what could have been a partial differential equation to an ordinary one.  This is because the lateral displacement of the rope (y) varies along the vertical rope's length (x).  A time-varying displacement would make the solution vary according to y(x,t), a multivariable function.  Now, I can begin my search for the single variable function, y(x).  I will also assume what appears to be true: the rope is uniform in terms of its properties and cross-sectional geometry across its whole length.

At this point I could probably Google "steady-state lateral deformation of a vertically suspended uniform rope exposed to a uniform and constant lateral wind", but I strongly doubt anything useful will turn up.  So, because I can (I hope) solve this problem, I am diving into it head first.

Yup, it's boxing day, but instead of looking for deals on stuff, I am entertaining myself for free as my wife shakes her head (well, she doesn't, but that is only because she thinks I am up to something more important).

Before beginning this analysis, I must determine what approach to take.  It is clear that with a constant wind speed at all locations of the rope at all times, the rope will reach a steady state y(x).  So, I am in search of an equilibrium position.  This simplifies things considerably from a typical first principles analysis.  Rather than applying Newton's second law for all of the infinitesimally small segments of rope, dx, I can do this using the first law.  That is because acceleration has been removed from the scenario.

I could do a quick first pass using an assumed modes shortcut.  If I assume that the shape of the rope will follow a specific y(x), I could quickly establish a single algebraic equation in which wind speed is the only unknown.  Here, I would effectively be starting with an assumed solution, but if my guessed shape happens to be good, the answer it gives could be surprisingly accurate.  It is a good moment to pause and ask ourselves the following question: If we were forced to assume the shape that a constant wind would impose upon the freely suspended rope, what mathematical function might it follow?

Four possibilities immediately come to mind:

(1) Linear

(2) 1/4 sine wave

(3) Some sort of polynomial 

(4) Some sort of exponential

I will not reveal here which of the above options my intuition leans towards.  I will leave that as a fun mental exercise for the reader.  I will dissect each of the above options in my solutions post.

Alternatively, I could use a more generalized approach - one that requires no shape assumption.  The downside here would be that an ordinary differential equation would require solving.  It would probably be solvable analytically, without the use of numerical tools.  The type of y(x) function that would result would either confirm or negate the choice of assumed function used in the shortcut approach described above.  I guess I should be rigorous, and explore both options.  I could then note the extent to which the quicker method is valid.

I suppose this is where I leave you, for now.  If you are a mechanics savvy reader and are up to the challenge, feel free to post your solution for y(x) in the comments (you can use any approach you like, but please state the one used).  To ensure we use the same variables, express your y(x) in terms of the following variables:

Surface gravity: g

Air density: p

Wind speed: v

Rope mass: M

Rope diameter: D

Rope length: L

Rope elastic modulus: E

Rope shear modulus: G

Rope shape (cylinder) coefficient of drag: Cd

It is likely than one or more of the above parameters will not appear in your solution for y(x).  It is also possible that some additional assumption on your part will be required, though at this time I cannot think of one.

Alright, now for the hard part, if I do not get overwhelmed by laziness: working out the detailed solutions.  See you on the other side of 2020.

Thursday, November 19, 2020

The Smoke of Wildfires Travels to Canada

The 2020 wildfires in California have devastated parts of that state, but they also serve as a symbol for the socio-political fire that is consuming America. One clear similarity between the two scenarios is that the current environmental conditions favour devastation. The likelihood of the physical fires increases as the globe warms, and high winds paired with low precipitation spurn it onward. The socio-political fire that rages on in the United States is a direct result of a tribalistic political environment, which threatens to overthrow democracy in that country.

There are, of course, some important differences between how these fires are being handled. The physical fires are combatted by waterbombers and firefighters. Fire chiefs have done everything in their power to prevent its spread. The divisions in the United States – the deep discord among its citizens – has been and continues to be exacerbated by the outgoing President. The fire chief is stoking the fire as he contemplates his next move. We are witnessing the fall of Rome.

As Canadians, I may consider myself a passive observer of the 24/7 drama channels, like CNN. But I have good reason for staying tuned, despite its obvious negative effects on my mental health. There are serious existential risks to our species that we must face, from the aforementioned climate change, to the current (and next) pandemic. Neither of these threats lead to our extinction in the short term, but they require our serious attention to limit their harmful effects in the long term.

Various branches of government must work alongside scientists to combat these fires from spreading. But instead of that, the leader (did I mention outgoing?) of the American government is questioning the expertise of the scientists, and even worse, causing his mob of supporters to doubt scientists. That is like halting efforts to stop the fires in California because we distrust that water molecules consist of two atoms of Hydrogen and one of Oxygen (I can actually imagine President Trump arguing that the Oxygen atom helps the fire breath as his minions nod their heads). Worse still, the smoke that blows from his mouth travels around the world, infecting non-Americans as well.

