Friday, August 5, 2022

The Journey that is 'Publishing a Book'

Have you ever tried to publish a book?  If yes, I wonder if your experience tested your might the way mine has.  I have nearly crossed the finish line, but what a long and arduous road it has been (the publishing part, not the writing part).  If you are thinking about publishing a book, perhaps the following bits of wisdom gleaned throughout my soul-crushing journey will be of some benefit to you.

There are two main avenues to publication: to work with a publishing house or to go about it independently.  In the case of my forthcoming book, Getting Physics, I experienced both.  That, in and of itself, is an indicator that things did not go smoothly...

My first mistake: Writing the book before choosing the avenue for publication

If you intend to publish with a publishing house, it is far more efficient to make a book proposal, which includes a synopsis, proposed table of contents, marketing ideas, and perhaps two chapters, before completing a manuscript.  It turns out that even if you decide to publish independently, a book proposal is an excellent idea.  If you will be your own boss, you ought to provide yourself with a roadmap that considers the big picture.

Having written the book first, I backtracked and prepared a book proposal, and this process led me to modify my manuscript.  Armed with my book proposal, I now wished to find a publisher that was interested in my book.

I did some research, and found that the best way to get a good publisher is to get a literary agent.  They work on your behalf, meeting with established publishers, many of whom only consider works that arrive via such an agent.  It turns out that enticing a literary agent is about as hard as enticing a good publisher.

My second mistake: Having pride

I diligently prepared a list of literary agents that fit my work (non-fiction, popular science), got their contact info, and noted the package they wished to receive (usually a book proposal or a query, which is a much shorter synopsis of the project).  I sent in five tailored packages and waited for a response.  And waited.  I checked my email and junk mail more often than I care to admit.  Nothing.

I moved on from getting representation and began reaching out directly to publishers.  Again, I researched publishing houses with non-fiction pop-sci experience.  This time, I had a list of fifteen.  I sent out packages in groups of three.  I sometimes did get a reply, but it was never the green light I wanted.  There were some helpful back and forth exchanges, including brief explanations of why my book was not the right fit.  The main issue was that it was too lay to be a textbook, but too technical to be a lay book.  Well, that is exactly what I was going for: a book that would be challenging but accessible for a physics novice, and a light, enjoyable read for seasoned physicists.  I wrote it because that type of book did not exist, and it was the kind of thing my students needed; but the fact that it did not exist made publishers hesitant to sign a contract with a first-time science author not named Bill Nye.

With my pride swallowed and humble pie consumed, I remained committed to the project, and to working with a publishing house.  This set me up for my biggest mistake.

My third mistake: Signing on with an unestablished publisher

A colleague told me about her friend who had recently published with a new publishing house who shall go nameless.  I sent them a package and received a contract offer shortly thereafter.  I examined the contract and sent it to an author friend, who saw no major red flags.  I contacted one of the publisher's authors who spoke highly of his experience (his book was in sociology, not a natural science, but still, this gave me confidence to move forward).  I signed the contract, and celebrated my victory.

I completed a bunch of paperwork and tailored my manuscript to the publisher's standards within a month.  I recorded a promo video per their request.  Then, I waited months with little contact.  Eventually, they admitted that they could not find a content editor for science!  In the meantime, they decided to copy-edit (format) the book, and worry about finding a content editor afterwards.  The formatting process went on for months.  The final look of the book was not bad, but getting there required so much input from me (they clearly did not know what they were doing).  Months later, they still did not have a content editor for me, and I decided to part ways with them.  Both the publisher and I wasted nearly 18 months that felt like 36 in this process that almost caused me to give up on the book entirely.

The only good thing to come out of all of this is that the publisher's ineptitude forced me to learn a lot about publishing books.  This positioned me well to take on my latest (and, knock on wood, final) avenue for publication: KDP (Kindle Direct Publishing), which is run by Amazon.  The support at KDL via both online tools and actual humans you can call and who call you back within minutes or hours, was incredible (feedback with publishers happens on timescales of weeks and months).  Within weeks, the paperback was completed, and as I write, a proof hardcopy is on its way to my home by way of, well, Amazon.

