For safety in climbing, a mountaineer uses a nylon rope that is 65 m long and 0.9 cm in diameter. When supporting a 92-kg climber, the rope elongates 2.0 m. Find its Young's modulus.
To find the Young's modulus of the nylon rope, we can use Hooke's Law, which states that the extension (elongation) of an elastic material is directly proportional to the applied force. The formula is given as:
Stress (σ) = Young's modulus (E) * Strain (ε)
Where:
Stress (σ) = Force (F) / Area (A)
Strain (ε) = Change in length (ΔL) / Original length (L₀)
In this case, the change in length (ΔL) is given as 2.0 m, and the original length (L₀) is 65 m. The diameter of the rope is given as 0.9 cm, so the radius (r) can be calculated as follows:
radius (r) = diameter / 2
= 0.9 cm / 2
= 0.45 cm
Now, let's convert the radius to meters:
radius (r) = 0.45 cm * (1 m / 100 cm)
= 0.0045 m
Next, we can calculate the area (A) of the rope:
Area (A) = π * r²
= 3.14 * (0.0045 m)²
= 3.14 * 0.00002025 m²
= 0.000063585 m²
Now, we can substitute the known values into the Stress formula and solve for the Young's modulus:
Stress (σ) = Force (F) / Area (A)
Force (F) = mass (m) * gravitational acceleration (g)
= 92 kg * 9.8 m/s²
= 901.6 N
Stress (σ) = 901.6 N / 0.000063585 m²
Strain (ε) = 2.0 m / 65 m
Therefore, Young's modulus (E) can be calculated as follows:
Stress (σ) = Young's modulus (E) * Strain (ε)
901.6 N / 0.000063585 m² = E * (2.0 m / 65 m)
To solve for E, we rearrange the equation:
E = (901.6 N / 0.000063585 m²) * (65 m / 2.0 m)
E = 901.6 N * 65 m / (0.000063585 m² * 2.0 m)
Now, let's calculate the value of Young's modulus (E).
To find the Young's modulus of the nylon rope, we need to use Hooke's Law, which states that the strain (elongation or compression) of an object is directly proportional to the force applied to it.
Hooke's Law equation:
Stress = Young's modulus × Strain
First, we need to find the strain. Strain is defined as the change in length divided by the original length.
Strain (ε) = Change in length / Original length
In this case, the change in length is given as 2.0 m, and the original length of the rope is 65 m.
ε = 2.0 m / 65 m
Next, we can calculate the stress. Stress is defined as the force applied divided by the cross-sectional area of the object.
Stress (σ) = Force / Area
The force applied is the weight of the climber, which can be calculated by multiplying the mass (92 kg) by the acceleration due to gravity (9.8 m/s^2).
Force = Mass × Acceleration due to gravity
Force = 92 kg × 9.8 m/s^2
Now, let's find the cross-sectional area of the rope. The cross-sectional area of a cylindrical object can be calculated using the formula:
Area = π × (radius)^2
The diameter of the rope is given as 0.9 cm, so the radius would be half of that, which is 0.9 cm / 2 = 0.45 cm. However, it is more convenient to work with meters, so we need to convert the radius to meters.
Radius = 0.45 cm / 100 cm/m
Now we have all the values needed to calculate the stress.
Substitute the values into the stress equation:
Stress = (92 kg × 9.8 m/s^2) / (π × (0.45 cm / 100 cm/m)^2)
Now, we can substitute the calculated stress and the strain value into Hooke's Law equation and solve for Young's modulus.
Young's modulus = Stress / Strain
Young's modulus = (Stress) / (Strain)
Young's modulus = ((92 kg × 9.8 m/s^2) / (π × (0.45 cm / 100 cm/m)^2)) / (2.0 m / 65 m)
By plugging in the values and calculating, you can find the Young's modulus of the nylon rope.