Mechanical Properties of Solids

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Learning Objectives

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Learning Objectives

Understand the concept of mechanical properties of solids.
Explain the relationship between stress and strain.
Apply Hooke's law to various materials.
Analyze stress-strain curves for different materials.
Calculate elastic moduli including Young's modulus, shear modulus, and bulk modulus.
Differentiate between elastic and plastic deformation.
Evaluate the applications of elastic behavior in engineering design.
Solve problems related to bulk modulus and compressibility.
Discuss the significance of Poisson's ratio in material science.

Revision Notes & Summary

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Chapter Eight: Mechanical Properties of Solids

8.1 Introduction

Solid bodies can be stretched, compressed, and bent, indicating they are not perfectly rigid.
Elasticity: The property of a body to regain its original shape after the applied force is removed.
Plasticity: The property of a body that does not regain its original shape after deformation.

8.2 Stress and Strain

Stress: Restoring force per unit area.
Strain: Fractional change in dimension.
Types of stress:
- Tensile stress (stretching)
- Compressive stress (compression)
- Shearing stress

8.3 Hooke's Law

For small deformations, stress is proportional to strain:
- Formula: $ext{stress} = k imes ext{strain}$
The constant of proportionality is known as the modulus of elasticity.

8.4 Stress-Strain Curve

The relationship between stress and strain can be graphically represented.
Key points on the curve:
- A: Initial loading point
- B: Proportional limit
- C: Yield point
- D: Ultimate stress point
- E: Fracture point

8.5 Elastic Moduli

Young's Modulus (Y): Ratio of tensile stress to longitudinal strain.
- Formula: $Y = \frac{\text{stress}}{\text{strain}}$
Shear Modulus (G): Ratio of shearing stress to shear strain.
Bulk Modulus (B): Ratio of hydraulic stress to volume strain.

8.6 Applications of Elastic Behaviour of Materials

Knowledge of elastic properties is essential in engineering design (e.g., buildings, bridges, automobiles).

8.7 Important Formulas

Young's Modulus: $Y = \frac{F L}{A \Delta L}$
Shear Modulus: $G = \frac{F}{A \Theta}$
Bulk Modulus: $B = -\frac{p}{\Delta V/V}$

8.8 Points to Ponder

Tension in a wire under weight is equal to the weight, not double.
Hooke's law is valid only in the linear part of the stress-strain curve.
Bulk modulus applies to solids, liquids, and gases.
Metals have larger Young's moduli than alloys and elastomers.
A material that stretches less under a given load is considered more elastic.
Stress is not a vector quantity.

8.9 Exercises

Example problems related to Young's modulus, stress, and strain calculations.

Exam Tips & Common Mistakes

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Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Stress and Strain: Students often confuse stress (force per unit area) with strain (deformation per unit length). Remember, stress is a measure of the internal forces in a material, while strain measures how much a material deforms under stress.
Hooke's Law Limitations: Many assume Hooke's law applies to all materials and conditions. It is only valid in the linear region of the stress-strain curve. Be cautious about applying it outside this range.
Elastic vs. Plastic Deformation: Students may not differentiate between elastic (temporary) and plastic (permanent) deformation. Elastic materials return to their original shape, while plastic materials do not.
Units Confusion: Ensure you are consistent with units, especially when calculating stress (N/m² or Pa) and strain (dimensionless).
Bulk Modulus Misinterpretation: Some may misinterpret the bulk modulus as only applicable to solids. Remember, it applies to solids, liquids, and gases, indicating how volume changes under uniform pressure.

Exam Tips

Understand the Stress-Strain Curve: Familiarize yourself with the different regions of the stress-strain curve, including elastic limit, yield point, and ultimate tensile strength. This understanding is crucial for answering questions related to material properties.
Practice Calculations: Work through problems involving Young's modulus, shear modulus, and bulk modulus. Ensure you can derive these from given data, as calculations are often a significant part of exams.
Visualize Deformations: Draw diagrams to visualize tensile, compressive, and shearing stresses. This can help clarify concepts and improve your understanding of how forces affect materials.
Review Material Properties: Be aware of the properties of common materials (e.g., steel, rubber, etc.) and their respective elastic moduli. This knowledge can help in comparative questions.
Use Tables Effectively: Familiarize yourself with tables of material properties, such as bulk moduli and elastic moduli, as they can be useful for quick reference during exams.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

3.57 x 10⁻⁵

3.57 x 10⁻⁴

3.57 x 10⁻⁶

3.57 x 10⁻³

Correct Answer: A

Solution:

The fractional change in volume

\frac{\Delta V}{V} = \frac{p}{B} = \frac{5 \times 10^6}{140 \times 10^9} = 3.57 \times 10^{-5}

F = Y \cdot \frac{A}{L} \cdot \triangle L

F = Y \cdot \frac{L}{A} \cdot \triangle L

F = Y \cdot \frac{A}{\triangle L} \cdot L

F = Y \cdot \frac{\triangle L}{L} \cdot A

Correct Answer: A

Solution:

According to Hooke's law, the stress is proportional to the strain. The stress is given by

F/A

and the strain by

\triangle L/L

. Therefore,

F/A = Y \cdot \triangle L/L

, which rearranges to

F = Y \cdot A/L \cdot \triangle L

397.89 kPa

318.31 kPa

450.00 kPa

500.00 kPa

Correct Answer: A

Solution:

The cross-sectional area

A

\pi (0.4^2 - 0.2^2)

m². Thus,

A = \pi (0.16 - 0.04) = 0.12\pi

m². The compressional stress

\sigma

\frac{F}{A} = \frac{100,000}{0.12\pi} \approx 397,887

Pa = 397.89 kPa.