Assuming that President-elect Joe Biden is indeed sworn into office in January, the next question becomes: Can America reassert its place in the world, become a voice of reason, and help us homo sapiens become responsible custodians of Earth?

Perhaps Canadians can instead ask themselves: "What can we do to help make this possible?" I have a few suggestions. Let's:

1) Be our best selves. We need not enter into Facebook yelling contests with conspiracy theorists. Let the FBI worry about them. Just act kindly - be respectful and act with integrity.

2) Promote the virtues of democracy and show up to vote when it is our turn. We can also encourage our leaders to be their best selves.

3) Clean our own house. Canada is great but it is far from perfect. We must lead by example and do what is undeniably right. We must continue to strive for equality across race and gender.

4) Act responsibly towards the environment as individuals and ask our leaders to hold corporations and institutions to this same standard.

Who knows? The age of reason and enlightenment may not be out of reach. Getting there, however, will mean fighting back against the many fires that threaten to consume us. It means trusting the experts who have earned such distinction. It means electing officials who serve our best interests. Most of all, it means thinking locally and globally; it is not and has never been a choice between one or the other. A fire anywhere on this planet is a fire everywhere on this planet; we have just one planet.

I fear that Americans may be so divided that no leader can sew the country back together in four or even eight years. I mitigate this fear by turning off CNN and going for a walk.

Monday, November 2, 2020

The Benefits of a Decade Blogging

 I founded this blog in October, 2010, which feels like an eternity ago.

After ten years of blogging, I can speak to the positive impacts it has had on my career.  Though I originally started the blog as an outlet for my ideas on science and engineering, it quickly evolved into a space where I write for my students; after all, I became a physics professor at Vanier College in 2010 also.

Blogging has:

(1) Helped me consolidate my own ideas

(2) Made me a better teacher

(3) Given me a bank of topical content at the right level to pass on to my students (incidentally, this has become a cornerstone of my teaching approach, whereby new content is introduced before class, conceptually - students then write their reflections and submit them - and I then assign their best questions back to them so they may discuss them in groups)

(4) Been a launchpad to some speaking and radio gigs

(5) Made me a better writer

This last benefit has enabled me to produce a manuscript for my first book about the fundamentals of physics.  A couple of publishers have shown some interest in it recently.  I am hopeful to sign a publishing deal sometime in 2021.

Ah, 2010 was long ago indeed.  The blog has aged with me, but also my children - my eldest had just turned one when I wrote my first post, which was fittingly entitled "My Daughter the Physicist".  Reminiscing further, humanity seemed in better shape then.  Two years into President Obama's first term, there was reason for optimism.  I still remain hopeful that we can right the fragile ship on which we sail, but along with many of you, I am growing worried of the future (like, two days in the future, when a racist, misogynistic, reason-defying narcissist is hoping to see his time in the White House extended by four more years).

I pray that in 2030, humanity will have found its stride, and have become responsible custodians of this planet.  It is within our capability.  Until then, I will continue to write, for there is indeed a sixth benefit to doing so: it is so much fun.

Friday, October 2, 2020

How do I Feel About This?

Humans on this planet woke up to the very not fake news that the sitting President of the United States has contracted the COVID-19 virus.

Like anyone with a reasonable amount of education and any semblance of a moral compass, I have no desire to witness, albeit from across a border, four more years of lies, deceit and fear-mongering.  Let me also clearly state that I sincerely hope the President recovers from this illness.  What is more, I do not even think he deserved to get the virus - no one deserves it.  Some might invoke karma, because after all, this is a man who hid the dangers of the disease from those he claims to represent.  In my head, I am not really going there.  My thoughts this morning are moving in a completely other direction...

Obviously, this only hurts his re-election campaign.  Even if his symptoms are mild, his supporters will have a hard time calling this a total hoax (it was announced by his highness on his royal twitter platform).  One way or another, his diagnosis gives the optics of weakness - his perceived strength and bravado is the one thing some might say he had 'going for him'.  Of course, if his health suffers for a couple of weeks or more, then it seems to me that the election is a moot point (though we have all been proven wrong by this man in the past).

If indeed, this virus spells the end of the sitting President's reign, that is, without question, a good thing from the point of view of the vast majority of humans on this planet.  But as this particular human sits typing away this morning, he cannot help but feel strangely about it.

