For safety in climbing, a mountaineer uses a nylon rope that is 65 m long and 1.1 cm in diameter. When supporting a 94-kg climber, the rope elongates 1.4 m. Find its Young's modulus.
To find the Young's modulus of the nylon rope, we need to use Hooke's Law, which states that the stress (σ) on a material is directly proportional to the strain (ε) applied to it. Mathematically, this relationship can be expressed as:
σ = E * ε
Where σ is the stress, E is the Young's modulus, and ε is the strain.
In this case, the strain is the elongation of the rope, given as 1.4 m. The stress can be calculated by dividing the weight (force) applied to the rope by the cross-sectional area. The formula for stress is:
σ = F / A
Where F is the force and A is the cross-sectional area.
To calculate the force, we need to find the weight of the climber. The weight (force) can be calculated using the formula:
F = m * g
Where m is the mass of the climber and g is the acceleration due to gravity (which is approximately 9.8 m/s^2).
Let's calculate the force:
m = 94 kg
g = 9.8 m/s^2
F = 94 kg * 9.8 m/s^2
F = 921.2 N
Now, we need to calculate the cross-sectional area of the rope. The formula for the cross-sectional area of a cylinder is:
A = π * r^2
Where A is the cross-sectional area and r is the radius of the rope.
The diameter of the rope is given as 1.1 cm, so the radius can be calculated as:
r = 0.011 m (converting centimeters to meters)
Now, let's calculate the cross-sectional area:
A = π * (0.011 m)^2
A = 0.00038 m^2
Finally, substitute the values of force (F) and cross-sectional area (A) into the formula for stress (σ) and rearrange the equation to solve for the Young's modulus (E):
σ = E * ε
E = σ / ε
Let's calculate the Young's modulus:
σ = F / A
E = (F / A) / ε
E = (921.2 N / 0.00038 m^2) / 1.4 m
E ≈ 1,498,854 N/m^2
Therefore, the Young's modulus of the nylon rope is approximately 1,498,854 N/m^2.
To find the Young's modulus of the nylon rope, we can use Hooke's Law, which states that the stress on a material is directly proportional to the strain it undergoes. The equation is given as:
Stress = Young's modulus x Strain
First, we need to calculate the strain using the given information. The strain is defined as the change in length divided by the original length:
Strain = (Change in length) / (Original length)
Change in length = elongation = 1.4 m
Original length = 65 m
Strain = 1.4 m / 65 m
Now, we can calculate the Young's modulus using the stress-strain relationship:
Stress = Force / Area
Force = weight of the climber = mass x gravity
Mass of the climber = 94 kg
Gravity = 9.8 m/s^2
We can assume that the weight of the climber acts as a force on the rope.
The area of the rope can be calculated using the formula for the area of a circle:
Area = π x (radius)^2
The radius of the rope would be half of its diameter:
Diameter = 1.1 cm
Radius = 1.1 cm / 2 = 0.55 cm = 0.0055 m
Now we can substitute the values into the equations:
Strain = 1.4 m / 65 m ≈ 0.0215
Stress = Force / Area
Force = mass x gravity = 94 kg x 9.8 m/s^2
Area = π x (radius)^2 = 3.14 x (0.0055 m)^2
Now we can rearrange the equation to solve for Young's modulus:
Young's modulus = Stress / Strain
Young's modulus = (Force / Area) / Strain
Substituting the values:
Young's modulus ≈ (94 kg x 9.8 m/s^2) / (3.14 x (0.0055 m)^2) / 0.0215
Calculating this expression gives us the Young's modulus of the nylon rope.