My friends and I were watching Tangled the other day and we got to the scene where Flynn Rider climbs up Rapunzel’s hair. Although her hair is magic, we all couldn’t help but comment on how much that would it. This got me thinking about how strong her hair must be in order to support a full blown adult man. According to IMDB, Rider is 165 lbs (74.8 kg), Rapunzel’s hair is 70 ft (21.3 meters, 840 inches) long, and she has about 100,000 strands. An average inch of hair apparently has a mass of 50 µg (5 * 10-5 g), meaning that her hair has a mass of approximately 4200 g (4.2 kg). To determine if Rapunzel’s hair is strong enough to support Flynn, we must consider the tensile strength of hair, which is how much the hair can support without breaking. According to UCSB, the typical tensile strength of hair is 200 MPa (N/m2) or 200 * 106 Pa. In order to determine how much each hair follicle can support, we must multiply the tensile strength by the cross sectional area of each hair follicle. According to The Physics Factbook, the diameter of a human hair is 25.4 µm (2.54 * 10-5 m), which means the radius is 1.27 * 10-5 m. Given that each hair is a cylinder, the area is 5.07 * 10-10 m2. Therefore, each hair can support a maximum load of about 200 * 106 N/m2 * 5.07 * 10-10 m2 = 0.101 N. If she has around 100,000 strands of hair, this means her entire head of hair can support a load of about 0.101 N * 100,000 = 10134 N. The downward force acting on her hair consists of the force of gravity, which is equal to the mass of Rapuzel’s hair (4.2 kg) plus the mass of Flynn (74.8 kg) times the acceleration due to gravity (9.8 m/s2). This means her hair experiences a downward force of 774.2 N, which is much less than what her hair could handle. Therefore, Rapunzel’s hair can easily allow Flynn to climb up. Other factors may make it hard however, for example, her hair looks really shiny and like it wouldn’t have much friction. Additionally, if Rapunzel were to have Flynn climb up her hair without anything taking the force off of her scalp, her hair would likely rip out and her neck would likely not be able to support the load either. Disney accounts for this to some extent by having her wrap her hair around a metal hook like thing. Ultimately, Rapunzel’s could support way more than just one man climbing up her hair so I’d love to see a sequel where the title of the movie becomes relevant because all of her suitors get tangled in her long hair. Of course, she’d have to grow it back out after Flynn cut it at the end of Tangled.
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