I am not saying that all small publishers are bad, or that established ones only deal with established authors.  Everyone's publication journey is unique, and not all are fiascos like the one I have detailed here.  Still, I hope that some of this information will benefit another budding author on their road to publication.

My book has been a labor of love along a dirt road littered with shards of broken glass.  I hope that many will enjoy it once it becomes available.

Friday, January 7, 2022

David Suzuki's 'The Sacred Balance'

Like many, I have seen an uptick in my reading quota over the past couple of years.  My diet has included about 25% fiction, 25% biography, and 50% science non-fiction.  My favourite fiction was Matt Haig's The Midnight Library and my favourite non-fiction was probably David Suzuki's The Sacred Balance.  I had never read any of Suzuki's work, and although this one is more than a decade old, it seemed to be his defining work, so I went with it.

Suzuki is a prominent figure in Canada; he has been a leader in the sustainability movement for most of my life.  While my personal interest is in space exploration, there is no question that sustainability is the most pressing issue of our time.

The premise of the book is quite simple: while science is a powerful tool and a culmination of our collective creativity and curiosity, it has a tendency to be fragmented, failing to view ecosystems as a whole.  The findings of science has led to short-term increases in standard of living, increasing lifespan and comfort, but it has come at a major cost to the prosperity of our species in the long-term.  We are simply not thoughtful enough to use science conservatively; our economic system is based on unsustainable growth, and all political systems, thus far, have failed to prioritize the long-term.  Science, when perverted by runaway capitalism, is nothing short of a slowly burning fire on the global scale with nothing to put it out.  So, you know, this was a fun read in the midst of a global pandemic.

The thesis of the book is that we will not be able to control our planet with science for the foreseeable future.  If we wish to have a foreseeable future, we need to model our behaviour after civilizations that have lived in harmony with the sustaining features of Earth for hundreds of years: namely, indigenous people.  This does not mean we must abandon science and technology.  It simply means we must refocus it.  We must rethink our socio-political and economic systems; they must have sustainability sewed into their fabric.  In a finite system, growth is madness.  Growth is suicide.

The first half of the book focuses on the science of our sustaining systems and their interconnections: air, water, soil, solar energy, and biodiversity.  It is in this latter chapter that the writing flourishes.  A strong case is made that decrease in biodiversity hurts all species in the long run - it is a precursor to mass extinction.  Biodiversity becomes a measure of the long-term prosperity of our species, like placing a stethoscope to our existence on this planet.

The second half of the book is where its strength lies.  It talks about love and spirituality, the joys of being alive, the vitality that we are granted once our requirements of air, water, food, and warmth are met.  The final chapter is about restoring balance, not with further attempts to engineer our planet, but by allowing the ecosystems of Earth time to fix themselves - by getting out of the way.  We will need engineering to allow comfortable lives for our roughly eight billion population.  But it must be long-term-focused.  It must get out of the way.  This final chapter is about how we can get there.  It highlights stories of individuals, who become grassroot movements, who have come to effect macroscopic change.  Their stories must become a beacon for us.  They are truly motivational.  This motivation will be crucial in the way forward.

Last semester, a colleague of mine taught a sustainability course.  The experience left him disheartened because the students in the course did not believe humanity had the wherewithal to change.  They lacked faith in our species, and who can blame them?  In their lifetimes, world leaders have only set us in the wrong course, and these leaders often reflect the wants of the societies they represent.  I understand my colleague's sadness.  As a teacher, the students' morale is our morale.  And frankly, if today's young people have thrown in the towel, we are indeed a lost species.

