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Chapter 9: Problem 3

Column-I \(\quad\) Column-II (a) Ratio of longitudinal or tensile stress to longitudinal strain (p) Moldulus of Rigidly (b) Ratio of normal or hydrostatic stress to volumetric strain (q) Poisson's ratio (c) Ratio of lateral surain to longitudinal strain (r) Bulk modulus (d) Ratio of tangential stress to shear strain (s) Young's modulus

Short Answer

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(a) 鈥� (s), (b) 鈥� (r), (c) 鈥� (q), (d) 鈥� (p)

Step by step solution

01

Identify Definitions

Identify the definitions associated with each quantity provided in Column-I. - The ratio of longitudinal or tensile stress to longitudinal strain is known as Young's Modulus. - The ratio of normal or hydrostatic stress to volumetric strain is called Bulk Modulus. - The ratio of lateral strain to longitudinal strain is known as Poisson's Ratio. - The ratio of tangential stress to shear strain is called Modulus of Rigidity.
02

Match Each Term

Match each term from Column-I with the correct physical quantity from Column-II based on definitions given in Step 1. - (a) Ratio of longitudinal or tensile stress to longitudinal strain corresponds to (s) Young's modulus. - (b) Ratio of normal or hydrostatic stress to volumetric strain corresponds to (r) Bulk modulus. - (c) Ratio of lateral strain to longitudinal strain corresponds to (q) Poisson's ratio. - (d) Ratio of tangential stress to shear strain corresponds to (p) Moldulus of Rigidly.

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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 the study of the mechanical properties of solids. It is named after the British scientist Thomas Young. This modulus measures the ability of a material to withstand changes in length when subjected to tensile (stretching) or compressive (squeezing) forces. The formula for Young's modulus is expressed as:\[ E = \frac{\sigma}{\varepsilon} \]where:
  • \( E \) is the Young's modulus of the material,
  • \( \sigma \) is the tensile stress,
  • \( \varepsilon \) is the longitudinal strain.
A high Young's modulus indicates that a material is stiff and can bear a lot of stress before deforming. It plays a crucial role in engineering and construction, helping us select materials that won't easily bend or break under load.
Bulk modulus
The bulk modulus represents a material's response to uniform pressure applied in all directions. Think of squeezing a rubber ball evenly from all sides. This modulus indicates a material's ability to resist changes in its volume without changing shape. It is calculated using:\[ K = -V \frac{\Delta P}{\Delta V} \]where:
  • \( K \) is the bulk modulus,
  • \( V \) is the original volume,
  • \( \Delta P \) is the change in pressure,
  • \( \Delta V \) is the change in volume.
Materials with a high bulk modulus are incompressible and tend to retain their volume under pressure. This is important in designing systems under significant external pressure, such as submarines or pressure vessels.
Poisson's ratio
Poisson's ratio describes how a material will deform in the directions perpendicular to the applied force. When you pull on a rubber band, it not only gets longer but also thinner. Poisson's ratio is the measure of this lateral thinning. It's defined as:\[ u = \frac{-\Delta Lateral\ Strain}{\Delta Longitudinal\ Strain} \]Key points to remember:
  • A positive Poisson鈥檚 ratio (most materials) means that when a material is stretched, it becomes thinner.
  • If a material has a negative Poisson鈥檚 ratio, it becomes "fatter" when stretched.
Most common materials have a Poisson's ratio between 0 and 0.5. Understanding this helps in predicting how materials behave under various forces, which is crucial in fields like civil engineering and biomechanics.
Modulus of Rigidity
The modulus of rigidity, also known as the shear modulus, measures a material's ability to resist forces that cause it to twist or shear. Unlike stretching or compressing, shear involves forces parallel to a surface. Formally, it is defined by:\[ G = \frac{\tau}{\gamma} \]where:
  • \( G \) is the modulus of rigidity,
  • \( \tau \) is the shear stress,
  • \( \gamma \) is the shear strain.
A higher modulus of rigidity means the material is more resistant to shearing deformations. This property is essential for materials used in structures and machinery that experience a lot of twisting or sliding forces.

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Most popular questions from this chapter

\(\Lambda\) thin uniform metallic rod of length \(0.5 \mathrm{~m}\) and radius \(0.1 \mathrm{~m}\) rotates with an angular velocity \(400 \mathrm{rad} / \mathrm{s}\) in a horizontal plane about a vertical axis passing through one of its ends. Calculate (a) the cension in the rod and (b) the clongation of the rod. The density of the material of the rod is \(10^{4} \mathrm{~kg} / \mathrm{m}^{3}\) and the Young's modulus is \(2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\).

A capillary is dipped in water vessel kept in a lift going up with acceleration \(2 g\). Then (a) The water will rise in the tube to the height observed under normal condition (b) Water will rise to the maximum available height of the tube (c) Water will rise to the height one third of the height of normal condition (d) Water will rise to the height double the height of normal condition

\(\Lambda\) closed tank is completely filled with a liquid of density \(\rho\) and placed on a car. The cart is given an acecleration \(a\). The pressure at the point \(P\) is (a) \(\rho(g h+a l)\), if the acceleration is horizontally towards right (b) \(\rho h(g+a)\), if the acceleration is vertically upward (c) \(\rho \mid h(g+a \sin \alpha)+l a \cos \alpha]\), if acceleration is inclined upward to the horizontal by an angle \(\alpha\) (d) \(\rho h(g-a)\), if acceleration is vertically downward

A metal picce is trapped in an ice cube floating in water. If icc melts and motal sinks, the levol of water will (a) Increase (b) Decrease (c) Ramain unchanged (d) Cannot be predicted

\(\Lambda\) body of mass \(M\) is attached to the lower end of a metal wire, whose upper end is fixed. 'Ihe clongation of the wire is \(l\) (a) loss in gravitational polential cnergy of \(M\) is \(M g l\) (b) the clastic potential encrgy stored in the wirc is \(M g l\) (c) the clastic potential energy swred in the wire is \(1 / 2 M g l\) (d) heal produced is \(1 / 2 \mathrm{Mgl}\)
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