Stress on a Mountaineer's Rope. A nylon rope used by mountaineers elongates 1.10 m under the weight of a 65.0-kg climber. If the rope is 45.0 m in length and 7.0 mm in diameter, what is Young's modulus for nylon?

Verified step by step guidance

1. Calculate the stress on the rope. Stress (\(\sigma\)) is defined as the force per unit area. Use the formula \(\sigma = \frac{F}{A}\), where \(F\) is the force applied (weight of the climber) and \(A\) is the cross-sectional area of the rope. The force can be calculated using \(F = mg\), where \(m\) is the mass of the climber and \(g\) is the acceleration due to gravity (approximately 9.8 m/s^2). The cross-sectional area of the rope can be calculated using the formula for the area of a circle, \(A = \pi r^2\), where \(r\) is the radius of the rope.
2. Calculate the strain on the rope. Strain (\(\epsilon\)) is defined as the change in length per original length. Use the formula \(\epsilon = \frac{\Delta L}{L_0}\), where \(\Delta L\) is the change in length of the rope (elongation) and \(L_0\) is the original length of the rope.
3. Calculate Young's modulus for nylon. Young's modulus (\(E\)) is a measure of the stiffness of a material and is defined as the ratio of stress to strain. Use the formula \(E = \frac{\sigma}{\epsilon}\).
4. Substitute the values calculated for stress and strain into the formula for Young's modulus to find the value of \(E\).
5. Ensure all units are consistent when performing calculations (e.g., convert diameters to radii, convert mm to meters, etc.) to avoid errors in the final result.

Key Concepts

Young's Modulus

Young's modulus is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain. It quantifies how much a material will deform under a given load, providing insight into its elastic properties. A higher Young's modulus indicates a stiffer material that deforms less under stress.

Tensile Stress and Strain

Tensile stress is the force applied per unit area on a material, while tensile strain is the ratio of the change in length to the original length of the material. These concepts are crucial for understanding how materials respond to stretching forces, and they are foundational in calculating Young's modulus.

Elastic Limit

The elastic limit is the maximum amount of stress that a material can withstand without undergoing permanent deformation. Beyond this point, the material may not return to its original shape. Understanding the elastic limit is essential for applications like mountaineering, where safety depends on the material's ability to return to its original state after being stretched.