One shining light, from my point of view, has been some sweeping change that we have seen over the last couple of years, in our response to a very different existential crisis: COVID-19.  Damn it!  I almost completed an article without bringing it up!  Maybe next time... But seriously, we saw a threat, and pivoted.  It was not pretty, and not without hardship, but as a species confronting a dangerous threat, we tried to make changes to adapt to the situation.  

Perhaps you have heard of the frog-in-the-pot analogy... A pandemic, to us, is like a frog that is dropped into a pot of boiling water.  We are that frog, still trying to climb out, the hot droplets of water striking our tushies.  

Our present situation, where our finite resources are being exacerbated, represents a different threat.  In this one, we are a frog in slightly warm water that is continuously being warmed further.  It will eventually boil.  In this scenario, a frog would likely meet its demise.  It would not instinctively react and jump out of the pot.  But we have an advantage over the frog.  We have tools, like a thermometer, and we understand the reasons for the warming of the water.  We can forecast, with limited but reasonable accuracy, the rate of warming that will occur if conditions go unchanged.  Armed with this, we can be smarter than a frog.  We can evolve our thinking, act responsibly, and earn the right to wield the powerful tools that science has unleashed.

It is essential that we react to our biosphere crisis with the same resolve as we did the pandemic.  We can do it.  At the very least, we can try.  But a sweeping response will only happen if a critical mass of people at all levels of society truly understand the severity of the situation.  They need to embrace the obvious truth that this threat is every bit as serious as a pandemic.  Its solutions are less scientifically complex than engineering a vaccine.  We just need to learn to get out of the way.  We need to exist within nature rather than attempt to manipulate it.  It is less about new science than it is about smart design.

We all know that science and technology can be abused.  We usually focus on the upside: agriculture nourishes the masses, electricity gives us light, warmth and comfort, and modern medicine reduces suffering and extends life.  But these are the very things that have allowed our population to balloon.  This larger population then demands the same kind of comfort, which means more brut engineering.  While this ballooning sounds like the opposite of extinction, it has taken an unprecedented toll on our sustaining systems in the blink of an eye.  

One way or another, this graph will come down.  But how will that journey look?  Will the descent entail pain and hardship?  Will it end at zero?  Or will we allow Earth's natural mechanisms the time needed to stabilize itself?  Will we be here to see it happen?  Will today's children come to know a world whose sacred balance has been restored?  Countless humans today have not given up.  Please be one of them.

Tuesday, December 7, 2021

As 2021 Winds Down in My Classroom

As I rev up for that final push through exam season, I am reflecting on this past semester.

This semester followed more than one year of online learning.  I felt relieved to know we were coming back.  I could not spend another semester teaching from this chair in my home.  The experience is ultimately deflating, because you know how ineffective it is for most students.  Online learning was a necessary evil that I hope to never experience again.  The one silver lining was observed very clearly yesterday morning as I cleared snow and ice off my car for the drive to the college.

This semester saw some very apprehensive students show up at the college.  The incoming cohort was generally less prepared than usual for college.  The second year science students, having spent their entire college education to that point online, were perhaps even more unprepared.  College can feel overwhelming at times, even for students who had adequate high school preparation.  This semester saw more students give up than I have ever seen in my 12 years as a teacher.

As midterm assessments were returned, some students were crushed, and stopped attending.  I felt awful for them.  I do not fault them for it.  I look at it another way: realizing the adversity that all students have faced over the past couple of years leaves me feeling extra proud of the ones who have stuck around.  They are the ones who will be sweating it out in three-hour exams next week.  I hope they take pride in showing all they have learned.

As for me, I remain committed to what seems like my mission in life: to help people 'get physics'.  I look forward to the years ahead spent in the classroom.  Career-wise, I am also excited about two things happening outside the classroom.

Over the past week, SERG (the Space Elevator Research Group) has reformed.  It includes three Vanier College students.  Over the coming months, we will be undertaking a new space elevator dynamical study.  It will involve the addition of a station at the geosynchronous altitude.  I am excited for that to get underway.  More updates to come on this project in 2022.