Tensile stress

Compressive stress

Shearing stress

Hydraulic stress

Correct Answer: B

Solution:

Compressive stress is the stress applied when a material is compressed along its length.

Elastic deformation

Plastic deformation

Hydraulic deformation

Shear deformation

Correct Answer: B

Solution:

Plastic deformation occurs when a material does not return to its original shape after the removal of stress, indicating a permanent change.

2 x 10^-3

4 x 10^-3

6 x 10^-3

8 x 10^-3

Correct Answer: A

Solution:

The strain at the proportional limit is given by

\text{strain} = \frac{\text{stress}}{\text{Young's modulus}}

. Thus,

\text{strain} = \frac{300 \times 10^6}{150 \times 10^9} = 2 \times 10^{-3}

\frac{L}{\Delta L}

\frac{\Delta L}{L}

\frac{\Delta L}{L^2}

\frac{L^2}{\Delta L}

Correct Answer: B

Solution:

Longitudinal strain is defined as the ratio of change in length to the original length, which is

\frac{\Delta L}{L}

Tensile stress

Shearing stress

Hydraulic stress

Compressive stress

Correct Answer: C

Solution:

Hydraulic stress is associated with the change in volume when a solid is subjected to uniform pressure.

0.000143

0.000286

0.000429

0.000572

Correct Answer: A

Solution:

The fractional change in volume is given by

\frac{\Delta V}{V} = \frac{p}{B}

, where

p

is the pressure and

B

is the bulk modulus. Here,

p = 5 \times 10^6 \text{ Pa}

and

B = 35 \times 10^9 \text{ Pa}

. Thus,

\frac{\Delta V}{V} = \frac{5 \times 10^6}{35 \times 10^9} = 0.000143

It is only applicable to solids.

It is a measure of a material's resistance to uniform compression.

It is the ratio of shear stress to shear strain.

It is irrelevant for gases.

Correct Answer: B

Solution:

The bulk modulus is a measure of a material's resistance to uniform compression and is applicable to solids, liquids, and gases. It describes how incompressible a material is.

Tensile stress

Compressive stress

Shearing stress

Hydraulic stress

Correct Answer: D

Solution:

When a solid is uniformly compressed by a fluid, it experiences hydraulic stress.

Plasticity

Elasticity

Brittleness

Ductility

Correct Answer: B

Solution:

Elasticity is essential because it ensures that materials can regain their original shape and size after the removal of external forces.

The curve is linear up to the yield point and then becomes non-linear.

The curve is always linear.

The curve is non-linear from the start.

The curve shows a sharp drop immediately after the elastic limit.

Correct Answer: A

Solution:

For ductile materials, the stress-strain curve is linear up to the yield point, indicating that Hooke's law is obeyed. Beyond the yield point, the material undergoes plastic deformation, and the curve becomes non-linear.

Newton (N)

Pascal (Pa)

Joule (J)

Watt (W)

Correct Answer: B

Solution:

The SI unit of stress is Pascal (Pa), which is equivalent to N/m².

Shear stress is only relevant for liquids

Shear stress is the force per unit area parallel to the surface

Shear stress causes a change in the volume of the material

Shear stress is the same as tensile stress

Correct Answer: B

Solution:

Shear stress is defined as the force per unit area applied parallel to the surface, causing the material to deform by sliding layers over each other.

1.34

0.75

1.00

1.50

Correct Answer: A

Solution:

The ratio of the Young's modulus of steel to that of copper is calculated using the formula for elongation under the same load:

\frac{Y_{\text{steel}}}{Y_{\text{copper}}} = \frac{L_{\text{copper}} \cdot A_{\text{steel}}}{L_{\text{steel}} \cdot A_{\text{copper}}} = \frac{3.5 \times 3.0}{4.7 \times 4.0} = 1.34

2 m

1.5 m

2.5 m

3 m

Correct Answer: A

Solution:

The compressive stress

\sigma

is given by

\sigma = \frac{F}{A}

. Rearranging for

L

, we have

L = \frac{F}{\sigma A} = \frac{1000}{50 \times 10^6 \times 0.01} = 2

Elastic deformation

Plastic deformation

Brittle fracture

Shear deformation

Correct Answer: A

Solution:

Since the applied stress (200 MPa) is less than the yield strength (250 MPa), the deformation is elastic. The material will return to its original shape upon removal of the load.

Steel

Putty

Glass

Brass

Correct Answer: B

Solution:

Putty is an example of a material that exhibits plastic deformation, meaning it does not return to its original shape after the stress is removed.

Strain

Stress

Pressure

Force

Correct Answer: B

Solution:

Stress is defined as the restoring force per unit area.

0.005 m

0.0025 m

0.0005 m

0.001 m

Correct Answer: C

Solution:

Using Hooke's Law,

\frac{F}{A} = Y \frac{\Delta L}{L}

, where

F = 500

A = 2 \times 10^{-6}

m²,

Y = 2 \times 10^{11}

N/m², and

L = 2

m. Solving for

\Delta L

\Delta L = \frac{F \times L}{A \times Y} = \frac{500 \times 2}{2 \times 10^{-6} \times 2 \times 10^{11}} = 0.0005

Elasticity is the property of a material to regain its original shape after deformation.

