Built on Facts

The Physics of Rapunzel

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.

i-1a81feff127be0861c834f7c1c5e935d-barbie.png
Fig. 1: Probably no pregnancies in this version.

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:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

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:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

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:

i-d18063683dcc0d42b9be45451a84d1e3-3.png

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.

Comments

  1. #1 Uncle Al
    May 4, 2009

    Nominal hair growth rate is about 0.5 inch/month (0.4 mm/day). 90 feet would then require about 180 years. If you take the tale at face value, don’t think about her bikini line.

  2. #2 Left_Wing_Fox
    May 4, 2009

    That’s a change in velocity, which implies a force. How much?

    I had a similar thought about that a coupe years back. I was animating a cartoon that involved Rapunzel’s hair, and wondered if tying the hair off around an anchor post in the window might prevent the stresses on the tresses from transmitting to Rapunzel.

    A physics background can be a blessing and a curse as an animator. On the one hand, it makes it a lot easier to explain animation in a technical manner, and produce more realistic motion. On the other hand, it can be hard to turn off the brain and go with “It’s a cartoon, Rule of Cool/Funny/Awesome applies.”

    FWIW, the cartoon I was working on was a Happy Tree Friends episode, so the implausible physics tend towards the gruesome side.

  3. #3 Steinn Sigurdsson
    May 4, 2009

    Hair is elastic…

  4. #4 John
    May 4, 2009

    The Mythbusters did once successfully test that a rope made from human hair was sufficiently strong to allow someone to escape from prison.

  5. #5 Mark
    May 4, 2009

    Rapunzel could also braid her hair for extra strength–at the cost of requiring a greater length.

    Anyway, I like this version: http://archive.salon.com/comics/lay/2005/03/22/lay/print.html

  6. #6 Joshua Zelinsky
    May 4, 2009

    Re:Time travel and dark wizards. Yeah and no one on either side tries to make any new time travel devices. Harry Potter had so many consistency problems it wasn’t funny.

    We also have a separate set of issues which is that Rapunzel needs to have hair that is strong enough to do this. I suspect that the weakest part is where the hair meets the scalp. That could rip badly.

  7. #7 Kobra
    May 5, 2009

    Problem: It only takes like 20 pounds of force to snap her neck.

  8. #8 zak
    May 5, 2009

    Yeah, I always assumed she tied a crude rigging knot on the bed post or something before tossing the hair-rope over the ledge.
    @Mark: That alternate Rapunzel story is great.

  9. #9 Ty Marx
    May 5, 2009

    I appreciate this post.
    Speaking of the biological needs to grow things quickly, I often have similar thoughts about Wolverine.
    By which I of course mean the amount of caloric energy required to regrow/heal limbs and wounds.
    Wolverine has been shown to, on multiple occasions, withstand massive injuries, practical death, and, on more than one occasion in the comics, being obliterated to nothing but an adamantine skeleton. Following these events, and depending on the series, he would regenerate over the course of a few minutes or hours.
    If the average mass of a human body is 70 kg, and Wolverine can regenerate from near destruction in an hour, how much energy would that take?
    As neither a biologist or a physicist or a biophysicist or a physologist, I have no clue.
    On the same note, what of Cyclops? He must give Wolfie a run for his money on energy expenditure

    I’d have to assume they eat alot of pancakes.

  10. #10 Ian
    May 5, 2009

    You guys needed an etymologist, bnot a biologist.

    Her name wasn’t Rapunzel, it was Rappelzel. She didn’t need a prince charming to rescue her. She grew her hair, tied it at the top and rappeled down, cut off her hair at the bottom and escaped.

    In the same way, the song “Some Day My Prints Will Come” was actually written by someone waiting for their film to be developed and the prints returned by mail….

    Don’t you guys know anything?!

  11. #11 MartinB
    May 6, 2009

    If she really unrolls her hair (from curls), part of the energy will go into rotational energy of the hair-mass, so your force estimate is too high, isn’t it? Otherwise, she could just let it down hand over hand.

    BTW, hair as a tensile strength of roughly 50-100MPa (depending on humidity), about 1/4 or so of mild steel and comparable to wood, so it is not too bad. (Rhinoceros horn is made from the same stuff, keratin.)

  12. #12 -dsr-
    May 6, 2009

    Re: Wolverine.