And then, there is my book, Getting Physics.  The publication process has been, well, lengthy, thus far.  I was hoping it would be published by end of 2021.  At this point, I will settle for sometime in 2022.  I appreciate all of the words of encouragement I have received via email and LinkedIn.  I am so excited to get this book into the hands of readers in the near future.

After years of teaching, it has become evident to me that you do not need to be a physicist to get physics.  Physicists will dive deeper than the rest of us, but there is much depth to physics even at the surface.  The fundamentals of physics are accessible to nearly anyone who wants to know them.  Ask my students this semester, who can describe all kinds of phenomena, from a car crash to a vibrating guitar string.  They are only beginning their journey into the sciences.  They are not experts yet, but the seeds have been planted.

I want to wish you all a wonderful holiday season and much happiness in 2022.

Monday, November 1, 2021

Disrupting Earth's Orbital Mechanics

After today's physics class, which involved orbital mechanics, I began thinking about ways in which humans could affect the Earth's spin rate or its path around the Sun.

Jumping all at once:

If all humans congregated at one place on Earth (7+ billion people in one city, while maintaining social distancing, of course), and then jumped simultaneously, there would be some repercussions.  The energy of all that mass shifting in a short time could lead to an Earthquake, for example.  But, that is not the sort of effect I am interested in.

Would Earth's path around the Sun be affected?  The answer is, surprisingly, not in the slightest.  The problem is that we would eventually land back where we started.  The net mass of the system consisting of Earth and us will not have changed.  As we are part of the total system that is in orbit, the forces exchanged during both the jump and landing would be internal to that system.  It is not possible to change the system's velocity without a force exchange with something external to the system.  For example, an asteroid collision could have some small effect on the Earth's orbit.

Running all at once:

OK.  So, jumping failed.  Maybe by running, we can impact the planet's spin rate.  Imagine that we (the human population) were to gather somewhere on the equator, like Singapore.  We collectively decide that we wish to change the length of a day on this planet.  We decide to run along the direction of the Earth's spin with the expectation that it might slow the rotation down (there are not enough hours in a day, they say).

With our first step, we propel ourselves forward (the Earth pushes us in the direction we move via static friction), so we impart an equal static friction onto the surface of Earth in the opposite direction.  However, every subsequent time that our foot strikes the ground, it slows us down before speeding us up again.  In fact, if we maintain our jogging speed, each step results in a net linear impulse of zero (on us and the Earth), which means that each step has zero net effect on the angular momentum of either.

It seems we suffer from the same problem as we did while jumping.  Our initial acceleration from rest gives a tiny net angular impulse to Earth, but it will undo itself when we decelerate, just as our jump was only temporary in the previous scenario.

The only way to accomplish either of the intended effects (disrupt orbital path or spin rate) is to do something more permanent, like sending payloads to space.  These do indeed impart small net impulses onto the Earth.  I could calculate their magnitudes, but I don't feel like it.

Blowing up the planet:

Frustrated with our wasted efforts, we decide to blow the planet up from the inside.  It splits into two halves.  Each hemisphere will orbit the Sun, but the precise orbit of each half will depend on the direction in which the planet splits apart.  Regardless, the Earth gets the last laugh... The center of mass consisting of each of the hemispheres will remain in the original orbit, because again, the explosion is ultimately an exchange of forces that are internal to the system.

Monday, July 5, 2021

A Goldilocks Universe

Anyone with experience in astronomy has encountered the term 'Goldilocks planet'.  It pertains to a planet that is not too near a star, nor too far, such that it may have liquid water on the surface.  Many scientists believe that this is a necessary pre-cursor for life.  Earth is the only Goldilocks planet in our solar system, but exoplanet searches have identified others across this galaxy.

This morning I was thinking about the Universe, and noting that there could be no Goldilocks planets without, what we might call 'Goldilocks stars'.  I would define a Goldilocks star as one that has a main sequence that endures for billions of years at the least.  There are countless such stars in our galaxy.

