Spinning My Wheels

This week I changed the toilet paper in my bathroom, and it got me thinking about the rotational properties of a roll – When you try to spin something, you have to fight against rotational inertia, which depends both on mass, and how far from the center it's distributed.  That's why figure skaters speed up their spin when they pull their arms in:

What made me curious is that as the roll turns, the radius decreases, which lowers both the inertia and the torque that spins it.  Do these balance out?  I figured I'd try modelling what happens if you drop a roll while holding one end.

Since we're only interested in the roll rotating in the usual way, we can consider things in 2D:

We can find the rotational inertia by thinking of the roll as a disk with radius r(t), minus a smaller disk of radius r_i:

where ρ is the mass-density of the roll.  Now we can set up a bunch of other equations that we need.  By holding on to the end, we're applying a torque:

where α is the angular acceleration of the roll.  If we suppose each sheet has thickness T, we can relate the radius to the angle the roll has turned through:

Finally, we can integrate to connect the angle to the distance the roll has dropped:

Combining the torque and radius equations gives a differential equation:

This is fairly simple to solve by integration:

but I'm having trouble getting a plot.  I guess this is another week with complex equations leading to unsatisfying results.  Thanks for sticking through :)

MeteoRing

Shortly after I mentioned my girlfriend Marika, I'm delighted to announce we got engaged!  A few days later, we went shopping for my ring, which arrived earlier this week.  The metal used in the ring comes from a meteorite that hit the Earth during prehistoric times, and scattered over 8,000 square miles in the region of Africa that gives the meteorite its name: Gibeon, Namibia.

The unique design on the surface of the ring is called a Widmanstätten pattern, and is evidence of its extraterrestrial origins.  The pattern forms when a mixture of nickel and iron cool over a long period of time, on the order of 10 million years.  Such slow cooling isn't possible on Earth, where the rest of the planet can serve as a heat-sink.

The name "Widmanstätten" comes from Count Alois von Beckh Widmanstätten in 1808, but the first published study was actually from G. Thomson in 1804.  There's quite a story behind why he was not acknowledged (from Wikipedia):

Civil wars and political instability in southern Italy made it difficult for Thomson to maintain contact with his colleagues in England. This was demonstrated in his loss of important correspondence when its carrier was murdered. As a result, in 1804, his findings were only published in French in the Bibliothèque Britannique. At the beginning of 1806, Napoleon invaded the Kingdom of Naples and Thomson was forced to flee to Sicily and in November of that year, he died in Palermo at the age of 46.

It seems like a perfect choice for an astrophysicist, and a wonderful start to the next chapter of our relationship.  Thanks Marika!

Not Quite "Beam Me Up"

Last month, a group from Calgary announced they had broken the record for quantum teleportation over fiber-optics, so I thought I might talk a bit about the idea of quantum entanglement.

One of the main concepts behind quantum mechanics is that every particle has a set of states that can be measured.  It's possible for a particle to be in a mixture of several states, but once you measure a property, it "collapses" into a single state.  This idea is usually introduced with the property of spin, which you can think of as an arrow pointing out of the particle.  We can detect this spin with something called a Stern-Gerlach apparatus, which you can think of as a box with one input, and two outputs:

 

The two outputs tell you whether the spin is aligned with the angle of the box (+) or opposite the box (–).  If the particle goes in sideways, as shown above, then there is a 50% chance of it coming out each port.  However, when it comes out, it will have changed its spin to straight up or down, and all information on its previous sideways state will be lost.  In quantum mechanics, the initial sideways state is described as a superposition of the + and – states of the box: The particle is literally in both states at the same time.

Entanglement refers to two (or more) particles whose states depend on each other.  In the framework we've been discussing, we could imagine a device that puts out pairs of particles that have opposite spin.  Once we measure the spin of one, we immediately know what the spin of the other is.  This instant change of state for the unmeasured particle is called quantum teleportation.  If we put one of the particles in the box above, and it came out +, we would immediately know that the other particle was aligned opposite the box, no matter how far we had separated the two particles before making the measurement.

The idea of the new state being transmitted instantly over a great distance might worry you, since that would appear to be information traveling faster than light, which Einstein showed was impossible.  However, the state the two particles end up in is still random, even if they're guaranteed to be opposite each other.  The only way to observe this correlation is by comparing the spins conventionally at sub-light speeds.

Richard Feynman once said, "If you think you understand quantum mechanics, you don't understand quantum mechanics," but I hope I've given some insight into one of the frontiers of modern physics.  As always, questions are welcome!