Before this news, I, along with most Canadians that I know, felt a great deal of anxiety over the four weeks leading up to the U.S. election - these feelings were independent of the fact that the world seems to be on fire, sometimes in a literal sense.  If the chances that Joe Biden will be the President in 2021, instead of the alternative, increased dramatically overnight, that is undoubtedly a good thing, and it does provide me with a certain sense of relief, or even, dare I say, cautious optimism.  But the prevailing thought I am overcome with is that I would have preferred it not happen this way.

I would like to have seen sensible Americans do the right thing in massive numbers based on the information before them.  I wanted my faith in my neighbours to the South restored.  I wanted the overwhelming majority of Americans to choose the reasonable human instead of the bully, the ignorant fool, the small-minded racist, the petty mysoginist... Need I go on?

Alas, the scale may have tipped over last night.  Biden may just win this thing by default.  The right thing might happen, but for the wrong reason.  It reminds me of a completely analogous situation - an even more serious one that also affects us on a global scale: climate change.

I have no doubt that humans will eventually move away from fossil fuels to power their lives.  All powerplants will be solar, all combustion engines will convert to electric, and most food consumption will go vegan.  Sadly, these changes will happen for the wrong reasons.

These grandiose modifications, so necessary for the viability of life as we know it on this planet, will be made not because they are the right thing to do.  They will happen because they are economically preferable.  The cost of vegan will be cheaper than raising cattle only to kill and distribute their parts.  The cost of solar energy production will become far less than that obtained by the burning of coal (it is comparable today).

The fight against climate change may be a victory one day (that is, it may not be a runaway train with no end in sight, and be limited to a minor global catastrophe), yes, but it will reflect that our species cares deeply about our world economy and not our world.  It will leave me shaking my head at our backwardness.

As a hopeful person, I yearn to see my fellow humans step up to the plate, and make the noble choice.  The President's contraction of COVID may have, in a sense, denied Americans that opportunity.  The outcome may well be positive, because the more suitable candidate may win, but it may not be a victory that Americans can be truly proud of.

I sincerely hope that us fickle humans come to our senses - that we become responsible custodians of this planet.  I want it to happen so much, that I would accept almost any avenue required to get there.  But I would prefer that our path to responsibility, accountability and sustainability be paved by the good will of humans rather than their obsession with the almighty dollar.

Tuesday, August 18, 2020

Back to School in the Post-COVID World

Somewhere across the spectrum of difficulties our society faces in the post-COVID world lies the education of our children.  As a parent and a teacher, I am currently operating under the assumption that, come the Fall, Secondary V and lower will be attending classes in person, while the majority of those in CEGEP and higher will be resuming their studies online.  Given what we know about COVID-19, it appears that the socio-economic fallout associated with keeping children home is the worse of two evils when compared to the health risks attached to attending classes in person.

My primary concern with the resumption of learning activities that await our students in a matter of weeks is that the already wide chasm that exists between so-called strong and weak students will widen, perhaps dramatically.

In my ten years of CEGEP teaching, I have observed the following: our best students get better every year, and our struggling students struggle more.  Anecdotally, I attribute these changes to increased access to technology.  Where a strong student might use Wikipedia to examine the link between black holes and general relativity, a less motivated student might spend an afternoon on Instagram.

Indeed, the resources available to our children are mind-boggling.  Self-directed learners (there are a handful in every class) could arguably work their way through elementary and high school on their own armed with only a list of content, a tablet, and an internet connection.  In this thought experiment, such students suffer socially, but may emerge unscathed academically.

My fear is that going forward, our students’ academic diet will be dominated by screen learning.  While this is evident for online learning, our younger students who sit in classrooms by day could experience a similar, though less dramatic shift.  Consider a teacher who is mandated to bring their students to a hand-washing station once per hour.  This process eats up fifteen minutes each time.  Where is this lost hour per day recouped?  Kahn Academy YouTube videos from home?  Flashy learning Apps that utilize Smart Gaming?

The motivated student whose parent can spend time alongside them may well eat this content up.  But what of her classmate, who would, quite understandably, prefer to play street hockey or watch an entire season of Friends, and whose parents arrive home exhausted around dinner time?  Scenarios such as this make it clear that the educational landscape, which already favours wealthier families, is about to stratify even further. 

Oh, and what about the teachers?

A common word that echoes through school administrations is equity.  Equity across a given course means that regardless of which section of say, a Mechanics class that a student is registered in, they will experience a similar degree of difficulty, cover roughly the same content to the same depth, and ultimately have an equal chance of passing.  Many departments succeed in this by meeting regularly in curriculum committees and sharing teaching materials. 