    At a minimum, 3500 Kcal per pound. Logan is short but thick; the definitive source says he weighs 300 lbs including 105 lbs of adamantium. So, assuming 4 lbs of brain and spinal cord, 191 * 3500 = 668500 Kcal, or about 300 Big Macs with large fries and chocolate shakes.

  13. #13 jim
    April 20, 2010

    Temperature is often subjective.
    The porridge could all have been roughly the same measure temp, but to the subjective tastes of the individual it could easily have been too hot, too cold, or just right.
    and i actually experienced this during lunch today – also, the too hot and too cold individuals were quite finicky.

  14. #14 jim
    April 20, 2010

    the temp comment was meant for the Goldilocks bears… my apologies…
    but for the Rapunzel, if you’re going to get into the Fairy Fhysics then you need to think of the mass of hair as a rope… off the head, that’s too thick a rope, hair is quite strong – without getting into the incremental measures and formulations. The lady is going to be clipping her hair except for the main length of the central back and doing extensions, thus abbreviating the measure of time necessary to grow the hair, etc. etc. etc.
    also:
    if you’ve ever seen a stage performance of Rapunzel, they tend not to roll down or fling the hair. instead the actress lets down her hair, hand over hand feeding it out the window.

  15. #15 jim
    April 20, 2010

    RE: Wolverine.
    you folks seem to be misunderstanding the biology of Mutants.
    a central premise for mutants is that Mitchondrion ATP production for simple calories is something Mutants have transcended. They’re even past Jedi Midi-chlorian – basically force sensitive Mitochondria with perhaps a more effective molecular energy currency than ATP’s proton motive force.
    A Mutant Mitochondrion is a far more efficient form of energy synthesis at time even reaching the upper limit of E=MC^2.

    now that easily explains an energy blast, but the translation to Wolverine regeneration, replenishing lost blood, flesh and muscle burned to ash or seared from his adamantine bones… wow, simple instant mechanics of how do you suddenly add back that much mass. We once again go to E=MC^2, only the Wolverine dynamic is even more advanced than a mere energy projector mutant type. Wolverine must store vast amounts of big E (energy) in his body in a non-volatile state (like calories in Fat), which he converts back to big M (mass), E/C^2=M.
    this allows Wolverine to eat as many Big Macs as he wants, and as needed use the mass as mutant calorie energy, or regenerate it.
    and… have you ever seen Wolverine take a dump? maybe he doesnt need to? his body is just that efficient.

  16. #16 oyun havaları dinle
    June 14, 2010

    if you’ve ever seen a stage performance of Rapunzel, they tend not to roll down or fling the hair. instead the actress lets down her hair, hand over hand feeding it out the window.You guys needed an etymologist, bnot a biologist.

    Her name wasn’t Rapunzel, it was Rappelzel. She didn’t need a prince charming to rescue her. She grew her hair, tied it at the top and rappeled down, cut off her hair at the bottom and escaped.

  17. #17 Sally
    June 22, 2011

    Interesting theory. I am more interested in knowing how much mass or weight the braid could support or lift. Forget it that no one telling the story ever indicated the princess locked in the tower was young. She could have been there all her life, in which case she probably would not have a happy marrage to the prince due to lack of socialization and dysfunctional family. So much for the young beautiful princess who never had a bath and was 100 years old. How much work could that braid do?

  18. #18 Anonymous
    November 29, 2011

    The Mythbusters did once successfully test that a rope made from human hair was sufficiently strong to allow someone to escape from prison.

  19. #19 Guest
    January 15, 2012

    From IMDB about Tangled:

    “Rapunzel’s hair is just long enough to pull Mother Gothel up, but it is also long enough to lower Rapunzel to the ground, which would take twice as much hair. However, it is seen that Rapunzel loops her hair over hooks when pulling Gothel up, which would allow for half the hair to still be in the tower, with just enough lowered to serve Gothel.”

    Also:

    “An average inch of hair weighs about 50 µg – a light estimate, as blonde hair tends to be lighter than other colors. Animators have said that Rapunzel’s hair is approximately 70 feet (840 inches), and consists of about 100,000 strands. That yields 4,200,000,000 µg = 4,200,000 mg = 4,200 g = 4.2 kg (approx 10.4 lbs) of hair. We assume that its manageable weight in the movie is another innate magical property.”

    10 pounds isn’t really that much and she doesn’t ever hang or pull directly by the scalp. I think it’s kinda like when you grab hair at the base of the scalp to brush out the knots. You don’t really feel any pulling, you know?

    Just food for thought.

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