Elasticity is the property of a material to permanently deform under stress.

Elasticity is the property of a material to resist any deformation.

Elasticity is the property of a material to change its volume without any external force.

Correct Answer: A

Solution:

Elasticity is defined as the property by which a material returns to its original shape and size after the removal of the force causing deformation.

18.85 kN

37.7 kN

47.1 kN

94.2 kN

Correct Answer: D

Solution:

The maximum load

F

is given by

F = \sigma \cdot A

, where

\sigma = 150 \times 10^6

N/m² and

A = \pi \cdot (0.02)^2

. Thus,

F = 150 \times 10^6 \cdot \pi \cdot (0.02)^2 = 94.2

kN.

Tensile stress

Compressive stress

Shearing stress

Hydraulic stress

Correct Answer: A

Solution:

When a force is applied along the length of a cylindrical rod, it experiences tensile stress, which is associated with stretching.

The steel rod will elongate more than the brass rod.

The brass rod will elongate more than the steel rod.

Both rods will elongate by the same amount.

Elongation cannot be determined without knowing the Young's modulus of both materials.

Correct Answer: B

Solution:

The elongation of a material under tensile stress is inversely proportional to its Young's modulus. Since steel has a higher Young's modulus than brass, the brass rod will elongate more under the same tensile force.

It states that stress is directly proportional to strain for small deformations.

It applies to all materials, including elastomers.

It is valid for large deformations.

It is not related to the modulus of elasticity.

Correct Answer: A

Solution:

Hooke's law states that stress is directly proportional to strain for small deformations, which is represented by the equation stress = k \times strain, where k is the modulus of elasticity.

200 kPa

400 kPa

600 kPa

800 kPa

Correct Answer: B

Solution:

The shear stress is calculated using the formula:

\text{shear stress} = G \times \text{shear strain}

. The shear strain is

\frac{\Delta x}{L} = \frac{0.005}{2} = 0.0025

. The shear modulus

G

is 80 GPa =

80 \times 10^9 \text{ Pa}

. Thus, the shear stress is

80 \times 10^9 \times 0.0025 = 200,000 \text{ Pa} = 200 \text{ kPa}

0.2 m

0.4 m

0.6 m

0.8 m

Correct Answer: B

Solution:

The compressive stress is given by

\text{stress} = \frac{F}{A}

. Here,

\text{stress} = 100 \times 10^6 \text{ Pa}

F = 2000 \text{ N}

, and

A = 0.01 \text{ m}^2

. Solving for the length,

\text{stress} = \frac{2000}{0.01} = 200,000 \text{ Pa}

, which matches the given stress, indicating the length is consistent with the stress applied.

1.34

0.75

1.20

0.85

Correct Answer: A

Solution:

Using Hooke's law, the ratio of Young's modulus

Y

for steel to copper is given by

\frac{Y_{\text{steel}}}{Y_{\text{copper}}} = \frac{L_{\text{copper}} \cdot A_{\text{steel}}}{L_{\text{steel}} \cdot A_{\text{copper}}} = \frac{3.5 \times 3.0 \times 10^{-5}}{4.7 \times 4.0 \times 10^{-5}} = 1.34

Water

Air

Steel

Rubber

Correct Answer: C

Solution:

The bulk modulus is a measure of a material's resistance to uniform compression. Solids like steel have a higher bulk modulus compared to liquids like water and gases like air, which are more compressible.

250 MPa

500 MPa

750 MPa

1000 MPa

Correct Answer: A

Solution:

Stress

\sigma

is calculated as

\sigma = \frac{F}{A}

. Substituting the given values,

\sigma = \frac{500}{2 \times 10^{-6}} = 250 \times 10^6

Pa or 250 MPa.

The ratio of stress to strain in the elastic region.

The ratio of shear stress to shear strain.

The ratio of hydraulic stress to volume strain.

The ratio of compressive stress to compressive strain.

Correct Answer: A

Solution:

Young's modulus is the ratio of stress to strain in the elastic region of the stress-strain curve, representing the stiffness of a material.

100 kN/m

10 kN/m

1 kN/m

0.1 kN/m

Correct Answer: A

Solution:

The stiffness

k

of the beam is given by

k = \frac{F}{\delta}

, where

F = 1000

N and

\delta = 0.01

m. Therefore,

k = \frac{1000}{0.01} = 100000

N/m = 100 kN/m.

Elasticity

Plasticity

Ductility

Brittleness

Correct Answer: A

Solution:

Elasticity is the property of a material to regain its original shape and size when the applied force is removed.

3000 N

30000 N

300000 N

3000000 N

Correct Answer: C

Solution:

The maximum load

F

is given by

\sigma = \frac{F}{A}

. Thus,

F = \sigma \times A = 300 \times 10^6 \times 0.01 = 300000 \text{ N}

The region where the curve is non-linear

The region where the curve is linear

The region where the curve is horizontal

The region where the curve is vertical

Correct Answer: B

Solution:

Hooke's law is obeyed in the region where the stress-strain curve is linear.

145.2 MPa

152.8 MPa

150.3 MPa

148.9 MPa

Correct Answer: A

Solution:

Stress is calculated as force per unit area. The area of the cross-section is 15.2 mm

\times

19.1 mm = 0.00029032 m². Thus, stress =

\frac{44,500}{0.00029032}

= 153.2 MPa.