Why are billions of years of stable star output important?  It is because such is the timeframe that it takes for the development of life (itself an unlikely event) on a planet (which itself may take hundreds of millions of years to develop into a potential host for life).

A star's main sequence describes its stable state where the gravity that holds it together is in balance with the internal pressure that pushes it outward.  It is achieved during the period of time when the core of the star is largely a mass of protons zooming about (these protons are denoted as H-1, as they are hydrogen isotopes that lack a neutron, known as 'protium' as they are effectively just protons).  Energy is created via nuclear fusion when these protons collide and enter into what is known as the unfortunately named 'p-p cycle'.

A complete p-p cycle is a complex series of nuclear fusion reactions that eventually convert six protons into two protons and one Helium atom.  Each link in the fusion chain spits out other matter including positrons, neutrinos, and gamma particles.  Most importantly, the fusion reaction releases thermal energy because the nuclear by-products have less mass than the nuclear fuel - the fusion process produces energy E in the amount of dm multiplied by the speed of light squared (Einstein's uber famous equation) where dm is the quantity of annihilated mass.

The big picture is far less complex than the details: hydrogen fuel converts to helium and releases energy at a specified rate until it runs out.  The amount of time that this dance will play out for is determined by just one thing: the star's mass.

Red Dwarfs are small stars and are the most common; they can burn for trillions of years.  Yellow Dwarfs (like the Sun) are medium-sized and less common but not uncommon; these burn for billions of years.  Supergiants are far more massive than the Sun and are far less common; these burn for just millions of years before they exhaust their fuel supply.

Given the brief period of time (in cosmological terms) that Supergiants undergo their main sequence, it is unlikely that its planets can ever harbor life.  We can deem these stars too big.  We do not yet know whether Red Dwarfs can sustain life on the planets that orbit them.  These stars might be too small.  We do know for certain that planets orbiting Yellow Dwarfs can harbor life (we know of one clear example of this).  These stars, it seems, are just right: Goldilocks stars.

But it all comes back to that p-p cycle.  The rate at which our Sun burns through its fuel depends upon the probability that a p-p cycle can be completed.  Smashing two protons (H-1) together does not guarantee that a deuteron (H-2) will be synthesized (step one in the p-p cycle)... Far from it!  It is actually extremely unlikely.  The probability that it will occur is on the order of 1 in 10 to the power of 26!  The reason that the Sun produces energy at such a high rate is that despite the low fusion rate, there are some 10 to the power of 57 protons zooming about.

It is the 1 in 10 to the 26 rate that confounds me.  I mean, like, why that rate?  Each proton-proton collision is a quantum event.  The particular fusion rate seems so random, arbitrary even.  But it is ultimately critical to our existence.  If this rate were, say, ten times higher than it is, our Sun would have burned out long before life emerged on this planet.

Physics reveals many instances where the conditions of the Universe, its matter and the laws that govern how it interacts, seem to be just right.  If the strong nuclear force that binds the nucleus of an atom were slightly weaker, the electrostatic repulsion of protons would exceed it and prevent the existence of any atom not called Hydrogen.  No atomic variety means no life, just as no long-burning stars means no life.

One can imagine a universe not so perfectly tuned; a universe where life is impossible instead of improbable.  We may live on a Goldilocks planet that orbits a Goldilocks star, but if we widen our gaze, we see that we reside in a Goldilocks universe.  Not that it matters, but it is a funny coincidence that like Goldilocks herself, I ate porridge for breakfast today.  I mixed it with leftover brownies.  It became just right

Saturday, June 19, 2021

AATIP Reveals Compelling Videos of UFOs

Some weeks ago, as my class was discovering notions of relativity, a student asked what I thought of the bizarre videos that were making its rounds on the internet - they reveal what appears to be some kind of unusual aerial vehicle.  I watched these black and white videos with curiosity.  In the background, you can hear some excited voices expressing genuine confusion about what they are witnessing.  With final exams looming and little free time, I did not pursue this rabbit hole any further.  Then weeks later, a friend we'll call 'Phil', asked what I thought about the UFOs.

