Elasticity

SYLLABUS

Hooke’s law: Force constant, Stress; Strain; Elasticity and plasticity, Elastic modulus: Young modulus, bulk modulus, shear modulus, Poisson’s ratio, Elastic potential energy.

What is This Chapter About?

Press your finger against a rubber band and it deforms. Release it and it springs back. Stretch a steel wire and it elongates slightly. Squeeze a block of rubber and its volume decreases. These everyday observations point to a fundamental property of solid materials — their ability to deform under applied forces and recover their original shape when those forces are removed. This property is called elasticity.

Chapter 8 of NEB Grade 11 Physics explores the science of how solid materials respond to forces that try to change their shape or size. It introduces the concepts of stress, strain, and elastic moduli — the quantitative tools engineers use every day to design bridges, buildings, aircraft, and medical implants. Carrying 5 teaching hours, this chapter is compact but rich in both conceptual depth and practical relevance.

Key Concepts You Will Learn

Hooke’s Law

The foundation of elasticity is Hooke’s Law, which states that within the elastic limit, the extension of a spring (or deformation of any elastic body) is directly proportional to the applied force: [latex]F = kx[/latex], where k is the spring constant or force constant of the spring. The elastic limit is the maximum stress beyond which the material does not return to its original shape — it becomes permanently deformed.

Hooke’s Law applies not just to springs but to any elastic material within its elastic range. Beyond the elastic limit, materials enter the plastic region where permanent deformation occurs. The force-extension graph is linear up to the elastic limit, and the area under this graph gives the elastic potential energy stored:

[latex]E = \frac{1}{2}kx^2[/latex]

.

Stress and Strain

When a force is applied to a material, it creates an internal stress — the force per unit area acting within the material: [latex]\sigma = \frac{F}{A}[/latex]. Stress is measured in pascals (Pa) or N/m².

The resulting deformation is quantified by strain — the fractional change in dimension caused by the stress. Strain is dimensionless. Depending on the type of deformation, strain takes different forms: longitudinal strain is

[latex]\frac{\Delta l}{l}[/latex]

, volumetric strain is [latex]\frac{\Delta V}{V}[/latex], and shear strain is the angular deformation produced.

Elasticity and Plasticity

A material is elastic if it returns to its original shape after the deforming force is removed. It is plastic if it retains its deformed shape permanently. Most real materials are elastic up to a limit and plastic beyond it. Rubber is highly elastic over a large range of strain. Clay is highly plastic. Steel is elastic for small strains but plastic for large ones.

Elastic Moduli

The ratio of stress to strain within the elastic limit is a constant called the elastic modulus — it characterises the stiffness of a material. There are three types depending on the kind of deformation:

Young’s Modulus (Y) relates longitudinal stress to longitudinal strain:

[latex]Y = \frac{\sigma}{\epsilon} = \frac{Fl}{A\Delta l}[/latex]

. It measures the resistance of a material to being stretched or compressed along its length. Steel has a very high Young’s modulus; rubber has a very low one.

Bulk Modulus (B) relates volumetric stress (pressure) to volumetric strain:

[latex]B = -\frac{P}{\Delta V / V}[/latex]

. It measures resistance to uniform compression. Liquids and gases have bulk moduli but no Young’s modulus (they cannot sustain tensile stress). The negative sign ensures B is positive since volume decreases when pressure increases.

Shear Modulus (G) relates shear stress to shear strain. It measures resistance to shape change without volume change — relevant when layers of a material slide relative to each other.

Poisson’s Ratio

When a material is stretched longitudinally, it contracts in the transverse (lateral) direction simultaneously. Poisson’s ratio ([latex]\nu[/latex]) is defined as the ratio of lateral strain to longitudinal strain:

[latex]\nu = -\frac{\text{lateral strain}}{\text{longitudinal strain}}[/latex]

. For most materials, Poisson’s ratio lies between 0 and 0.5. Rubber has a Poisson’s ratio close to 0.5 (nearly incompressible), while cork has a ratio near 0 (does not expand laterally when compressed — which is why it works so well as a bottle stopper).

Elastic Potential Energy in a Stretched Wire

When a wire is stretched by a force, work is done against the internal restoring forces of the material, and this work is stored as elastic potential energy. The energy stored per unit volume of the wire is:

[latex]U = \frac{1}{2} \times \text{stress} \times \text{strain}[/latex]

. This result, derived from first principles using Hooke’s Law, has important applications in engineering — springs in machines, shock absorbers in vehicles, and tendons in the human body all store and release elastic energy.

Why This Chapter Matters for NEB Exams

Elasticity is examined consistently in NEB Grade 11 Physics. Definitions and distinctions between stress, strain, and the three elastic moduli are standard short-answer questions. Derivation of elastic potential energy stored in a stretched wire is a frequent long-answer question. Numerical problems involving Young’s modulus — finding extension, stress, or strain given material properties — appear in almost every paper. The distinction between elastic and plastic behaviour and the significance of Poisson’s ratio are common conceptual questions.

Important Topics to Focus On

• Statement of Hooke’s Law and its limitations
• Definitions of stress and strain with their SI units
• Distinction between elasticity and plasticity with examples
• Young’s modulus — definition, formula, and numerical problems
• Bulk modulus — definition, formula, and physical significance
• Shear modulus — definition and physical meaning
• Poisson’s ratio — definition, typical values, and significance
• Derivation of elastic potential energy stored per unit volume:

[latex]U = \frac{1}{2} \times \text{stress} \times \text{strain}[/latex]

Stress-strain graph — elastic limit, yield point, breaking point Numerical problems on Young’s modulus and elastic energy

Connection to Other Chapters

Elasticity connects backward to Chapter 4 (Dynamics) through the concept of restoring force, and to Chapter 5 (Work, Energy and Power) through elastic potential energy. The spring constant k from Hooke’s Law reappears directly in Grade 12 Chapter 2 (Periodic Motion), where the period of a mass-spring system is derived. The bulk modulus of a gas connects to Chapter 7 of Grade 12 (Mechanical Waves) — it is the elastic property that determines the speed of sound. The concept of elastic potential energy stored in materials is applied throughout engineering physics and materials science.