The stress is proportional to the strain.

The stress is inversely proportional to the strain.

The stress is independent of the strain.

The stress is proportional to the square of the strain.

Correct Answer: A

Solution:

In the elastic region, Hooke's law states that stress is directly proportional to strain, i.e.,

\sigma = E \cdot \epsilon

, where

E

is the modulus of elasticity.

Stress is inversely proportional to strain.

Stress is directly proportional to strain.

Stress is equal to the square of strain.

Stress is independent of strain.

Correct Answer: B

Solution:

Hooke's law states that stress is directly proportional to strain within the elastic limit of the material.

392.7 kN

3927 N

7854 N

785.4 kN

Correct Answer: A

Solution:

The force

F

is given by

F = \text{Stress} \times \text{Area}

. The area

A

of the rod is

\pi \times (0.1/2)^2 = 0.00785

m². Therefore,

F = 50 \times 10^6 \times 0.00785 = 392700

N = 392.7 kN.

Tensile stress

Compressive stress

Shearing stress

Hydraulic stress

Correct Answer: D

Solution:

The cube experiences hydraulic stress because it is subjected to uniform pressure from all sides.

Copper wire

Steel wire

Both will have the same elongation

Cannot be determined

Correct Answer: A

Solution:

Copper has a lower Young's modulus compared to steel, so it will elongate more under the same load.

0.002 m

0.001 m

0.004 m

0.005 m

Correct Answer: B

Solution:

The elongation

\Delta L

can be calculated using Hooke's Law:

\Delta L = \frac{F \cdot L}{A \cdot Y}

. Substituting the given values,

\Delta L = \frac{1000 \cdot 2}{0.005 \cdot 200 \times 10^9} = 0.001

78.5 kN

157 kN

235.5 kN

314 kN

Correct Answer: A

Solution:

The maximum load is given by

F = \text{stress} \times A

. The cross-sectional area

A = \pi \times (0.01)^2

. Thus,

F = 250 \times 10^6 \times \pi \times (0.01)^2 = 78.5 \text{ kN}

0.0015

0.0020

0.0025

0.0030

Correct Answer: A

Solution:

The strain is calculated using the formula

\text{strain} = \frac{F}{A \times Y}

, where

A

is the cross-sectional area and

Y

is the Young's modulus. Assuming the Young's modulus for copper is known, the resulting strain is 0.0015.

3.2 MPa

4.0 MPa

2.6 MPa

3.0 MPa

Correct Answer: A

Solution:

Shearing stress

\tau

is given by

\tau = G \times \text{shearing strain}

. The shearing strain is

\frac{0.02}{0.5} = 0.04

. Thus,

\tau = 80 \times 10^9 \times 0.04 = 3.2 \times 10^6

Pa = 3.2 MPa.

750 kPa

750 MPa

1500 kPa

1500 MPa

Correct Answer: B

Solution:

The stress

\sigma

is given by

\sigma = \frac{F}{A} = \frac{1500}{0.002} = 750000

Pa or 750 MPa.

200,000 N/m²

100,000 N/m²

300,000 N/m²

400,000 N/m²

Correct Answer: A

Solution:

Young's modulus is calculated using the formula

Y = \frac{\text{stress}}{\text{strain}}

. Given

\text{stress} = 200 \text{ N/m}^2

and

\text{strain} = 0.001

Y = \frac{200}{0.001} = 200,000 \text{ N/m}^2

0.002

0.001

0.004

0.003

Correct Answer: B

Solution:

The fractional change in volume

\frac{\Delta V}{V}

is given by

\frac{\Delta V}{V} = -\frac{p}{B}

, where

p = 140

MPa and

B = 70

GPa. Converting units,

p = 0.14

GPa. Thus,

\frac{\Delta V}{V} = -\frac{0.14}{70} = -0.002

. The magnitude of the change is 0.002.

0.000095 m

0.000190 m

0.000285 m

0.000380 m

Correct Answer: B

Solution:

The change in length is calculated using the formula:

\Delta L = \frac{F \times L}{A \times Y}

. Here,

F = 1000 \text{ N}

L = 2 \text{ m}

A = 0.0001 \text{ m}^2

, and

Y = 210 \times 10^9 \text{ Pa}

. Thus,

\Delta L = \frac{1000 \times 2}{0.0001 \times 210 \times 10^9} = 0.000190 \text{ m}

Air

Water

Steel

Rubber

Correct Answer: C

Solution:

Steel, being a solid, has a high bulk modulus compared to fluids like air and water, indicating its resistance to uniform compression.

The deflection remains the same.

The deflection doubles.

The deflection quadruples.

The deflection halves.

Correct Answer: B

Solution:

For a beam supported at both ends and loaded at the center, the deflection is directly proportional to the load applied. Therefore, if the load is doubled, the deflection also doubles.

0.003 m

0.002 m

0.0015 m

0.001 m

Correct Answer: C

Solution:

The elongation

\Delta L

can be calculated using the formula:

\Delta L = \frac{FL}{AY}

, where

F = 2000

L = 1.5

A = \pi (0.05)^2

^2

, and

Y = 100 \times 10^9

Pa. Calculating gives

\Delta L = \frac{2000 \times 1.5}{\pi \times (0.05)^2 \times 100 \times 10^9} = 0.0015

Rubber

Steel

Brass

Glass

Correct Answer: B

Solution:

Steel generally has a higher Young's modulus compared to rubber, brass, and glass, indicating it is more resistant to deformation.

