Built on Facts

Our excellent physics blogger Chad Orzel has a post up about the thermodynamics of Goldilocks. Seems it's a little questionable to have the porridge configured as it was in the old tale. A few wags in the comments complain that in a story with talking bears physics is the least of the concerns, but I think that misses the point. Suspension of disbelief requires that we grant the story the ability to say wild things so long as it does so in an internally consistent way. Don't, for instance, make time-travel commonplace enough so that a 13-year-old girl can use it to rearrange her school schedule and then say "BTW all the time machines broke" when the dark wizards start killing people. But I digress.

So there's this lady named Rapunzel and she's trapped in a tower. Being a thoughtful sort of person, she decides that a rope is just the kind of thing a lady trapped in a tower might need and she grows her hair out to preposterous lengths so that the handsome prince du jour can climb her hair to visit. In the original Brothers Grimm version, that would be "visit" - the witch discovers the prince when Rapunzel begins expanding about the midsection. Subsequent versions have tended to be more demure.

Regardless, the plot revolves around her letting down her hair. Hair has weight, and so she's going to have to have some strength to hold up the weight of all that hair. But that's not all. If you were to mark an X in black ink at about the halfway mark on her hair, you'd see that when she pitches her hair out the window that mark will accelerate downward with gravity until it jolts to a stop as the hair has finished unrolling to the halfway mark. That's a change in velocity, which implies a force. How much?

The change in momentum of each segment of hair will be equal to its mass times its velocity:

Force is change in momentum over the time it takes to make that change, and the mass of the hair segment being stopped per time will just be its density times its velocity:

We have from conservation of energy that the velocity of the falling hair will be the square root of 2*g*x, where x is the distance the hair has fallen. x has a maximum at h, so substituting we end up with:

as the maximum force required to hold her falling hair together. But you have to add to that the force required to hold up the stationary hair, itself equal to mgh, so the total force that her head must support during the unrolling peaks at 3mgh - triple the weight of her unsupported hair just hanging there.

So what does hair weigh? Some googling turns up a booming trade in human hair of which I was heretofore unaware, with one Rapunzelesque specimen clocking in at 14 ounces in 45 inches. The tower height is unknown, so taking the 90 foot height of one of the Towers of London as a representative ominous tower of confinement, that would be about 21 pounds of hair. The maximum weight experienced during unrolling should thus be 63 pounds. Not unreasonable, and certainly much less than your average handsome prince.

From a purely physical perspective we might have to call this one "myth plausible". Actually growing 90 feet of hair would be a considerable challenge, but I'll leave that one to the biology bloggers.