4 Most SAQ’s of Mechanical Properties of Solids Chapter in Inter 1st Year Physics (TS/AP)

4 Marks

SAQ-1 : Describe the behavior of a wire under gradually increasing load.

Introduction

Understanding the behavior of a wire under gradually increasing load is essential in materials science and engineering. This behavior highlights the wire’s response to stress and provides insights into its mechanical properties.

1. Elastic Region
   • In the elastic region, when a load is initially applied to the wire, it stretches in proportion to the force applied. This is in accordance with Hooke’s Law, which states that the extension of the wire is directly proportional to the load applied, as long as the elastic limit is not exceeded.
   • During this phase, the wire will return to its original length when the load is removed. The stress and strain are linearly related, and the wire exhibits elastic behavior.
2. Yield Point and Plastic Region
   • As the load increases, the wire reaches its yield point, where it begins to deform plastically. Beyond this point, permanent deformation occurs, and the wire will not return to its original length even when the load is removed.
   • In the plastic region, the wire continues to stretch without a significant increase in load. This behavior is known as plastic deformation. The wire becomes permanently elongated and weakened.
3. Necking and Fracture
   • Upon further increasing the load, the wire experiences necking, where a localized reduction in the cross-sectional area occurs. This typically marks the onset of impending failure.
   • Finally, if the load continues to increase, the wire reaches its breaking point and fractures. The point of fracture is where the material’s tensile strength is exceeded, leading to a catastrophic failure.

Summary

The behavior of a wire under gradually increasing load encompasses initial elastic deformation, followed by plastic deformation, necking, and ultimately fracture. Understanding this behavior is crucial for assessing the mechanical properties of materials and for applications where wires are subjected to various forces. This knowledge helps in determining the suitability of different materials for specific engineering applications.

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SAQ-2 : Define strain energy and derive the equation for the same.

Introduction

Strain Energy is the energy stored in a material when it undergoes deformation or strain due to an applied force. It is considered as potential energy because it is the energy stored by the material as it deforms.

Derivation of Strain Energy Equation

1. Initial Configuration: Initially, a rod is in its unstressed state with a length ‘L’ and a cross-sectional area ‘A’.
2. Applied Force: An external force ‘F’ is applied along the length of the rod, causing it to elongate.
3. Elongation and Strain:
   • The rod elongates by a small amount ‘ΔL’.
   • The strain (ε) produced is calculated as $$\epsilon = \frac{\Delta L}{L}$$
4. Stress and Young’s Modulus:
   • The stress (σ) is defined as the force per unit area, $$\sigma = \frac{F}{A}$$
   • Young’s Modulus (‘E’) is the ratio of stress to strain, $$E = \frac{\sigma}{\epsilon}$$
5. Work Done and Strain Energy:
   • Work done (W) due to elongation is W = F × ΔL.
   • This work is stored as strain energy (‘U’) in the rod.
6. Expression for Strain Energy: The strain energy (U) is given by $$U = \frac{1}{2} F \Delta L$$
7. Express Strain Energy in Terms of Stress and Strain: Rewriting the strain energy equation in terms of stress and strain gives $$U = \frac{1}{2} (\sigma A) (\frac{\Delta L}{L})$$

Summary

Strain energy is crucial in engineering and material science for designing structures capable of withstanding various loads. In the case of tensile deformation of a rod, the strain energy (‘U’) can be expressed as $$U = \frac{1}{2} (\text{stress} \times \text{cross-sectional area}) \times \text{strain}$$ This equation helps in understanding how materials store energy when deformed, which is essential for ensuring safety and efficiency in structural design.

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SAQ-3 : Define Stress and explain the types of stress.

Introduction

Stress is a measure of the internal resistance of a material to deformation under an applied external force. It quantifies the force per unit area of the material’s cross-section and is denoted by the symbol ‘σ’ (sigma). Stress is a critical concept in material science and engineering, essential for designing structures and understanding their mechanical behavior.

Types of Stress

1. Tensile Stress: Tensile stress occurs when a material is subjected to stretching or pulling forces, leading to elongation in the direction of the applied force. It is represented by a positive value of stress (‘σ’).
2. Compressive Stress: Compressive stress is experienced when a material faces compressive or squeezing forces, causing it to shrink or decrease in size along the direction of the force. This type of stress is represented by a negative value of stress (‘σ’).
3. Shear Stress: Shear stress arises when a material is subjected to parallel forces in opposite directions, leading to deformation by sliding one layer of particles over another. Common in structures like beams, shear stress is represented by stress (‘σ’).
4. Torsional Stress: Torsional stress occurs in materials under twisting or torsional forces, causing them to twist or deform along their axis. This type of stress is often encountered in shafts and springs and is also represented by stress (‘σ’).
5. Hydrostatic Stress: Hydrostatic stress is experienced when a material is subjected to equal forces acting in all directions, resulting in uniform volumetric deformation. It is represented by stress (‘σ’).
6. Thermal Stress: Thermal stress occurs due to changes in temperature, leading to thermal expansion or contraction of the material. The changes in dimensions due to temperature variations cause stress, represented by stress (‘σ’).

Summary

Stress is vital for understanding the internal resistance of materials to deformation under external forces. Different types of stress, including tensile stress, compressive stress, shear stress, torsional stress, hydrostatic stress, and thermal stress, affect materials in distinct ways. Knowledge of these stress types is crucial in engineering and material science for the safe and efficient design of structures and components.

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SAQ-4 : Explain the concept of Elastic potential Energy in a stretched wire and hence obtain the expression for it.

Introduction

Elastic Potential Energy refers to the energy stored in an object when it is stretched or compressed. In the context of a stretched wire, this energy is stored due to the wire’s deformation under an applied force. It is a key concept in understanding the behavior of materials under mechanical stress.

Concept of Elastic Potential Energy in a Stretched Wire

1. When a wire is stretched by an external force, it stores energy due to its elastic properties. This stored energy is the elastic potential energy.
2. The wire will return to its original shape when the force is removed, and the stored energy is released. This phenomenon follows Hooke’s Law for materials within their elastic limit.

Derivation of the Expression for Elastic Potential Energy

1. Initial State and Applied Force: Consider a wire initially at rest and unstressed. When an external force, F, is applied, it causes the wire to stretch.
2. Elongation and Hooke’s Law: Let the wire elongate by a distance, x. According to Hooke’s Law, the force applied is proportional to the elongation, represented as F = kx, where k is the spring constant or stiffness of the wire.
3. Work Done and Energy Stored:
   • The work done in stretching the wire, which is equal to the elastic potential energy stored, is given by the integral of the force over the displacement.
   • The work done (and hence, the elastic potential energy, U) is calculated as $$U = \int_0^x F \, dx$$
4. Expression for Elastic Potential Energy:
   • Substituting F = kx into the integral, we get $$U = \int_0^x kx \, dx$$
   • Solving this integral, the expression for elastic potential energy is $$U = \frac{1}{2} kx^2$$

Summary

The Elastic Potential Energy in a stretched wire is the energy stored due to its deformation under an applied force. This energy can be quantified by the expression $$U = \frac{1}{2} kx^2$$ where k is the spring constant of the wire, and x is the elongation. Understanding this concept is crucial in fields like material science and mechanical engineering, where the elastic behavior of materials plays a pivotal role in design and analysis.