According to one set of measurements, the tensile strength of hair is 196 MPa, which produces a maximum strain of 0.400 in the hair. The thickness of hair varies considerably, but let's use a diameter of 50.0 u(mu)m . What is the magnitude of the force giving this tensile stress? If the length of a strand of the hair is 12.0 cm at its breaking point, what was its unstressed length?
To find the magnitude of the force giving this tensile stress, we can use Hooke's law, which states that stress is proportional to strain. The formula for stress is given by:
Stress = Force / Area
Given that the tensile strength of hair is 196 MPa (which is equivalent to 196 × 10^6 Pa), and the maximum strain is 0.400, we can calculate the force using the equation:
Stress = Force / Area
Rearranging the equation, we have:
Force = Stress × Area
To calculate the area, we need to use the thickness of the hair, which is given as a diameter of 50.0 µm. The radius can be found by dividing the diameter by 2:
Radius = Diameter / 2
Radius = 50.0 µm / 2 = 25.0 µm
Now, we can calculate the area of the hair using the formula for the area of a circle:
Area = π × Radius^2
Area = π × (25.0 µm)^2
Area ≈ 1963.50 µm^2
Converting the area to square meters, we get:
Area = 1963.50 × 10^-12 m^2
Substituting the value into the force equation, we have:
Force = 196 × 10^6 Pa × 1963.50 × 10^-12 m^2
Force ≈ 384.07 × 10^-6 N
Therefore, the magnitude of the force giving this tensile stress is approximately 0.384 N.
Now, let's determine the unstressed length of the hair.
The strain is given as 0.400, which is the change in length (ΔL) divided by the original length (L0):
Strain = ΔL / L0
Rearranging the equation, we can solve for the unstressed length (L0) using the known values:
0.400 = (12.0 cm - L0) / L0
Cross-multiplying, we have:
0.400L0 = 12.0 cm - L0
1.400L0 = 12.0 cm
L0 = 12.0 cm / 1.400
L0 ≈ 8.57 cm
Therefore, the unstressed length of the hair is approximately 8.57 cm.