Thursday, September 25, 2014

Rapunzel's hair in the movie "Tangled": Length, mass, and volume

A. What is the length of Rapunzel's hair?

According to Disney's wiki, Rapunzel's hair is about $70\ ft$ or about $21.3\ m$.  Human hair grows at an average rate of $0.15\ m/yr$ or $0.5\ ft/yr$.  By Rapunzel's 18th year, her hair should be only about $18\times 0.15=2.70\ m$ or $18\times 0.5= 6\ ft$, which is far from the $70\ ft$ listed by Disney. We'll, Rapunzel's hair is indeed magical.  Dividing 70 feet by 6 feet, we get 11.  This means Rapunzel's hair is equivalent to that of a woman aged $11\times 18 = 198$ years who also did not cut her hair.  Since the witch Gothel lived hundreds of years, we can guess that about 200 years of the witch's longevity is somehow transferred to Rapunzel's hair, because the witch used to take care of the plant that gave her immortality, and this plant was used to cure Rapunzel's mother:
Once upon a time, a single drop of sunlight falls to Earth and becomes a golden flower filled with incredibly strong healing properties. An old enchantress named Mother Gothel discovers the flower and hoards its healing ability through a song to keep her young for hundreds of years. Centuries later, the pregnant Queen of a nearby kingdom falls deathly sick, leading the King to seek out the legendary mystical plant. Despite Gothel's efforts to hide it, the flower is found, uprooted, and boiled into medicine for the queen, which heals her. The Queen gives birth to a girl, Princess Rapunzel, who has beautiful golden hair that possesses magical healing properties. Gothel, wanting to reclaim her immortality and youth, breaks into the royal nursery to cut a lock of Rapunzel's hair, only to discover that doing so causes it to lose its magic. She kidnaps the Princess instead, raising her as her own in an isolated tower. The King and Queen honor their missing daughter by releasing floating lanterns on the night of her birthday each year in hopes that one day, their Lost Princess will return. (Wikipedia: Tangled)

B. What is the volume of Rapunzel's hair?

Caucasians have an average hair diameter of $70\times 10^{-6} m$ (some say it is 17 to 51 not 70) or a cross sectional area of $3.14\times (35\times 10^{-6})^2 = 3.9\times 10^{-9}\ m^2$, assuming circular cross-section (some hairs have elliptical cross-section).  The volume of Rapunzel's hair strand would be $21.3\times 3.9\times 10^{-9} = 8.3\times 10^{-8}\ m^3 $.  For blondes like Rapunzel, the average number of hairs is 146,000.  This means that the total volume of hair is $146,000\times 8.3\times 10^{-8} = 1.2\times 10^{-2} = 0.012 m^3$.  This is approximately equivalent to a cube with sides 0.23 m long or 3/4 feet or 9 inches, assuming a dense circle-packing density (which is 0.9 but we can round it off to 1).  Thus, a braided hair would more or less fit this cube, as seen in the movie.

C. What is the weight of Rapunzel's hair?

The relative density of human hair is 1.268 to 1.304, depending on relative humidity (RH).  Let us choose 1.3.  This means that density human hair is 1.3 times the density of water which is $1,000\ kg/m^3$.  Since the total volume of Rapunzel's hair is $0.012 m^3$, then the total mass of Rapunzel's hair is $1.3\times 10^3\times 1.2\times 10^{-2}=15.6\ kg$ or $34.4\ lbs$.  In comparison, the weight of the Imperial State Crown of the United Kingdom that Queen Elizabeth II wore during her coronation is $0.91\ kg$ or $2.0\ lbs$.  Thus, Rapunzel's hair is $15.6/0.91 = 17.1 $ times heavier that the Imperial State Crown.
Tangled DVD
Tangled DVD
Tangled
Tangled
The Art of Tangled
The Art of Tangled
Disney Tangled Exclusive 7 Piece Deluxe Mini PVC Figurine Set Rapunzel, Flynn, Maximus, Toddler,Rapunzel, Mother Gothel, Hookhand Thug Pascal
Disney Tangled Exclusive 7 Piece Deluxe Mini PVC Figurine Set Rapunzel, Flynn, Maximus, Toddler,Rapunzel, Mother Gothel, Hookhand Thug Pascal