Tom is a staunch believer in the scientific method and a skeptic when it comes to conspiracies and the like.  But he found these videos to be very compelling.  He informed me about Luis Elizondo and the Advanced Aerospace Threat Identification Program (AATIP) and suggested I watch his recent interviews.  I did.  I also came across a clip of Barack Obama giving credence to the notion that the highest levels of American intelligence have come across aerial vehicles whose origins confound them.  It appears that AATIP is indeed a genuine Pentagon program and they will issue an official response to the aforementioned videos.

I impressed upon Phil that I am typically not drawn into stories of this nature due to the extreme unlikelihood of alien visitation.  However, if these videos were real artifacts, free of manipulations, they reveal technology that is far beyond current human capability.  The aerial vehicle in the videos:

1. Has no visible means of propulsion and whatever does propel it shows no sign of interacting with the environment.

2. Transfers from air to water without disturbing the water.

3. Banks extremely sharp turns at impossibly high speeds.

Let us, for instance, analyze point 3.  The vehicle is tracked at speeds in excess of Mach 5 (five times the speed of sound in air, so about 1,650 m/s).  In order to not experience violent accelerations in excess of 5g (about 50 m/s/s), the minimum radius that its circular path would require is 36 km!  Points 1 and 2 are even more bewildering.

If these videos are authentic, how did the vehicles get here undetected by our radio astronomers?  Elizondo theorizes they emerged from the deep ocean.  Phil asked me where we should purchase our aluminum hats.

Passing on the hats for the moment, I went to the library later that day, and returned home with They Are Already Here: UFO Culture and Why we See Saucers, by Sarah Scoles.  The book is a historical account of the human obsession with UFOs and the possibility of alien intelligence, from Roswell and Area 51 to AATIP.  The title to the book is misleading: the author confides on the last pages that she remains unconvinced that any interplanetary intelligence has ever visited Earth, and that the plethora of reported human encounters with aliens are either honest mistakes or fabrications.

I am interested in honest mistakes, as they force us to apply the scientific method within this thought-provoking context.  These range from explainable celestial events, to high-tech military operations, and a wide range of optical illusions.  I also understand and do not fault claims of UFOs that are entirely psychological, whether they be drug-induced or convincing dreams.   

On the other hand, fabrications offend me.  They are an affront to my senses.  They degrade the entire process of discovery.  Muddying the evidence, manipulating the data, unfalsifiable claims masked as truths... These acts of dishonesty, whatever their motivation, highlight the fly in the ointment, which is human corruption.  Such acts of deceit serve only to spoil the earnest endeavor of identifying UFOs.  One key take-away from Scoles' book is that distinguishing genuine science from hoaxes is half the battle in the search for alien intelligence.

When it comes to the matter of extra-terrestrials, we must be extra skeptical of information emanating from sources who have a vested interest in making the first human contact with them.  One such player is Robert Bigelow, a wealthy American who has initiated numerous 'scientific teams' whose primary outputs have been UFO fabrications.  When I discovered that Bigelow has a connection to AATIP, I began to doubt the authenticity of the internet videos.

It is improbable that we have been or will ever be visited by interplanetary beings during our species' tenure on this pale blue dot.  The chances that intelligent life exists in our neighborhood of this galaxy during the small window of time comprising human existence are very low.  But not zero.  And that is what distinguishes the topic of aliens from other human obsessions, like paranormal activity.  The former is entirely conceivable according to our current understanding of nature.  

Evidence that confirms the existence of aliens would cause a dramatic shift in our understanding of the universe and our place in it.  That is why this conversation is so alluring.

