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.
To find the answer to the first question, we need to determine the elongation of Rapunzel's hair when the prince starts climbing at a constant speed.
The elongation can be found using Hooke's Law, which states that the stress on a material is directly proportional to the strain it experiences. The equation for Hooke's Law is:
Stress = Young's Modulus * Strain
We know the stress that the hair can support, which is 196 MPa. We are looking for the strain, which can be calculated using the formula:
Strain = Elongation / Original Length
We can rearrange this equation to solve for the elongation:
Elongation = Strain * Original Length
We know the original length of the hair is 20.0 meters. We need to find the strain, which is the change in length divided by the original length. Since the prince is climbing at a constant speed, the hair is experiencing a constant force, which creates a constant stress. This means the strain is also constant.
So, to find the elongation, we can calculate:
Elongation = Strain * Original Length = Stress / Young's Modulus * Original Length
Substituting in the given values:
Elongation = (196 MPa / 490 MPa) * 20.0 m
Now, we can calculate the elongation:
Elongation = (0.4) * 20.0 m = 8.0 m
Therefore, the prince stretches Rapunzel's hair by 8.0 meters.
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To find the answer to the second question, we need to determine the maximum mass of a prince that could climb up without exceeding the hair's maximum tensile stress.
The tensile stress on the hair can be calculated using the formula:
Tensile Stress = Force / Cross-sectional Area
We know the tensile stress the hair can support is 196 MPa. We are looking for the maximum mass, so we can calculate the Force using:
Force = Mass * Acceleration due to gravity
The cross-sectional area can be calculated using the formula for the area of a circle:
Cross-sectional Area = π * (Diameter/2)^2
Substituting these equations into the equation for tensile stress:
Tensile Stress = (Mass * Acceleration due to gravity) / (π * (Diameter/2)^2)
196 MPa = (Mass * 9.8 m/s^2) / (π * (0.02 m/2)^2)
We can rearrange this equation to solve for the mass:
Mass = (Tensile Stress * π * (Diameter/2)^2) / (Acceleration due to gravity)
Substituting in the given values:
Mass = (196 MPa * π * (0.02 m/2)^2) / (9.8 m/s^2)
Now, we can calculate the maximum mass:
Mass = (196 * 10^6 N/m^2 * 3.14 * (0.01 m)^2) / (9.8 m/s^2)
Mass = 1.018 kg
Therefore, the mass of the heaviest prince that could climb up without exceeding the hair's maximum tensile stress is approximately 1.018 kg. Keep in mind that this assumes Hooke's Law holds up to the breaking point of the hair, which is not actually the case.