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Chapter 7: Problem 946

What is the relationship between Young's modulus Y, Bulk modulus \(\mathrm{k}\) and modulus of rigidity \(\eta\) ? (A) \(\mathrm{Y}=[9 \eta \mathrm{k} /(\eta+3 \mathrm{k})]\) (B) \(\mathrm{Y}=[9 \mathrm{Yk} /(\mathrm{y}+3 \mathrm{k})]\) (C) \(\mathrm{Y}=[9 \eta \mathrm{k} /(3+\mathrm{k})]\) (D) \(\mathrm{Y}=[3 \eta \mathrm{k} /(9 \eta+\mathrm{k})]\)

Short Answer

Expert verified
The correct relationship between Young's modulus (Y), Bulk modulus (k), and modulus of rigidity (畏) is: \[Y = \frac{9畏k}{畏+3k}\] Hence, the correct answer is (A).

Step by step solution

01

Understand the relation between the moduli

The relationship between Young's modulus (Y), Bulk modulus (k) and modulus of rigidity (畏) can be expressed as follows: \[Y = \frac{9畏k}{3k + 畏}\] Step 2 - Identify the correct formula
02

Identify the correct formula

Now, we will check each option, one by one and compare it to the established relationship: (A) Y = [9畏k /(畏+3k)] Comparing to the correct formula, this option matches the established relationship. (B) Y=[9Yk /(y+3k)] This option contains incorrect terms and doesn't match the correct relationship. (C) Y=[9畏k /(3+k)] This option does not match the established relationship due to wrong placement of the terms in the denominator. (D) Y=[3畏k /(9畏+k)] This option also has wrong coefficients and doesn't match the correct relation. So, the correct answer is (A). Y = [9畏k /(畏+3k)]

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Key Concepts

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

Young's Modulus
Young's Modulus is a fundamental concept in materials science and physics. It measures a material's ability to withstand changes in length under lengthwise tension or compression. Essentially, it describes how stiff or elastic a material is. The higher the Young's Modulus, the stiffer the material.To determine Young's Modulus mathematically, you use the formula:\[ Y = \frac{\text{Stress}}{\text{Strain}} \]Where stress is the force applied per unit area and strain is the deformation experienced by the material. Typically, it's measured in Pascals (Pa).
  • Stress is the force acting per unit area.
  • Strain is the relative change in shape or size.
Young's Modulus plays a critical role in engineering and construction, as it helps in selecting the right materials that ensure structures remain strong yet flexible enough to handle stressors.
Bulk Modulus
Bulk Modulus is another key measurement of a material's elasticity. Unlike Young's Modulus which focuses on stretching and compression along one dimension, Bulk Modulus considers the material's response to changes in pressure in all directions - essentially volumetric compression.The Bulk Modulus (\( k \)) is defined as:\[ k = -V \cdot \frac{\Delta P}{\Delta V} \]Where:
  • \( V \) is the initial volume of the object.
  • \( \Delta P \) is the change in pressure.
  • \( \Delta V \) is the change in volume.
A higher Bulk Modulus indicates that a material is incompressible or less susceptible to pressure changes. In applications, this concept is vital for materials used in high-pressure environments, such as the hull of a submarine.
Modulus of Rigidity
The Modulus of Rigidity, also known as the Shear Modulus (\( \eta \)), is an essential property describing a material's response to shear stress (think twisting or shearing).The formula to find the Modulus of Rigidity is:\[ \eta = \frac{\text{Shear Stress}}{\text{Shear Strain}} \]Where:
  • Shear Stress is the force causing the layers of the material to slide past each other.
  • Shear Strain is the deformation perpendicularly to the force's direction.
A material with a high Modulus of Rigidity will not easily change shape when a shear force is applied.
This property is critical when dealing with materials subjected to twisting forces, as in shafts or beams bearing torque.

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Most popular questions from this chapter

Melting point of ice (A) Increases with increasing pressure (B) Decreases with increasing pressure (C) Is independent of pressure (D) is proportional of pressure

When liquid medicine of density \(\mathrm{S}\) is to be put in the eye. It is done with the help of a dropper as the bulb on the top of the dropper is pressed a drop forms at the opening of the dropper we wish to estimate the size of the drop. We dirst assume that the drop formed at the opening is spherical because the requires a minimum increase in its surface energy. To determine the size we calculate the net vertical force due to surface tension \(\mathrm{T}\) when the radius of the drop is \(\mathrm{R}\). When this force becomes smaller than the weight of the drop the drop gets detached from the dropper. If \(\mathrm{r}=5 \times 10^{-4} \mathrm{~m}, \mathrm{p}=10^{3} \mathrm{~kg} \mathrm{~m}^{-3}=10 \mathrm{~ms}^{-2} \mathrm{~T}=0.11 \mathrm{~N} \mathrm{~m}^{-1}\) the radius of the drop when it detaches from the dropper is approximately (A) \(1.4 \times 10^{-3} \mathrm{~m}\) (B) \(3.3 \times 10^{-3} \mathrm{~m}\) (C) \(2.0 \times 10^{-3} \mathrm{~m}\) (D) \(4.1 \times 10^{-3} \mathrm{~m}\)

By sucking through a straw, a student can reduce the pressure in his lungs to \(750 \mathrm{~mm}\) of \(\mathrm{Hg}\) (density \(\left.=13.6\left(\mathrm{gm} / \mathrm{cm}^{2}\right)\right)\) using the straw, he can drink water from \(\mathrm{a}\) glass up to a maximum depth of (A) \(10 \mathrm{~cm}\) (B) \(75 \mathrm{~cm}\) (C) \(13.6 \mathrm{~cm}\) (D) \(1.36 \mathrm{~cm}\)

The surface tension of a liquid is \(5 \mathrm{~N} / \mathrm{m}\). If a thin film of the area \(0.02 \mathrm{~m}^{2}\) is formed on a loop, then its surface energy will be (A) \(5 \times 10^{-2} \mathrm{~J}\) (B) \(2.5 \times 10^{-2} \mathrm{~J}\) (C) \(2 \times 10^{-1} \mathrm{~J}\) (D) \(5 \times 10^{-1} \mathrm{~J}\)

The relation between surface tension T. Surface area \(\mathrm{A}\) and surface energy \(\mathrm{E}\) is given by. (A) \(\mathrm{T}=(\mathrm{E} / \mathrm{A})\) (B) \(\mathrm{T}=\mathrm{EA}\) (C) \(\mathrm{E}=(\mathrm{T} / \mathrm{A})\) (D) \(\mathrm{T}=(\mathrm{A} / \mathrm{E})\)
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