The Physics of "Gravity" and "Interstellar"

Hanukkah is a Jewish holiday celebrating a miracle whereby oil sufficient for just one day burned for eight.  Over this past week, an entirely different miracle took place: I, a father of two young girls watched two movies, none of which contained a single princess.

I really got my nerd on: I rented "Gravity" and then went to the theater to see "Interstellar".  Both films feature a lot of physics, but they could not be more different.  "Gravity" is a ninety minute expedition featuring orbital mechanics, while general relativity weaves its way through the three hours of "Interstellar".

You are probably expecting me to dissect these two films and expose all of their scientific inaccuracies.  In truth, I enjoyed both of them immensely, but would like to address one major problem with each one.

Deep Breaths During Exam Time

Over the past weeks and months, I have witnessed some distraught students.  Young adults who walked into the semester cool as a cucumber are hitting the finish line exasperated - not all of them, but many.  A few have asked me for advice, and all I can seem to muster is "Breath".

I thought that I would strive here for something deeper and perhaps more helpful than this.  At the same time, I'll toss in a science metaphor, cause that is sort of my thing.

We are all aware of the need to manage our stress level.  Everyone has their own coping mechanisms for this, so I won't say "Go to the gym" or "Get more sleep" - it's good advice, but I want to focus instead on the bigger picture.

One thing that has been important for me is setting reasonable expectations.  I was not always good at this.

Rosetta Mission and Noodle Bridges

A couple of unrelated engineering feats occurred yesterday, and I'd like to address them.

First off, the European Space Agency (ESA) succeeded in landing a probe (Philae) on a comet (67P) for the first time in history.  The level of difficulty of accomplishing such a thing is truly off the scale.

A comet is tiny, 'celestially speaking'.  The mass of 67P is about 10^13 kg.  That's a one with 13 zeros attached to it.  But that makes its mass about 100,000,000,000 times lower than that of a typical planet.  Another aspect that makes docking with such a body tricky is its irregular, peanut-like shape.  It is far from a sphere (largest dimension about 5 km, and smallest about 2 km).  Still, regardless of where you land on this comet, the surface gravity is on the order of 0.001 m/s/s (about one ten-thousandth that of the Earth).  So, if you want to dock, you'd better hang on, because escape velocity is only about 1 m/s (jump, and you now orbit the Sun).

My Halloween Costume: A Free-Body-Diagram

Last week, on Halloween day, I came to work dressed in a costume...

As you can possibly tell, I was dressed up as a free-body-diagram.  All the forces acting on me were identified by force vectors (large red arrows) at their respective points of application.

I had the gravitational force, mg, acting at my center of mass, and then a reaction load at each foot.  In the picture above, I also posed with a reaction coming from the wall I was leaning against.  Hanging around my neck is a list of assumptions (this costume is valid under the following conditions...).

So, there you have it.  If you were wondering if I'm a nerd, you now have your answer.  I think even my physics teacher colleagues found it slightly nerdy - that speaks volumes.

I got some nice reactions from my students that day, though as I walked through the halls, non-science folks gave me some strange looks.  I spared my family any embarrassment by removing my costume before I returned home.

IAC 2014 - Day 4: My Presentation

Today, I give my presentation about the static deformation of the space elevator tether due to the presence of a climber.  I have decided to open the talk with a picture of the CN tower.  You see, its been a foggy week of weather in Toronto, and I have taken some time to stare up at the tower from its base.  The top of the tower disappears into the fog - no end in sight.  The sight appeals to me for obvious reasons, so I have to mention it to the other space elevator aficionados in attendance.  Imagine a time, decades from now, when there will be no end in sight to such a structure, even on the clearest day.

I present a summary of my most recent research.  One of the most surprising things is that this fundamental mechanical analysis had not been documented yet.  You station a climber at some location on the tether, and what will the new equilibrium state of the tether be?  Will it stretch more or less and in what locations?  What are the changes in stress and tension throughout the tether?

The results are as follows:

IAC 2014 - Day 3: An Independent Review of the Mars One Proposal

I know I should be writing about day 3 here, but yesterday, after submitting my day 2 entry, I attended perhaps the most compelling symposium of the conference; it was all about manned missions to Mars.  I can say with some confidence that nothing today will top it, so this entry will focus entirely on the exciting presentations that took place yesterday afternoon.

Most of the conference rooms hold about 150 people in it, and it is rare that a room is more than half full.  But I had a feeling that this symposium about Mars missions would draw interest, and sure enough, it spilled out the back of the room.  The first few talks were rather benign: A plan to land on Mars by 2030, recommendations by several experts in the field on planning the first mission, and a commentary on the social and legal structure of the first Martian society (nothing more compelling than reading the novel Red Mars).