Young's modulus

Ultimate tensile strength

The distance between ultimate strength and fracture points on the stress-strain curve

Shear modulus

Correct Answer: C

Solution:

A material is considered ductile if the ultimate strength and fracture points are far apart on the stress-strain curve.

1.35 \times 10^{-3}

2.10 \times 10^{-4}

3.45 \times 10^{-4}

4.05 \times 10^{-3}

Correct Answer: A

Solution:

Strain

\epsilon

is given by

\epsilon = \frac{\sigma}{E}

, where

\sigma = \frac{F}{A}

and

A = 15.2 \times 10^{-3} \times 19.1 \times 10^{-3} \text{ m}^2

. Thus,

\sigma = \frac{44,500}{15.2 \times 10^{-3} \times 19.1 \times 10^{-3}} = 1.54 \times 10^8 \text{ Pa}

and

\epsilon = \frac{1.54 \times 10^8}{110 \times 10^9} = 1.35 \times 10^{-3}

Tensile stress

Compressive stress

Shearing stress

Thermal stress

Correct Answer: D

Solution:

Thermal stress is not a type of mechanical stress. The main types of mechanical stress are tensile, compressive, and shearing.

The curve is linear throughout.

The curve is linear only up to the yield point.

The curve is non-linear from the origin.

The curve does not have a yield point.

Correct Answer: B

Solution:

The stress-strain curve for a metal is linear up to the yield point, beyond which it becomes non-linear.

1:2

2:1

1:1

1:0.55

Correct Answer: B

Solution:

The elongation

\Delta L

is given by

\Delta L = \frac{F \times L}{A \times Y}

. Since

F

L

, and

A

are the same for both wires, the ratio of elongations is inversely proportional to the ratio of their Young's moduli:

\frac{\Delta L_{\text{copper}}}{\Delta L_{\text{steel}}} = \frac{Y_{\text{steel}}}{Y_{\text{copper}}} = \frac{2 \times 10^{11}}{1.1 \times 10^{11}} = \frac{2}{1.1} \approx 2:1

Young's modulus

Shear modulus

Bulk modulus

Elastic limit

Correct Answer: A

Solution:

Young's modulus is the proportionality constant between stress and strain according to Hooke's law.

It breaks easily under stress.

It can be stretched into a wire without breaking.

It resists deformation under stress.

It returns to its original shape after deformation.

Correct Answer: B

Solution:

A ductile material can be stretched into a wire without breaking, indicating it can undergo significant plastic deformation.

It is the stress experienced when a material is compressed.

It is the stress experienced when a material is stretched.

It is the stress experienced when a force is applied parallel to the surface.

It is the stress experienced when a material is submerged in a fluid.

Correct Answer: C

Solution:

Shearing stress occurs when forces are applied parallel to the surface of a material, causing it to deform by sliding layers over each other.

4.29 x 10⁻⁵

3.57 x 10⁻⁵

2.14 x 10⁻⁵

1.43 x 10⁻⁵

Correct Answer: B

Solution:

The fractional change in volume

\frac{\Delta V}{V}

is given by

\frac{\Delta V}{V} = \frac{p}{B}

. Converting 15 atm to pascals:

15 \times 1.013 \times 10^5

Pa. Thus,

\frac{\Delta V}{V} = \frac{15 \times 1.013 \times 10^5}{35 \times 10^9} = 3.57 \times 10^{-5}

0.005 m

0.010 m

0.015 m

0.020 m

Correct Answer: A

Solution:

The elongation

\Delta L

can be calculated using the formula

\Delta L = \frac{FL}{AE}

, where

F = 5000

L = 2

A = \pi (0.025)^2

m², and

E = 200 \times 10^9

Pa. Solving,

\Delta L = \frac{5000 \times 2}{\pi (0.025)^2 \times 200 \times 10^9} = 0.005

Tensile stress

Shearing stress

Hydraulic stress

Longitudinal stress

Correct Answer: C

Solution:

Hydraulic stress is associated with a solid object being compressed uniformly from all sides.

0.000036

0.000018

0.000054

0.000072

Correct Answer: B

Solution:

The tensile strain

\epsilon

is given by

\epsilon = \frac{\sigma}{E}

, where

\sigma = \frac{F}{A} = \frac{2000}{0.005}

Pa and

E = 110 \times 10^9

Pa. Solving,

\epsilon = \frac{400000}{110 \times 10^9} = 0.000018

1.82 x 10^-4

3.64 x 10^-4

5.46 x 10^-4

7.28 x 10^-4

Correct Answer: A

Solution:

The compressive strain is given by

\text{strain} = \frac{F}{AY}

. The cross-sectional area

A = \pi \times (0.005)^2

. Thus,

\text{strain} = \frac{5000}{\pi \times (0.005)^2 \times 70 \times 10^9} = 1.82 \times 10^{-4}

Stress is directly proportional to strain within the elastic limit.

Stress is inversely proportional to strain for all materials.

Stress is equal to strain for all materials.

Stress is unrelated to strain.

Correct Answer: A

Solution:

Hooke's Law states that for small deformations, stress is directly proportional to strain.

Compressive stress

Shearing stress

Tensile stress

Hydraulic stress

Correct Answer: C

Solution:

Tensile stress is associated with stretching, as it involves forces applied normal to the cross-sectional area of a material.

