Questions and Answers
What is the property of a body that tends to regain its original size and shape when the applied force is removed?
elasticity
What is the deformation caused by a body that does not tend to regain its previous shape?
plastic deformation
What is the property called when a body is pressed with a hand and pushed horizontally?
Hooke's Law states that stress is not proportional to strain for small deformations.
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What is the typical term for a material if the ultimate strength and fracture points are close?
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What is the term used to describe a material if the ultimate strength and fracture points are far apart?
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What is the ratio defined as Young's Modulus?
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What is the unit of Young's Modulus?
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What is the name given to materials like rubber and tissue of aorta that exhibit large elastic regions?
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What is the ratio of hydraulic stress to the corresponding hydraulic strain called?
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What is the symbol used to denote Bulk Modulus?
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The negative sign in the bulk modulus formula B = - p/(∆V/V) indicates that with an increase in pressure, a decrease in _____ occurs.
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Which of the following materials has the highest bulk modulus?
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Bulk moduli for liquids are generally smaller than for solids.
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What is the formula for the amount a bar sags when loaded at the centre by a load? Write it in terms of the load (W), length (l), breadth (b), depth (d), and Young's modulus (Y).
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What is the elastic limit for a typical rock if the shear modulus is 30 x 10^7 N m^-2?
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What is the relationship between stress and strain under tension or compression based on Hooke's law?
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What are the three types of stresses mentioned in the text?
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Hooke's law is only valid in the linear part of the stress-strain curve.
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Calculate the bulk modulus of water given: Initial volume = 100.0 litre, Pressure increase = 100.0 atm, Final volume = 100.5 litre. Compare the bulk modulus of water with that of air at constant temperature. Explain why the ratio is so large.
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What is the density of water at a depth where the pressure is 80.0 atm, given that its density at the surface is 1.03 × 10^3 kg m^3?
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Calculate the fractional change in volume of a glass slab when subjected to a hydraulic pressure of 10 atm.
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Determine the volume contraction of a solid copper cube with 10 cm edges when subjected to a hydraulic pressure of 7.0 × 10^6 Pa.
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How much should the pressure on a litre of water be changed to compress it by 0.10%?
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Study Notes
Mechanical Properties of Solids
- Mechanical properties of solids are important in engineering design and everyday life.
- Solids can be stretched, compressed, and bent, and their properties can be changed by applying external forces.
Stress and Strain
- When a force is applied to a solid, it deforms, and the body develops a restoring force to counteract the applied force.
- Stress is the restoring force per unit area, and its SI unit is N m-2 or Pascal (Pa).
- Strain is the ratio of change in dimension to the original dimension, and it has no units or dimensional formula.
Types of Stress and Strain
- There are three types of stress: tensile, compressive, and shear stress.
- Tensile stress occurs when a force is applied normal to the cross-sectional area of a body, causing it to elongate.
- Compressive stress occurs when a force is applied parallel to the cross-sectional area of a body, causing it to compress.
- Shear stress occurs when a force is applied tangentially to the surface of a body, causing it to deform by an angle.
Hooke's Law
- Hooke's law states that stress and strain are proportional to each other for small deformations.
- The law is an empirical law and is valid for most materials.
- The proportionality constant is known as the modulus of elasticity.
Stress-Strain Curve
- The stress-strain curve is a graph that shows the relationship between stress and strain for a given material.
- The curve can be used to determine the elastic limit, yield strength, and ultimate tensile strength of a material.
- The elastic region of the curve is where the stress and strain are proportional to each other.
Elastic Moduli
- Elastic moduli are measures of the stiffness of a material.
- Young's modulus is a measure of the stiffness of a material under tensile or compressive stress.
- Shear modulus is a measure of the stiffness of a material under shear stress.
Applications of Elastic Behavior
- The elastic behavior of materials is important in engineering design, such as in the design of buildings, bridges, and machines.
- The elastic behavior of materials is also important in biological systems, such as in the structure of bones and tissues.
Examples and Problems
- A structural steel rod is stretched by a 100 kN force, and its elongation is calculated.
- A copper wire and a steel wire are stretched by a load, and the net elongation is calculated.
- The compression of a thighbone under an extra load is calculated.
Tables
- Table 8.1 shows the values of Young's moduli and yield strengths of some materials.
- Table 8.2 shows the values of shear moduli of some common materials.### Mechanical Properties of Solids
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Stress and Strain:
- Stress: the restoring force per unit area, measured in N m^-2 (Pa) or GPa.
- Strain: the fractional change in dimension, a pure number with no dimensions or units.
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Young's Modulus (Y):
- The ratio of tensile stress to tensile strain, measured in N m^-2 (Pa) or GPa.
- Symbol: Y, unit: N m^-2 (Pa) or GPa.
