On the Rack

Lately I've been watching the anime series One Piece, about a man named Monkey D. Luffy who wants to be the Pirate King.  In the story's world, there is something called Devil's Fruit, which gives those who eat it different supernatural abilities.  In Luffy's case, it made him a rubberman, giving him the ability to stretch his body as if it were made of rubber.  I won't get into the details, but in the episodes I've recently watched, Luffy ended up with one arm encased in a giant gold sphere, leading to this scene:

Since it's a little hard to see out of context, Luffy is holding onto a beanstalk with one arm, while the one with the gold dangles below.  I decided it was the perfect opportunity to see how stretchy Luffy is.

First, I needed to figure out the scale in the picture above.  The white line shows Luffy's height, which I assumed to be the average male height in Japan.  Using this, I found the distance his arms stretched (red) and the diameter of the sphere (blue).  From this, we can calculate that the sphere has a volume of about 1500 cubic meters.  The density of gold is about 19 grams per cubic centimeter, so the sphere weighs 29 million kilograms.

Supposing Luffy's arms follow Hooke's Law (the basic spring force), we can use the weight to find the spring constant.  Setting the weight of the sphere equal to the Hooke's Law force gives a spring constant of 10 million Newtons per meter.  For comparison, the spring constant for the front suspension of a Triumph sports car is about 35,000 Newtons per meter.

Of course, using Hooke's Law is an approximation, since rubber has some unusual stretching properties.  The polymers that make up rubber are best modeled with statistical mechanics, where temperature and entropy play significant roles.  If you're interested, this is a nice explanation.

Lazy Light

After a grueling semester and a half of graduate work, I've finally found some time to come back here.  I was inspired by a question posed by one of my fellow Michigan physics students to the rest of us.  He asked a rather philosophical question about the principle of least time, a shockingly universal rule about how nature chooses one path out of all the possibilities.  That's a little big for me to talk about here, so I thought I'd restrict myself to a corollary, Fermat's principle.

Fermat's principle states that light will take the path between two points that minimizes the time, rather than the distance, required to travel between them. In most cases, light simply moves in a straight line, but when it moves from one material to another, it refracts, bending its path according to the properties of the two materials.  The important factor is the difference in refractive index, a number determined by the material's molecular structure, conductivity, and other properties.  Vacuum has a refractive index of 1, and air is quite close, with 1.0003.  However, water has a refractive index of about 1.33.  This is why objects appear shifted when you dip them partway into water.

 

It might seem perplexing that this shift has anything to do with the speed of light, but it's a little clearer looking at a diagram.  If we plot the light's path according to position, it will look like this:

The shortest distance between two points is a straight line, so this certainly doesn't look right.  However, if we instead plot according to time, taking the speed of the light into account,

 

The light moves more slowly in the second material, so we stretch out the line we had.  Looking at it this way, it's much clearer why the light is bent in space.

I always enjoy making these write-ups, and I'm sure it's good practice, so I'll be back here as often as my workload allows. As always, questions are welcome, on this or any of the older posts.