The fourth presentation was introduced by the session chair as 'something special', and he was right.  Five students from M.I.T. wrote a paper challenging the proposal for Mars One.

IAC 2014 - Day 2: An Ode to my Yoda

Today, I will abandon the approach of summarizing my experience at the IAC, and focus solely on the morning session on space structures.  In particular, I want to highlight the talk that was given by Professor Arun K. Misra, my mentor.

He is the Yoda to my Skywalker, the Dumbledore to my Potter.  Much of my knowledge of space research and the reason for my initial interest in space elevators came from Misra.  He supervised my Masters project from 2004 to 2006, and we have published several papers together since then.  He is kind and acts with integrity, and has served as a model for professionalism in my eyes.

I will never forget my initial meetings with Professor Misra in 2004, when I expressed my concern that the project will be difficult for me to carry out.  He quietly laughed, and said "You know, building the first space elevator will be hard... Your project, studying its dynamics, will be easy."  He curbed my definition of hard work, and in so doing, helped me mitigate a common fear that young professionals commonly face - that of daunting tasks.

IAC 2014 - Day 1: Blast Off

"Dude, Buzz Aldrin just walked by..."  This type of thing happens at a major space conference, but as someone who has only been to a few such events, it stopped me in my tracks.  I paused, and considered the significance of the moment when that man walked on the moon forty-five years ago.  Aldrin is 84 years old now, but he still moves well (fast enough that I decided not to chase after him for his autograph).

Today, the International Astronautical Congress kicked off with an exceptional opening ceremony featuring a couple of astronaut MCs and Cirque de Soleil performances.  However, it was the musical performance of Peter Katz that really stole the show - he spoke to the 3,000 audience members about the importance of dreaming and following those dreams.  He then went on to blow everyone in attendance away with his amazing voice and songwriting; that was my friend Peter Katz, upstaging Cirque de Soleil.  The event closed with a fine speech by astronaut/rock star Chris Hadfield.

At the convention centre, it is hard to keep up with the fascinating information that constantly flies in all directions.  During the head of agencies event, I learned that only 40% of spacecrafts sent to Mars actually make it there.  It takes most countries many tries to succeed.  India's space agency (ISRO) recently made history by succeeding on a Mars mission on their first try.

The 65th International Astronautical Congress

Next week, I will be attending and presenting at the 65th International Astronautical Congress (IAC).  It is the annual space conference with the highest attendance in the world.  Thousands of experts in industry and academia from all corners of the world with gather to share expertise in anything and everything space-related.  The event will take place between Sept. 29 and Oct. 3 in Toronto, Ontario, at the Metro Toronto Convention Centre; the location changes every year - it's like the Olympics for nerds.

The scale of the event is kind of mind-boggling.  There will be 180 symposiums, each on a specific space topic.  Each symposium consists of anywhere from five to ten 14-minute talks.  Therefore, there will be about one thousand presenters and more than one thousand presentations, as some speakers will be giving several talks.

It is impossible to physically attend more than 6% of these talks.  At any given time, there are eighteen symposiums taking place.  When attending such a conference, one must make choices; there will be interesting talks missed because something more interesting is taking place at the time.  Spinning it more positively: there is always something amazing going on.

How do you Pee in Space?

I recently finished an autobiography of sorts penned by Canada's most famous astronaut, entitled Chris Hadfield: An Astronaut's Guide to Life on Earth.  It was a Christmas gift that I ever so slowly moved through - this reflects not my enthusiasm for the book, but rather the life of a father of two young children.

If I take just one thing from this book it is that the life of an astronaut is not for me.  While I would love to see the world from their orbiting eyes for a day, I cannot fathom dedicating my life to achieving such a goal.  In any case, it's too late for me.  At 32 years old, I have a better chance of becoming a professional athlete than an astronaut (and I'm already a mechanical engineer).

The truth is that any kid has a much better chance of being a professional athlete than an astronaut.  At any point in time over the last decades, there were merely tens of active astronauts in the world cleared for flight (slightly more than 500 people have ever been to space in history).  Compare that to the thousands of currently active professional athletes, and the case is settled.  I guess what I'm saying is that we should stop stomping on the dreams of kids who want to pursue sport, and stomp instead on those of would-be astronauts.  I'm joking of course, but those longing to be astronauts should know that the odds of it happening are slim.

