Friday, August 19, 2011

Monorail Kitteh Physics

With the economy in the crapper, it's been suggested that the United States should invest in a large-scale public works project to reduce unemployment and kick start the economy.  Some have argued for the construction of a high speed rail similar to the ones found in Europe and Asia.  These trains utilize magnetic levitation or "maglev" technology for smoother and potentially faster travel.  China's high-speed line travels at a record 217 mph.  In contrast to a typical four hour Boston-New York bus ride, at this speed, you could cover the same distance in a little over an hour.  Sadly, the only American advances in rail technology over the last 10 years have been of the "kitteh" variety.  How large of a field would one need to levitate a monorail kitteh?1 

While current technology uses magnetic fields to levitate trains, the same effect could be accomplished using electric fields.  As anyone who's ever petted a cat and then shocked it by touching its nose can attest, cats easily become electrically charged.  Since like charges repel, one can levitate a cat by charging it and then placing a sufficient amount of like charges on the surface beneath the cat.  These charges will create an electric field E that will push up on the cat with a force

F = q · E,

where q is the net charge on the cat.  In order to levitate, this force must cancel the downward gravitational force on the cat,

F = mg.

Here, m ≈ 10 lbs is the mass of the cat and g = 9.8 m/s2 is the acceleration of gravity.  Setting the two equal, one can solve for the electric field,

E = m · g / q.

You may notice that we've yet to specify to specify the charge q on our feline friend.  Estimating this value is not trivial, but one may arrive at a suitable estimate by using the following logic.  Let's say you rubbed a balloon on a cat. Some electrons will be transferred from the cat's fur to the balloon.  If you repeat this with a second balloon, you'll find that the two balloons repel each other.  In fact, they repel so much after you do this, that you can levitate one balloon about 10 cm above the other one.  The charged balloons will repel each other with a force similar to that between two charged point particles,

F = k · q2 / r2,

where k = 9×109 N · m2/C2 is a constant and r ≈ 10 cm.  A balloon with a mass of 10 g feels a gravitational force of about 0.1 N.  Setting this gravitational force equal to the electrical force, we can solve for the charge on the balloon,

q =(F · r2 / k)1/2,
= [(0.1 N) · (10 cm)2 / (9×109 N m2/C2)]1/2
= -3×10-7 C.

That's about 2 trillion electrons worth of charges that have been transferred to the balloon.  Since that charge had to come from the cat, it has been left with a positive charge 3×10-7 C.  Plugging this into the equation for the electric field, we get

E = m · g / q
= (10 lbs) · (9.8 m/s2) / (3×10-7 C)
= 1.5×108 V/m. 

That's 150 million volts per meter.  Air starts to spark at about 3 million volts per meter.  Unless you want lightning bolts shooting out of your cat Star-Wars-Emperor-style, it's probably not a good idea to try and levitate him with an electric field.

[1] My more sophisticated readers will no doubt recognize the similarities between monorail kitteh and the well-known jelly-toast rocket.  For those that are unfamiliar, the logic goes as follows.  It is a physical fact that jellied toast always lands jelly-side down.  Likewise, it is a physical fact that cats always land on four feet.  By tying or in some other humane way affixing the jellied toast to the back of the cat and letting it fall from a modest height, the pair must never touch the ground.  Presumeably, the cat-toast is left oscillating between jelly-side down and cat-side down states.  If one wishes to utilize this scheme as a means of transportation, simply wedge a bottle of diet coke between the cat and the toast and insert some Mentos into the bottle.  This should serve as a suitable propulsion mechanism.

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