Friday, February 7, 2014

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)

As shown in Fig. 1, the difference in direction between the gravitational field and the normal acceleration vectors is actually equal to the angle of latitude.  As a result, the net effect of gravitation and acceleration is deviated ever so slightly from vertically downwards.  But, as the centripetal effect is so small by comparison, this deviation is negligible.

With all of this in mind, let us speed up the spin rate by a factor of ten and examine what happens...

The normal acceleration is proportional to the square of the angular velocity, so a factor of ten increase in spin rate actually causes the normal acceleration to increase one hundredfold.  On the equator, this acceleration becomes 3.37 m/s/s, and the apparent gravity reduces dramatically.  The apparent g-force becomes 0.66g or 6.43 m/s/s.  If such a change occurred today, everyone along the equator would suddenly feel 33% lighter!  I could finally realize my childhood dream of dunking a basketball on a ten-foot rim (maybe).

The effect of the faster spin rate would be of little consequence to anyone residing on either geometric pole.  The reading on their scale would be the same as always (they could, however, hitch a flight to Singapore, step on the same scale, and observe a weight reduction of 33%).  The only real difference to life at these poles would be the frequency at which the features in the sky revolve around.

As for the rest of us, that is, the vast majority of us, who live a reasonably large distance from the equator, well... Life would become much more complex.  We would feel lighter, but we would also feel unbalanced, as though we were living on a slant.  Here in Montreal, the normal acceleration would become 2.67 m/s/s.  The net g-effect would be 8.13 m/s/s, but it would not point downward.  It would actually rotate 13.4 degrees to the South.  To counter this, Montrealers would feel compelled to lean 13.4 degrees to the North.

Buildings would be intentionally constructed along a latitude dependent critical angle.  Otherwise, a large torque would manifest at their base, much like it would if we built leaning buildings in a vertical-g environment.  Buildings would be north leaning in the northern hemisphere and south leaning in the southern hemisphere.  Who knows, someone might build a "Vertical Tower of Pisa" to attract tourists.  

I don't wish this fate upon our planet.  I feel lucky to live on a planet where the centripetal effect can be ignored at all surface locations.  On the other hand, I am certain that my students and the general public would have a deeper understanding of vectors if our planet spun faster.


Special note for mechanics students... Note how the normal acceleration points inward, yet the effect is to make us feel lighter, akin to an outward acting force.  This direction confusion can be explained by Newton's Second Law, where forces and inertial terms ('ma' terms) appear on opposite sides of the equal sign.  Sometimes, the term associated with normal acceleration is brought to the other side of the equation and referred to as a pseudo-force.  An inward acceleration leads to an outward pseudo-force, which, in effect, reduces the normal force acting on our feet when we stand up.

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