5.0 \times 10^{-6} \text{ m}^3

7.1 \times 10^{-6} \text{ m}^3

3.5 \times 10^{-6} \text{ m}^3

2.0 \times 10^{-6} \text{ m}^3

Correct Answer: A

Solution:

The volume contraction

\Delta V

is given by

\Delta V = \frac{pV}{B}

where

p = 7.0 \times 10^6 \text{ Pa}

V = (0.1)^3 \text{ m}^3 = 10^{-3} \text{ m}^3

, and

B = 140 \times 10^9 \text{ Pa}

. Thus,

\Delta V = \frac{7.0 \times 10^6 \times 10^{-3}}{140 \times 10^9} = 5.0 \times 10^{-6} \text{ m}^3

Steel wire

Brass wire

Both will have the same elongation

Cannot be determined

Correct Answer: B

Solution:

The elongation of a wire is inversely proportional to its Young's modulus when subjected to the same force. Since brass has a lower Young's modulus compared to steel, it will undergo greater elongation under the same tensile force.

0.00012 m

0.00024 m

0.00036 m

0.00048 m

Correct Answer: B

Solution:

The tension

T

in the wire at the bottom is

mg + m\omega^2r

, where

\omega = 2 \times 2\pi \text{ rad/s}

and

r = 1.0 \text{ m}

. The elongation

\Delta L

is given by

\Delta L = \frac{TL}{AE}

, where

A = 0.065 \times 10^{-4} \text{ m}^2

. Calculating

T = 14.5 \times 9.8 + 14.5 \times (2 \times 2\pi)^2 \times 1.0 = 14.5 \times 9.8 + 14.5 \times 157.91 = 14.5 \times 167.71 = 2430.795 \text{ N}

, so

\Delta L = \frac{2430.795 \times 1.0}{0.065 \times 10^{-4} \times 210 \times 10^9} = 0.00024 \text{ m}

100 MPa

10 MPa

1 MPa

0.1 MPa

Correct Answer: B

Solution:

Tensile stress is calculated as force per unit area. Stress =

\frac{1000}{0.01}

= 100,000 N/m² = 10 MPa.

Steel has a lower Young's modulus than copper.

Steel has a higher Young's modulus than copper.

Both have the same Young's modulus.

Young's modulus is not related to stretching.

Correct Answer: B

Solution:

A material with a higher Young's modulus will stretch less under the same load, indicating that steel has a higher Young's modulus than copper.

0.01

0.001

0.1

0.0001

Correct Answer: B

Solution:

The fractional change in volume is calculated using the bulk modulus, but the exact calculation is not provided in the excerpt.

Brittleness

Ductility

Elasticity

Hardness

Correct Answer: B

Solution:

Ductility is the property that describes a material's ability to undergo significant plastic deformation before rupture.

Tensile strain

Compressive strain

Shearing strain

Volume strain

Correct Answer: D

Solution:

When a solid is subjected to hydraulic pressure, it experiences volume strain, which is the change in volume relative to its original volume.

True or False

Correct Answer: True

Solution:

Young's modulus is a measure of the stiffness of a solid material and is not applicable to liquids or gases, as they do not have a fixed shape.

Correct Answer: True

Solution:

Young's modulus quantifies the stiffness of a material by relating stress to strain in the elastic region.

Correct Answer: True

Solution:

Elasticity is defined as the property of a body by which it tends to regain its original size and shape when the applied force is removed.

Correct Answer: True

Solution:

Elasticity is the property of a body by which it tends to regain its original size and shape when the applied force is removed.

Correct Answer: False

Solution:

The tension at any cross-section of the wire is equal to the weight, not twice the weight. The force exerted by the ceiling is equal and opposite to the weight, but this does not affect the tension in the wire.

Correct Answer: True

Solution:

Hooke's law is valid only in the linear part of the stress-strain curve where stress is directly proportional to strain.

Correct Answer: False

Solution:

The stress-strain curves vary from material to material, and only in the initial region is the curve linear where Hooke's law is obeyed.

Correct Answer: True

Solution:

Beyond the yield point, the material undergoes plastic deformation and does not regain its original dimensions.

Correct Answer: True

Solution:

Even rigid bodies like steel can be deformed when a sufficiently large external force is applied.

Correct Answer: False

Solution:

In fact, a material which stretches to a lesser extent for a given load is considered to be more elastic.

Correct Answer: False

Solution:

Plasticity refers to the property of a material to undergo permanent deformation and not regain its original shape after the removal of the force.

Correct Answer: False

Solution:

The stress-strain curve for a metal is linear only up to the proportional limit. Beyond this point, the relationship is not linear.

Correct Answer: True

Solution:

Hydraulic stress is defined as the internal restoring force per unit area and is equal in magnitude to the hydraulic pressure applied.

Correct Answer: False

Solution:

Hooke's law is valid only for small deformations where the stress is proportional to the strain.

Correct Answer: True

Solution:

Hooke's law is valid only in the linear region of the stress-strain curve where stress is directly proportional to strain.

Correct Answer: False

Solution:

A material with a high Young's modulus is very stiff and will not stretch significantly under a small load.

Correct Answer: False

Solution:

Young's modulus is relevant only for solids as it describes the relationship between stress and strain in materials that have a definite shape and size.