- Young's modulus is a measure of the stiffness of a material.
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Compression in Thighbone:
- The weight of the body is supported by the legs, and the thighbone is compressed.
- The compression in each thighbone (∆L) can be computed using the formula: ∆L = (F × L)/(Y × A).
- The fractional decrease in the thighbone is ∆L/L = 0.000091 or 0.0091%.
Shear Modulus
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Shear Modulus (G):
- The ratio of shearing stress to the corresponding shearing strain, measured in N m^-2 (Pa) or GPa.
- Symbol: G, unit: N m^-2 (Pa) or GPa.
- The shear modulus is a measure of the rigidity of a material.
- The shear moduli of a few common materials are given in Table 9.2.
Bulk Modulus
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Bulk Modulus (B):
- The ratio of hydraulic stress to the corresponding hydraulic strain, measured in N m^-2 (Pa) or GPa.
- Symbol: B, unit: N m^-2 (Pa) or GPa.
- The bulk modulus is a measure of the compressibility of a material.
- The bulk moduli of a few common materials are given in Table 8.3.
Elastic Potential Energy
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Elastic Potential Energy in a Stretched Wire:
- When a wire is put under a tensile stress, work is done against the inter-atomic forces.
- The work is stored in the wire in the form of elastic potential energy.
- The elastic potential energy per unit volume of the wire (u) is given by the formula: u = (1/2) × σ ε.
Poisson's Ratio
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Poisson's Ratio:
- The ratio of the lateral strain to the longitudinal strain in a stretched wire.
- A pure number with no dimensions or units.
- Poisson's ratio is a measure of the lateral strain response to a longitudinal tensile strain.
Applications of Elastic Behaviour of Materials
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Crane Design:
- The design of a crane requires knowledge of the elastic behaviour of materials.
- The rope should be able to withstand the weight of the load without deforming permanently.
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Beam Design:
- The design of a beam requires knowledge of the elastic behaviour of materials.
- The beam should be able to withstand the load without bending or breaking.
- The formula for the sagging of a beam under a load is: δ = W l^3/(4bd^3Y).
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Bridge Design:
- The design of a bridge requires knowledge of the elastic behaviour of materials.
- The beam should be able to withstand the load of the flowing traffic, the force of winds, and its own weight.
- The cross-sectional shape of the beam is critical in reducing the bending.
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Mountain Height:
- The maximum height of a mountain on earth is limited by the elastic properties of rocks.
- The stress due to the weight of the material on top should be less than the critical shearing stress at which the rocks flow.
- The formula for the maximum height of a mountain is: h = 30 × 10^7 N m^-2/(3 × 10^3 kg m^-3 × 10 m s^-2) = 10 km.### Types of Stresses
- There are three types of stresses: tensile stress, shearing stress, and hydraulic stress
- Tensile stress is associated with stretching, while compressive stress is associated with compression
- Shearing stress occurs when a pair of forces is applied parallel to the upper and lower faces of an object
- Hydraulic stress occurs when an object undergoes compression due to a surrounding fluid
Hooke's Law
- Hooke's law states that for small deformations, stress is directly proportional to strain for many materials
- The constant of proportionality is called the modulus of elasticity
- There are three elastic moduli: Young's modulus, shear modulus, and bulk modulus
- Young's modulus is used to describe the elastic behavior of objects under tension or compression
- Shear modulus is used to describe the elastic behavior of objects under shearing stress
- Bulk modulus is used to describe the elastic behavior of objects under hydraulic stress
Young's Modulus
- Young's modulus (Y) is used to calculate the stress and strain of an object under tension or compression
- The formula for Young's modulus is F/A = Y∆L/L, where F is the applied force, A is the cross-sectional area, and ∆L/L is the tensile or compressive strain
Shear Modulus
- Shear modulus (G) is used to calculate the stress and strain of an object under shearing stress
- The formula for shear modulus is F/A = G × ∆L/L, where F is the applied force, A is the cross-sectional area, and ∆L/L is the displacement of one end of the object in the direction of the applied force
Bulk Modulus
- Bulk modulus (B) is used to calculate the pressure and volume change of an object under hydraulic stress
- The formula for bulk modulus is p = B (∆V/V), where p is the pressure, ∆V/V is the volume strain, and B is the bulk modulus
Points to Ponder
- Hooke's law is only valid in the linear part of the stress-strain curve
- Young's modulus and shear modulus are only relevant for solids, as they have lengths and shapes
- Bulk modulus is relevant for solids, liquids, and gases, and refers to the change in volume under uniform stress
- Materials with larger values of Young's modulus require a larger force to produce small changes in length
- Materials that stretch more are not necessarily more elastic; in fact, materials that stretch less are considered more elastic
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Description
This quiz covers the mechanical properties of solids, including stress and strain, and the motion of rigid bodies.