However, this does not ensure that the learning experience is equal across different sections in all courses. Academic freedom means that each teacher is free to select their preferred pedagogical approaches for the courses they teach.  This freedom is crucial to the teaching profession, as it allows a teacher to tailor the learning experience to their own unique strengths.  There is, however, a downside to this necessary freedom.

Back in March, when education abruptly moved online, the learning experience became, for lack of a better term, the Wild West.  Some teachers threw massive streams of video content at their students, some gave their students regular feedback, and some teachers, I can only assume, replaced their morning coffee with gin and held wildly entertaining Zoom sessions.

Fortunately, us teachers have had more time to prepare for this Fall (although it must be said that, with weeks to go, there has been little flow of information from the provincial government thus far).  Many educators will adapt to the new boundaries inherent to teaching in 2020 and beyond.  My hope is that our students, regardless of their socio-economic class, can adapt along with us. 

Wednesday, June 24, 2020

Things Must not Return 'Back to Normal'

I had written a short post regarding the return to somewhat normal life after three months of quarantine.  As I was about to post it here on my blog, I decided it would benefit from a larger audience, and so I sent it as an op-ed to the Montreal Gazette.  They decided to publish it (it is in the June 25, 2020, print edition); the online link is here.

In summary, my hope is that as certain aspects of life get back to normal, I sincerely hope that some key lessons learned on our three month long quarantine reflection will endure.  Things must not go back to how they were.  Black Lives Matter, buying local, minimal commuting... These positive movements were inevitable, but COVID-19 accelerated them.  The hurt brought upon by the pandemic cannot be undone, but it can stand for something if we maintain the positive changes that it has brought about.

I extend my best wishes to you and your families and friends.  I remain hopeful that our species can learn to live justly among ourselves and sustainably atop this beautiful planet.

Monday, May 6, 2019

Meet S.E.R.G.

Towards the end of 2018, I decided to try to form a small research team of college students to join in on my space elevator research.  I managed to recruit three such students.  This turned out to be the beginning of S.E.R.G. (Space Elevator Research Group), a lab that aims to study various mechanical aspects of the Space Elevator.

From left to right: Professor Arun Misra (McGill University),
Richard Ziegahn, Raffael Rinaldi, Tristan Vieira, and Stephen Cohen (Vanier College)

Over the course of the past four months, I have met with the three students every so often to help guide them through our research.  Besides learning some fundamentals associated with space elevator mechanics, their focus has been to write a code that will explore something new.  On my end, I have been delighted with how little support they have needed to get as far as they have.

Without getting into the details of our research, I just feel the need to express that they are doing the work I would be doing if I had the time to do it.  It is not unusual for students to aid researchers in this way; it is unusual that the students are 18-19 years old.  I was 24 when I published my first paper - these students will be published before they start university.  They will all begin undergraduate studies in various engineering disciplines later this year.

Last week, the three students shared their contributions to our research project at a tech/science fair.  Also in attendance, was my mentor-turned-collaborator, Professor Arun Misra.  Indeed, the five of us will be jointly published in the proceedings of the International Astronautical Congress that will be held in Washington D.C. this coming October.  Misra and I are sure to be there - the other three will join us, but only if they can cut classes for a few days.  This unique opportunity may actually justify doing so.

I must admit that I was initially unsure as to how much any college-level students could contribute to real-world research.  The 21st century seems to have equipped these resourceful students with enough tools to contribute a great deal.  I am encouraged by this, and am seeking funding that will allow me to grow and maintain S.E.R.G. for years to come.

Friday, February 1, 2019

How Powerful we Have Become

The bad news is:

1. Homosapiens are fallible
2. Our planet's stability is fragile
3. We have attained a significant level of technological maturity

Our technology is powerful enough to alter our biosphere.  It did not happen in one year - it took one century.  The most dramatic changes occurred over the past few decades.  Unquestionably, if we continue along, status quo, this planet will become both unstable, and unrecognizable in a few more decades.  We will render ourselves powerless to stop it.  We will reduce in population size, not by choice, but by attrition.

Bottom line: our social and ethical maturity has not kept pace with that of our technology.

The good news is:

1. The future is unwritten
2. We can learn from our mistakes
3. New technologies can actually help if they are tempered with reasonable behaviour

I have faith in this species to which I belong.  As individuals, we are, by and large, quite dignified in our actions most of the time.  As a collective, we can be better, though we are, at times, worse.  Even when 10% act horrifically, if some of them command significant sway, it takes the other 90% to simply hold them at bay.

Bottom line: we must allow civilization to mature from a sociological standpoint, and make the kinds of decisions that give us the right to deem ourselves an intelligent species.