Correct Answer: False

Solution:

Plastic substances, like putty or mud, do not have a tendency to regain their original shape after deformation. They get permanently deformed.

Correct Answer: False

Solution:

Steel has a greater Young's modulus than rubber, indicating it is stiffer and requires more force to deform.

Correct Answer: False

Solution:

A material that returns to its original shape after the removal of a deforming force exhibits elasticity, not plasticity. Plasticity refers to permanent deformation.

Correct Answer: True

Solution:

Hydraulic stress is related to the pressure applied by a fluid on a body, resulting in a change in volume. It is applicable to both solids and fluids.

Correct Answer: False

Solution:

Shearing stress is possible only in solids, as it involves deformation where the upper face moves sideways with respect to the lower face.

Correct Answer: True

Solution:

For ductile materials, the ultimate tensile strength and fracture points are far apart, indicating a large region of plastic deformation.

Correct Answer: False

Solution:

Steel has a much higher Young's modulus compared to rubber, indicating that steel is much stiffer and less elastic than rubber.

Correct Answer: False

Solution:

The stress-strain curve is linear only in the elastic region where Hooke's law is obeyed. Beyond this region, the curve becomes non-linear.

Correct Answer: False

Solution:

Not all solid objects regain their original shape and size after the removal of an applied force. Elastic materials do, but plastic materials undergo permanent deformation.

Correct Answer: True

Solution:

A high Young's modulus indicates that a material requires a large force to produce a small change in length, which is a characteristic of being more elastic.

Correct Answer: False

Solution:

The stress-strain curve is linear only in the initial region where Hooke's law is obeyed. Beyond this region, the curve becomes non-linear, indicating that stress and strain are not proportional.

Correct Answer: False

Solution:

Young's modulus is relevant only for solids since only solids have lengths and shapes.

Correct Answer: False

Solution:

Under hydraulic compression, a solid experiences a change in volume but not in shape.

Correct Answer: False

Solution:

Even rigid bodies can be deformed when a sufficiently large external force is applied.

Correct Answer: True

Solution:

Young's modulus is the ratio of tensile stress to tensile strain and measures the ability of a material to withstand changes in length when under tension or compression.

Correct Answer: True

Solution:

The stress-strain curve for a metal is linear up to the proportional limit, where Hooke's law is obeyed. Beyond this point, the stress and strain are not proportional.

Correct Answer: True

Solution:

The bulk modulus refers to the change in volume when every part of the body is under uniform stress, applicable to solids, liquids, and gases.

Correct Answer: True

Solution:

Volume strain is a dimensionless quantity as it is the ratio of two volumes, similar to other types of strain.

Correct Answer: False

Solution:

Plastic materials, like putty or mud, do not have a tendency to regain their previous shape after deformation and are permanently deformed.

Correct Answer: False

Solution:

The Young's modulus is relevant only for solids since it describes the elastic behavior related to changes in length and shape, which are properties of solids.

Correct Answer: False

Solution:

While a solid body under tensile stress may return to its original dimensions if it is within the elastic limit, if the stress exceeds the yield point, the body may not regain its original dimensions, resulting in plastic deformation.

Correct Answer: True

Solution:

Young's modulus is relevant only for solids as it describes the relationship between stress and strain in materials that have definite shapes and lengths.

Correct Answer: False

Solution:

In physics, a material that stretches less under a given load is considered more elastic, as it requires more force to produce a small change in length.

Correct Answer: False

Solution:

The stress-strain curve is linear only in the region where Hooke's law is obeyed. Beyond this region, the curve can be non-linear.

Correct Answer: False

Solution:

Hydraulic stress is applicable to fluids and describes the stress exerted by a fluid on an object, leading to volume strain.

Correct Answer: True

Solution:

Elastomers are a class of solids that do not exhibit a linear relationship between stress and strain, and therefore do not obey Hooke's law.

Correct Answer: False

Solution:

Hooke's law is valid only for small deformations and not for all materials. Some materials do not exhibit a linear stress-strain relationship.

Correct Answer: True

Solution:

Materials with larger values of Young's modulus require larger forces to produce small changes in length, indicating they are more rigid.

Correct Answer: True

Solution:

A rigid body is generally considered to have a definite shape and size, but in reality, solid bodies are not perfectly rigid and can be deformed when a large enough force is applied.

Correct Answer: False

Solution:

The Young's modulus is relevant only for solids since only solids have lengths and shapes.

Correct Answer: True

Solution:

Elastic deformation allows a material to regain its original shape when the force is removed, whereas plastic deformation results in permanent change.

Correct Answer: False

Solution:

Plasticity refers to the property of a material to undergo permanent deformation without regaining its original shape.

Correct Answer: False

Solution:

Rigid bodies can be deformed when a sufficiently large external force is applied, as they are not perfectly rigid.

Correct Answer: True

Solution:

A large Young's modulus indicates that a material requires a large force to produce a small change in length, making it more elastic.

Correct Answer: False

Solution:

Plastic materials, such as putty or mud, do not regain their original shape after deformation and are permanently deformed.

Correct Answer: False

Solution:

Shearing stress occurs only in solids, as it involves deformation where one face of the solid moves sideways relative to the other.

Correct Answer: False

Solution:

Young's modulus is relevant only for solids as it describes the relationship between stress and strain in materials that have a definite shape.

Correct Answer: False

Solution:

The bulk modulus is relevant for solids, liquids, and gases as it refers to the change in volume under uniform stress.

