Some fishing line is made out of nylon with Y =3.0!109Pa and breaking stress
7.7!107
Pa . It is 5.0 m long, you are holding one end, and it hangs down vertically.
(a) When a 3.0-kg fish is attached to the bottom end, the fishing line elongates by 1.0 cm. What is its cross sectional area? What is its diameter (assuming the cross section is circular)?
(b) What is the largest mass that can be supported by the fishing line before it
breaks?
Indicate your specific subject in the "School Subject" box, so those with expertise in the area will respond to the question.
(a) To find the cross sectional area and diameter of the fishing line, we can use the stress-strain relationship.
The stress (σ) is given by the equation:
σ = F/A
Where:
σ is the stress,
F is the force applied,
A is the cross sectional area.
We are given the breaking stress (σ_b = 7.7 * 10^7 Pa) and the elongation (ΔL = 1.0 cm = 0.01 m). We can calculate the force using the weight of the fish:
F = mg
Where:
m is the mass of the fish,
g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values, we have:
F = 3.0 kg * 9.8 m/s^2 = 29.4 N
Now, we can calculate the cross sectional area:
σ = F/A
A = F/σ
A = 29.4 N / 7.7 * 10^7 Pa = 3.81 * 10^-7 m^2
To find the diameter, we can use the formula for the area of a circle:
A = πr^2
r = √(A/π)
r = √(3.81 * 10^-7 m^2 / π) = 8.78 * 10^-5 m
Finally, we can calculate the diameter:
d = 2r = 2 * 8.78 * 10^-5 m = 1.76 * 10^-4 m
So, the cross sectional area of the fishing line is 3.81 * 10^-7 m^2 and the diameter is 1.76 * 10^-4 m.
(b) To find the largest mass that can be supported by the fishing line before it breaks, we need to consider the breaking stress of the material.
The maximum force that can be applied to the fishing line without breaking is given by:
F_max = σ_b * A
Substituting the values, we have:
F_max = 7.7 * 10^7 Pa * 3.81 * 10^-7 m^2 = 29.357 N
To find the mass that corresponds to this force, we use the equation:
F_max = mg
m = F_max / g = 29.357 N / 9.8 m/s^2 = 2.999 kg
Therefore, the largest mass that can be supported by the fishing line before it breaks is approximately 2.999 kg.