For safety in climbing, a mountaineer uses a nylon rope that is 55 m long and 0.9 cm in diameter. When supporting a 82-kg climber, the rope elongates 2.0 m. Find its Young's modulus.
I got 34737.433 but that is not the right answer please help!
To find the Young's modulus of the nylon rope, we can use the formula:
Young's modulus (Y) = (F/A) / (ΔL/L0),
where:
F is the force applied to the rope,
A is the cross-sectional area of the rope,
ΔL is the change in length of the rope,
L0 is the original length of the rope (55 m in this case).
First, let's find the force applied to the rope by the climber using the formula:
Force (F) = mass × gravity,
where
mass = 82 kg (given)
gravity = 9.8 m/s².
F = 82 kg × 9.8 m/s² = 803.6 N.
Next, let's calculate the area from the diameter. The diameter is given as 0.9 cm, so the radius (r) can be found by dividing it by 2 and converting it to meters:
r = 0.9 cm / 2 = 0.45 cm = 0.0045 m.
Now, we can calculate the cross-sectional area (A) of the rope using the formula for the area of a circle:
A = πr² = π(0.0045 m)².
A ≈ 0.000063617 m².
Next, we can calculate the change in length of the rope (ΔL) using the given elongation:
ΔL = 2.0 m.
Finally, we can substitute all the values into the Young's modulus formula:
Y = (F/A) / (ΔL/L0) = (803.6 N / 0.000063617 m²) / (2.0 m / 55 m).
Y ≈ 3.61 × 10^9 N/m².
Therefore, the Young's modulus of the nylon rope is approximately 3.61 × 10^9 N/m².
To find the Young's modulus of the nylon rope, we can use the equation:
Stress (σ) = (Force (F) / Area (A))
The force in the equation is the weight of the climber, which can be calculated using the formula:
Force = mass * gravity
Considering the given mass of the climber is 82 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the force:
Force = 82 kg * 9.8 m/s² ≈ 803.6 N
To find the area (A) of the rope, we need to use the formula:
Area = π * (diameter / 2)²
Given that the diameter of the rope is 0.9 cm, we first need to convert it to meters:
Diameter = 0.9 cm = 0.9 cm * 0.01 m/cm = 0.009 m
Now we can calculate the area:
Area = π * (0.009 m / 2)² ≈ 6.364 × 10^(-5) m²
Next, we can calculate the stress:
Stress (σ) = Force (F) / Area (A) = 803.6 N / 6.364 × 10^(-5) m² ≈ 1.26 × 10^7 Pa
The stress is the force per unit area applied to the rope. To find the strain, we can use the formula:
Strain (ε) = Change in length (ΔL) / Original length (L)
In this case, the change in length is given as 2.0 m, and the original length (L) of the rope is given as 55 m. Thus, we can calculate the strain:
Strain (ε) = 2.0 m / 55 m ≈ 0.0364
Finally, we can calculate the Young's modulus (E) using the equation:
Young's modulus (E) = Stress (σ) / Strain (ε)
Young's modulus (E) = (1.26 × 10^7 Pa) / 0.0364 ≈ 3.46 × 10^8 Pa
So, the correct value for the Young's modulus of the nylon rope is approximately 3.46 × 10^8 Pa.