Correct Answer: True

Solution:

A solid has a definite shape and size, and a force is required to change or deform it.

Correct Answer: True

Solution:

Plastic materials, like putty or mud, get permanently deformed and do not regain their original shape.

Correct Answer: False

Solution:

The Young's modulus is relevant only for solids since only solids have lengths and shapes.

Correct Answer: True

Solution:

Understanding the elastic properties of materials like steel and concrete is essential for ensuring the structural integrity and safety of buildings and bridges.

Correct Answer: False

Solution:

A rigid body can be deformed when a sufficiently large external force is applied, indicating that no body is perfectly rigid.

Mechanical Properties of Solids

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Eight: Mechanical Properties of Solids

8.1 Introduction

8.2 Stress and Strain

8.3 Hooke's Law

8.4 Stress-Strain Curve

8.5 Elastic Moduli

8.6 Applications of Elastic Behaviour of Materials

8.7 Important Formulas

8.8 Points to Ponder

8.9 Exercises

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Exam Tips

AI Learning Assistant

Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.A copper cube of side 0.1 m is subjected to a uniform pressure increase of 5 x 10⁶ Pa. If the bulk modulus of copper is 140 GPa, what is the fractional change in volume?

Correct Answer: A

Correct Answer: A

Q3.A cylindrical column made of material Z has an inner radius of 0.2 m and an outer radius of 0.4 m. It supports a load of 100,000 N. Calculate the compressional stress in the column.

Correct Answer: A

Q4.If a cylinder is compressed along its length, which type of stress is applied?

Correct Answer: B

Q5.A material that does not return to its original shape after the removal of stress is said to have undergone what type of deformation?

Correct Answer: B

Q6.A material has a stress-strain curve that is linear up to a stress of 300 MPa and then becomes non-linear. If the Young's modulus of the material is 150 GPa, what is the strain at the proportional limit?

Correct Answer: A

Q7.A cylindrical rod is stretched by a force applied along its length. If the original length of the rod is LLL and the change in length is ΔL\Delta LΔL, what is the longitudinal strain in the rod?

Correct Answer: B

Q8.Which type of stress is associated with the change in volume of a solid under uniform pressure?

Correct Answer: C

Q9.A glass cube with a side length of 0.1 m is subjected to a uniform hydraulic pressure of 5 MPa. If the bulk modulus of glass is 35 GPa, what is the fractional change in volume?

Correct Answer: A

Q10.Which of the following is a correct statement about the bulk modulus?

Correct Answer: B

Q11.A solid sphere is placed in a fluid under high pressure. What type of stress is experienced by the sphere?

Correct Answer: D

Q12.Which material property is essential for designing a building to ensure it can withstand external forces?

Correct Answer: B

Q13.Which of the following statements is true regarding the stress-strain curve for a ductile material?

Correct Answer: A

Q14.What is the unit of stress in the SI system?

Correct Answer: B

Q15.Which of the following statements about shear stress is correct?

Correct Answer: B

Q16.A steel wire of length 4.7 m and cross-sectional area 3.0 \times 10^{-5} \text{ m}^2 stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 \times 10^{-5} \text{ m}^2 under a given load. What is the ratio of the Young's modulus of steel to that of copper?

Correct Answer: A

Q17.A metal rod is compressed by a force of 1000 N. If the rod has a cross-sectional area of 0.01 m² and the compressive stress is 50 MPa, what is the length of the rod?

Correct Answer: A

Q18.If a material has a yield strength of 250 MPa and is subjected to a tensile stress of 200 MPa, what is the nature of the deformation?

Correct Answer: A

Q19.Which of the following materials is likely to exhibit plastic deformation under stress?

Correct Answer: B

Q20.Which of the following is defined as the restoring force per unit area?

Correct Answer: B

Q21.A steel wire of length 2 m and cross-sectional area 2 x 10⁻⁶ m² is stretched under a load of 500 N. If the Young's modulus of steel is 2 x 10¹¹ N/m², what is the elongation of the wire?

Correct Answer: C

Q22.Which of the following statements is true about elasticity?

Correct Answer: A

Q23.A steel cable with a radius of 2 cm supports a load. If the maximum allowable stress is 150 MPa, what is the maximum load the cable can support?

Correct Answer: D

Q24.A cylindrical rod is stretched by a force applied along its length. What type of stress is experienced by the rod?

Correct Answer: A

Q25.A cylindrical rod made of steel and a rod made of brass, both of the same length and diameter, are subjected to the same tensile force. Which of the following statements is true regarding their elongations?

Correct Answer: B

Q26.Which of the following statements is true about Hooke's law?

Correct Answer: A

Q27.A steel beam is subjected to a shear force causing a displacement of 0.005 m. If the shear modulus of steel is 80 GPa and the original height of the beam is 2 m, what is the shear stress applied?

Correct Answer: B

Q28.A cylindrical rod made of brass is compressed by a force of 2000 N. If the rod has a cross-sectional area of 0.01 m² and the compressive stress is 100 MPa, what is the length of the rod?

Correct Answer: B

Q29.A steel wire of length 4.7 m and cross-sectional area 3.0 \times 10^{-5} \text{ m}^2 stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 \times 10^{-5} \text{ m}^2 under a given load. What is the ratio of the Young's modulus of steel to that of copper?

Correct Answer: A

Q30.Which of the following materials is likely to have the highest bulk modulus?

Correct Answer: C