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Ch 11: Equilibrium & Elasticity
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 11: Equilibrium & ElasticityProblem 30
Chapter 11, Problem 30

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?

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First, understand the concept of Young's modulus, which is a measure of the stiffness of a material. It is defined as the ratio of stress (force per unit area) to strain (relative deformation) in the linear elastic region of the material.
Calculate the cross-sectional area of the rope. The diameter is given as 7.0 mm, which needs to be converted to meters (7.0 mm = 0.007 m). The cross-sectional area A of the rope can be calculated using the formula for the area of a circle: A = π * (d/2)^2, where d is the diameter.
Determine the force exerted by the climber on the rope. This force is equal to the weight of the climber, which can be calculated using the formula F = m * g, where m is the mass of the climber (65.0 kg) and g is the acceleration due to gravity (approximately 9.81 m/s²).
Calculate the stress on the rope using the formula: stress = F / A, where F is the force exerted by the climber and A is the cross-sectional area of the rope.
Calculate the strain on the rope using the formula: strain = ΔL / L, where ΔL is the change in length of the rope (1.10 m) and L is the original length of the rope (45.0 m). Finally, use the formula for Young's modulus: E = stress / strain, to find the Young's modulus for nylon.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Young's Modulus

Young's modulus is a measure of the stiffness of a material. It is defined as the ratio of tensile stress to tensile strain in the linear elasticity regime of a uniaxial deformation. It is a fundamental property that indicates how much a material will deform under a given load, and is expressed in pascals (Pa).
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Stress and Strain

Stress is the force applied per unit area within materials and is measured in pascals (Pa). Strain is the deformation or displacement of material that results from an applied stress, expressed as a dimensionless ratio of change in length to original length. Understanding these concepts is crucial for calculating Young's modulus, as it involves the relationship between stress and strain.
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Elastic Deformation

Elastic deformation refers to the reversible change in shape of a material under stress. When the stress is removed, the material returns to its original shape. This concept is important in the context of Young's modulus, as it describes the behavior of materials within the elastic limit, where the stress-strain relationship is linear and reversible.
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Related Practice
Textbook Question

A solid gold bar is pulled up from the hold of the sunken RMS Titanic. The bulk modulus of lead is one-fourth that of gold. Find the ratio of the volume change of a solid lead bar to that of a gold bar of equal volume for the same pressure change.

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Textbook Question

Two circular rods, one steel and the other copper, are joined end to end. Each rod is 0.750 m long and 1.50 cm in diameter. The combination is subjected to a tensile force with magnitude 4000 N. For each rod, what are (a) the strain and (b) the elongation?

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Textbook Question

In the Challenger Deep of the Marianas Trench, the depth of seawater is 10.9 km and the pressure is 1.16×108 Pa (about 1.15×103 atm). If a cubic meter of water is taken from the surface to this depth, what is the change in its volume? (Normal atmospheric pressure is about 1.0×105 Pa. Assume that k for seawater is the same as the freshwater value given in Table 11.2.)

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Textbook Question

A nonuniform beam 4.50 m long and weighing 1.40 kN makes an angle of 25.0° below the horizontal. It is held in position by a frictionless pivot at its upper right end and by a cable 3.00 m farther down the beam and perpendicular to it (Fig. E11.20). The center of gravity of the beam is 2.00 m down the beam from the pivot. Lighting equipment exerts a 5.00-kN downward force on the lower left end of the beam. Find the tension T in the cable and the horizontal and vertical components of the force exerted on the beam by the pivot. Start by sketching a free-body diagram of the beam.

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Textbook Question

A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 700 N is applied to each end of the wire. What minimum diameter is required for the wire?

Textbook Question

A specimen of oil having an initial volume of 600 cm3 is subjected to a pressure increase of 3.6×106 Pa, and the volume is found to decrease by 0.45 cm3. What is the bulk modulus of the material and the compressibility?

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