Astronauts are the center of attention of the global space initiative pursued by thousands of engineers and technicians.  To become an astronaut, one needs to meet an exhaustive list of criteria, which hundreds of other applicants do, and then be among the best of them in every conceivable metric.

What if the laws of the universe are not constant?

I have been fascinated with nature for as long as I can remember.  How can one not be?  When I first understood that it is possible to understand and even predict its behaviour, I was hooked.  When considering these laws that appear to govern our universe, this code that nature follows, science assumes that it is all static - they neither fluctuate in space, nor time.  But, what if this is not the case?  What if the laws of nature themselves are transient?

Let me begin by saying that this is not an attractive notion.  The practice of science would be dramatically complicated by this.  But at its heart, science is a search for truth; this must never be sacrificed for the sake of convenience.

It is in this spirit that a 2012 study out of California State University set out to check whether or not Planck's constant is truly constant in space.  Using atomic clocks aboard various GPS satellites, the maximum variation found for Planck's constant was 0.7%, which, due to the tiny absolute value of the constant, might be attributable to measurement error.

The Potential Collapse of our Civilization

A controversial paper concerning the not so distant future of our civilization has been published this past month (May, 2014) in Ecological Economics.  The paper is entitled "Human and nature dynamics (HANDY): Modeling inequality and use of resources in the collapse or sustainability of societies". 

In summary, the paper uses a predator-prey modelling approach to predict the well-being of people (predator) and nature (prey) in the coming decades.  The numerical tool (HANDY) is applied not only to our civilization, but to a wide variety of potential civilizations.  The study concludes that our current civilization is on the brink of collapse, and identifies two particular causes: (1) over-exploitation of natural resources and (2) strong economic stratification (large gap between rich and poor).

The paper is controversial for a few reasons.  The first is that the media caught wind of this research before it was published, and erroneously attributed it to NASA.  The paper, now published, has three authors, none of whom are NASA representatives.  Still, their work has been peer-reviewed by experts in the field and has been approved for publication in a reputable journal.  This is why this paper should be met with controversy: we need to be talking about this because it is relevant to us all.

The Famous Twin Paradox

Maybe you have heard about the fantastic story of twin brothers who part ways on their twentieth birthday.  Let's call them Billy and Timmy.  Billy stays on Earth, while Timmy travels at 99.9% the speed of light on his fancy space ship.  Timmy returns to Earth on Billy's fortieth birthday after having explored the universe.  Timmy is visibly younger than Billy.  In fact, he has only been travelling for 326 days as far as he is concerned.  He is yet to turn 21 - he is not legal to consume alcohol in the United States though his ID says he is middle-aged.

Why the time discrepancy?  It has everything to do with speed.  Special relativity, as theorized by Einstein in 1905 and confirmed countless times by modern experiments, says that the faster we move, the slower time passes for us.  This is not science fiction; this is science.

The time ratio, also known as the Lorenz factor, which gives a ratio of the passage of time between stationary and moving observers, is given as follows...

 

"Daddy, Who Put the Whole World Here?"

My daughter and I had a brief discussion this morning, on the way to daycare, about the origins of the universe.  I can't imagine that I asked such deep questions as a four-year-old.

It was a true relief for me when she asked where the world came from, because before that, her deep questions were all about death.  While neither the birth of the universe or the death of living things is easy territory to navigate with a Pre-K child, I am far more comfortable discussing the former.

In answer to her question, I first described the idea of a Big Bang, a critical event where the entire universe is restricted to a tiny point at one instant, and then blowing up at the next.  Eventually, atoms form (we have talked about those before), and come together to make stars and planets.  I could not get into the forces that govern these processes - maybe when she's in elementary school.

At some point in my description, I paused to point out that there are parts of this that are still a big mystery.  I was worried that this might disturb her, but she seemed fascinated and unbothered that mommy and daddy don't know absolutely everything.  I explained that we don't know where the matter from the Big Bang came from, or what caused it to bang when it did.

Politics Through a Scientific Lens

There are many parallels between the world of politics and that of the physical sciences, but some major differences, of course, too.  The upcoming election in my province of residence within Canada (Quebec), as well as the unstable and too often brutal reality in many other jurisdictions across the globe, has brought politics to the forefront of this engineer's brain.

To be honest, when I turned eighteen, and obtained the right to vote, I had little interest in politics (I probably could not have named my province's Premier at the time).  My feeling is that today, this age group is more worldly than I was.  However, until we truly grow roots where we live, be it through having kids or buying real estate, politics are usually quite far from one's mind.

