Posted by Christina on .
In the Grimms' fairy tale Rapunzel, she lets down her golden hair to a length of 20.0 yards (we'll use 20.0 m, which is not much different) so that the prince can climb up to her room. Human hair has a Young's modulus of about 490 MPa, and we can assume that Rapunzel's hair can be squeezed into a rope about 2.00 cm in cross-sectional diameter. The prince is described as young and handsome, so we can estimate a mass of 60.0 kg for him.
Just after the prince has started to climb at constant speed, while he is still near the bottom of the hair, by how many centimeters does he stretch Rapunzel's hair?
What is the mass of the heaviest prince that could climb up, given that the maximum tensile stress hair can support is 196 MPa? (Assume that Hooke's law holds up to the breaking point of the hair, even though that would not actually be the case.)
(delta L)/L = (stress)/E
Stress = 4 M g /(pi d^2)
E = 490*10^6 N/m^2
M = 60 kg
d = 0.0200 m)
L = 20 m
g = 9.8 m/s^2
Solve for delta L
For the maximum load question, set
4 Mg/(pi d^2)= max stress = 196*10^6 N/m^2
Since a single hair that is tugged on is often more likely to be pulled out than break, it is quite likely that the overweight prince will pull all her hair out rather than break it.