I await the Pentagon's response to the videos that have captured the attention of so many.  If their  assessment does not support the alien intelligence theory (and I highly doubt that it will), conspiracy-theorists will be unmoved.  Government history does include cover-ups, which merely confirms the general prevalence of human weakness.  This history of dishonesty injects doubt into the UFO conversation.  

I will not be buying an Aluminum hat just yet.

Tuesday, June 8, 2021

Enforced Rotation of Tarzan Rope (Solution)

The semester has ended, and alas, nobody posted a solution to the difficult problem I posed months ago (see problem here).  In short, we have a rope that is suspended from the top and is being moved along a circular path in the horizontal plane with constant angular velocity.  Aerodynamic effects shall be neglected.  We are seeking a lateral deflection function.  Here is my solution...

With a problem such as this, we must begin with a physical model.  My hand drawing is seen below (I apologize for the crude sketch, but the summer me exerts less effort):


The solid blue line represents the rope whose profile we aim to determine.  At some location (x, y), we will apply Newton's second law to a single mass element dm.  My free body diagram is on the right side.  There are two external forces acting on the element; one is real and the other, a pseudo-force.  The real force, dFg, is gravitational, while the centrifugal load, dFc, is a pseudo-force as it is effectively an inertial term.  Finally, tension acts internally, pulling this element in both directions tangent to the rope's profile at (xy).  The upward pointing tension is (correctly) assumed slightly higher than the downward one, by some amount dT.  One useful, though limiting facet of the assumed model, is that, at a given vertical location x, each element simply displaces horizontally - in reality, it also shifts up vertically, ever so slightly.  This simplification allows an elegant solution, but whose accuracy is limited as we shall see.

Applying Newton's second law to that element on both axes, we get:

dFc = dTsinθ                                                                                                    (1)

dFg = dTcosθ                                                                                                   (2)

We can express the elemental forces as:

dFc = dm(ω2y)                                                                                                 (3)

dFg = dm(g)                                                                                                     (4)

The angular velocity of the enforced circular motion is denoted by ω.  If we divide equation 1 by equation 2 and then divide equation 3 by equation 4, we get the relationship

tanθ = ω2y/g                                                                                                     (5)

The key realization to move forward is that the derivative dy/dx = tanθ.  This yields the governing equation:

dy/dx = ω2y/g                                                                                                    (6)

The particular solution to equation 6, after having applied the boundary condition y(0) = R0, the radius of the enforced circular path, is given by:

y(x) = R0exp(xω2/g)                                                                                          (7)

This solution is quite interesting.  We first notice that the density and area of cross-section of the rope have no effect on the shape it takes.  This is not surprising because both external forces were proportional to the elemental mass.  The more important takeaway here is that the lateral deflection becomes exponential.  The faster we spin the top of the rope, the more dramatic the curve.  This makes sense, but there is a serious flaw: the rope has a finite length.  As this function is exponential, there is no limit to the lateral deflection it describes.  As the imposed angular velocity increases, the lateral deflection can quickly become greater than the total length of the rope, which is physically impossible.  

I suspect that I ran into this problem because, in my original model, I neglected the gain in altitude that a particle driven laterally inevitably experiences.  For fun, I included this effect in a subsequent attempt.  After a page of work, I saw that numerical tools would be required to solve.  Again, it's summer, and I am content to move on and not pursue this problem further, especially when a closed solution appears impossible. 

Equation 7 may be a good approximation of the rope's profile for fairly slow rotation rates.  An experiment is difficult to conduct for multiple reasons. While air effects lead to a three dimensional profile, so to would inertial effects when it comes to establishing planar motion.  In principle, it may be possible to enforce the theoretical equilibrium configuration as well as a uniform angular velocity for all string elements, but it is not practical.  Failure to do this would inevitably lead to a helical 3D profile.

You may be thinking I did all that work for nothing.  It is important to realize that simplified approaches teach us a lot about complex problems.  They give confidence to the more strenuous, complex solutions that follow them.

And now, out of my cave.  Summer beckons.

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:

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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!

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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.