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Chapter 7: Problem 7

Which of the following are correct? (1) The shear modulus of a liquid is infinite. (2) Bulk modulus of a perfectly rigid body is infinity. (3) According to Hooke's law, the ratio of the stress and strain remains constant. (4) None of the above.

Short Answer

Expert verified
Statements (2) and (3) are correct. The shear modulus of a liquid is zero.

Step by step solution

01

Understanding Shear Modulus for Liquids

The shear modulus is a measure of a material's ability to resist shear deformation. Liquids cannot sustain shear forces without flowing; hence, they do not have a well-defined shear modulus. Therefore, the shear modulus of a liquid is not infinite; it is, in fact, zero.
02

Evaluating Bulk Modulus for Perfectly Rigid Bodies

The bulk modulus of a material measures the material's resistance to uniform compression. In a perfectly rigid body, there is no compressibility, making it incapable of any volume change. Thus, the bulk modulus in this ideal case is infinite.
03

Analyzing Hooke's Law

Hooke's law states that, within the elastic limit of a material, the stress is directly proportional to strain. Therefore, the ratio of stress to strain, also known as the modulus of elasticity, remains constant for a given material's elastic region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Shear Modulus
The shear modulus, denoted by \( G \), measures a material's ability to resist shape changes at a constant volume. It defines how a material responds to shear stress, where forces are applied parallel to a surface causing sliding.

Shear modulus is crucial in understanding different phases of matter, especially solids versus liquids. Solids resist shear deformation and have a defined shear modulus.
  • Liquids, however, cannot resist shear forces. They flow instead of resisting deformation.
  • As a consequence, the shear modulus of a liquid is essentially zero, not infinite or even defined, since liquids do not retain shape under shear stress.
This property distinguishes fluids from solids, emphasizing the variability of elastic properties among different states of matter.
Bulk Modulus
The bulk modulus, represented by \( K \), quantifies a material's resistance to uniform compression. It is pivotal in illustrating how substances react to changes in pressure.

When a substance is subject to pressure, its volume decreases. The bulk modulus helps in understanding this resistance to volume change.
  • A perfectly rigid body exhibits no volume change regardless of pressure applied. Thus, its capacity to resist compression is infinite.
  • Consequently, the bulk modulus of such an ideal rigid body is infinite, showcasing its total incompressibility.
Real-world materials generally aren't perfectly rigid, but this concept helps in modeling and understanding material behaviors under pressure.
Hooke's Law
Hooke's Law characterizes the elastic properties of materials, primarily focusing on the relationship between stress and strain. Stress is the force applied to a material, while strain is the deformation experienced.

Hooke's Law states that within the elastic limit, stress \( \sigma \) and strain \( \epsilon \) are directly proportional. Mathematically, this relationship is expressed as \( \sigma = E \times \epsilon \), where \( E \) is the modulus of elasticity.
  • This implies that the ratio \( \frac{\sigma}{\epsilon} \) remains constant as long as the material does not exceed the elastic limit.
  • This constancy maintains the predictability of material behavior under stress, crucial in engineering and material science.
Hooke's Law is foundational for materials that return to their original shape post deformation, provided the limits aren't surpassed, ensuring reliable applications in structural analysis.

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Most popular questions from this chapter

A cube is shifted to a depth of \(100 \mathrm{~m}\) is a lake. The change in volume is \(0.1 \%\). The bulk modulus of the material is nearly (1) \(10 \mathrm{~Pa}\) (2) \(10^{4} \mathrm{~Pa}\) (3) \(10^{7} \mathrm{~Pa}\) (4) \(10^{9} \mathrm{~Pa}\)

A piece of copper wire has twice the radius of a piece of steel wire. Young's modulus for steel is twice that of the copper. One end of the copper wire is joined to one end of the steel wire so that both can be subjected to the same longitudinal force. By what fraction of its length will the steel have stretched when the length of the copper has increased by \(1 \%\) ? (1) \(1 \%\) (2) \(2 \%\) (3) \(2.5 \%\) (4) \(3 \%\)

A copper bar of length \(L\) and area of cross section \(A\) is placed in a chamber at atmospheric pressure. If the chamber is evacuated, the percentage change in its volume will be (compressibility of copper is \(8 \times 10^{-12} \mathrm{~m}^{2} / \mathrm{N}\) and \(\left.1 \mathrm{~atm}=10^{5} \mathrm{~N} / \mathrm{m}^{2}\right)\) (1) \(8 \times 10^{-7} \%\) (2) \(8 \times 10^{-5} \%\) (3) \(1.25 \times 10^{-4} \%\) (4) \(1.25 \times 10^{-5} \%\)

A copper wire and a steen wire of the same cross-sectional grea and length are joined end-to-end (at one end). Equal gind opposite longitudinal forces are applied to the free ends giving a total elongation of \(l .\) Then the two wires will have (l) same stress and same strain (2) same stress and different strains (3) different stresses and same strain (4) different stresses and different strains

A wire is stretched \(1 \mathrm{~mm}\) by a force of \(1 \mathrm{kN}\). How far would a wire of the same material and length but of four times that diameter be stretched by the same force? (1) \(\frac{1}{2} \mathrm{~mm}\) (2) \(\frac{1}{4} \mathrm{~mm}\) (3) \(\frac{1}{8} \mathrm{~mm}\) (4) \(\frac{1}{16} \mathrm{~mm}\)
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