As an adolescent with an interest in science, I had a kind of skewed view of what people should focus on.  My father explained to me that not everyone needed to know science, but everyone needed to be involved in democracy, be politically literate, and cast a vote.  I thought that if the choice was one or the other, people should be scientifically literate.

Now that I am older and hopefully wiser, I see that we were both wrong.  People should have a minimum literacy with both science and politics.  If we do not understand the basic science behind an energy crisis or global warming, how can we establish who has the best energy or environmental policies?  We certainly cannot trust most journalists to inform us - for two reasons: (1) they often work for a media company with a political agenda and (2) they rarely possess a basic background in science themselves.

The most interesting contrast between science and politics that I see is as follows...

What if the Earth Spun Faster?

Yes, these are the kinds of questions that sail through my brain.  Deal with it.

By this question, I do not mean, "What if some orbital event occurred, say, a collision with a large asteroid, and it imparted a significant angular impulse on our planet?"  After all, if such an event occurred, our new spin rate would be of little concern, because if anyone did survive, they'd be preoccupied with the task of finding their next meal.

My question is more along the lines of, "What if, in the Earth's early formation, the net angular momentum of its particles about the center of spin had been a lot greater?"  How would life be different?

To perform analyses, let us pretend that the spin rate were ten times faster, resulting in a 2.4 hour day.

How would such a change have impacted biological evolution on this planet?  If nothing else, our sleep cycles would be different.  There would be far worse and more frequent hurricanes.  And, we'd need to invent more holidays to fill the 3,652 days of the year.

My real interest, however, is the impact that such a change in angular velocity would have on the mechanics of life.

At present, the Earth's angular rotation leads to a normal acceleration at the surface of the equator of about 0.0337 m/s/s.  This acceleration makes the normal force on the bottom of our feet when we stand on the equator slightly less than our weight force.  It brings our apparent weight down by about 0.4%, which is not really noticeable.  This is because the surface gravity is about 9.8 m/s/s, and clearly dominates any rotational effects.

If we stand on the geometric north or south pole, we experience no such effect, as we are standing on the axis of spin, and the radial arm is zero.  Consider, however, the mechanics of standing at an intermediate latitude, as most of us do...

In Montreal, Quebec, where I reside, the latitude is about 45 degrees.  The magnitude of the normal acceleration felt here is actually 71% of that felt along the equator, so 0.0267 m/s/s.  The interesting thing is that this acceleration does not point parallel to the gravitational field here as it does on the equator.

Fig. 1: Gravitation and acceleration vectors in Montreal
('ac' denotes the normal acceleration)

The 'Square Circle' Puzzle (Part 2)

I have outlined one way to solve the 'Square Circle' puzzle originally posed in my last post.

Before jumping into the solution, I want to share the origins of the bizarre expression used frequently in the worlds of boxing and wrestling.  In short, boxing rings were circular centuries ago.  In 1838, the first square ring was introduced (thanks Wikipedia!).  So, despite the fact that modern boxing rings are square, they are often referred to as square circles for historical reasons.

Now, what ratio of the identical areas of concentric square and circle shapes is shared?  One straight forward method is shown below:

The 'Square Circle' Puzzle (Part 1)

On occasion, I watch televised boxing.  It motivates me to exercise, because the two guys in the ring are often lighter than me and could kick my butt.  Last week, as I watched two sweaty guys with twisted noses and not an ounce of fat try to beat the snot out of the other guy, the commentator referred to the boxing ring as the 'square circle'.  Now, I've heard this expression before, but never considered just how ridiculous an expression it is.  I mean, to me, it's a square.

Though the expression is odd, it caused me to stop focusing on the bout in the ring, and instead on the fun little geometry problem I concocted in the corner of my brain that loves math.  Can you solve this puzzle?

The square circle puzzle:

Consider two shapes, one a square, the other a circle.  Both shapes have the same area, individually.  When the shapes are concentric, what ratio of their areas overlap (are shared)?

  The red portion of the concentric shapes is shared

This is the sort of math problem I really enjoy: no numbers, as pure as can be.  Whatever the answer, it is independent of the size of the shapes.

I have yet to sit down and attempt to solve the problem, but my gut feeling is that there are many ways to go about it.  The 'obvious' way would be to use integration, but that seems hard due to the discontinuity.  Which kind of coordinates would be best, polar or rectangular?  I believe this can instead be solved without Calculus.  Pythagorean level math should suffice.  Again, that's just my gut feeling.

I will try to solve this without Calculus, and put up the solution to this (Part 2) within a couple of weeks.  Give it your best shot.  Oh, and if anyone knows the origin of the ridiculous 'square circle' expression, please leave it in a comment.