Final answer:
The stretch of a 0.800-cm diameter nylon rope with a 65.0-kg climber can be calculated using the stress-strain relationship and Hooke's Law, but exact numbers aren't possible without specific material properties like Young's modulus. The observed stretch of nylon won't be extensive. However, bungee cords, designed for elasticity, will stretch significantly more.
Step-by-step explanation:
Stretching of Nylon Rope by a Mountain Climber
To determine by how much a 65.0-kg mountain climber stretches her 0.800-cm diameter nylon rope, we must consider the physical properties of nylon as well as the force exerted by the climber due to gravity. This problem involves concepts from classical mechanics, specifically the stress-strain relationship in materials and the force of gravity.
(a) The extension of the rope depends on the climber's weight, which exerts a force due to gravity (65.0 kg × 9.81 m/s²), and the rope's elastic properties, described by its Young's modulus. Applying Hooke's Law and the definition of stress and strain to these values, one can calculate the elongation. However, without the Young's modulus of nylon, an exact number cannot be provided.
(b) The observed stretch of a nylon rope would not be substantial due to its high tensile strength and relatively low elasticity, which is why the answer should not indicate a large extension. For a bungee cord, which is made to be highly elastic, the expected stretch would be much greater, to allow for a safe deceleration during a bungee jump.
(c) Using twice the length of nylon rope can affect the extension differently, as doubling the length of the rope could potentially double the stretch, but this also depends on other factors such as rope thickness and its tensile properties.
(b) If the climber free-falls for 2.00 m before the rope takes up the slack, the stretch can be calculated using conservation of energy, where the potential energy lost during the fall equals the elastic potential energy stored in the stretched rope. The calculation would require the spring constant or Young's modulus of the rope.