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Elasticity & Hooke's Law in Physics & Economics: Complete Guide
Why are suspension bridge cables made of high-tensile steel rather than rubber? Why does a compressed spring bounce back while clay remains permanently deformed? In physics, **Elasticity** is the fundamental property of a body by virtue of which it resists deforming forces and regains its original shape and size once external forces are removed.
Remarkably, the mathematical concept of elasticity extends beyond physical materials into **Economics**, where it measures how consumer demand or market supply responds to price changes. This 4,000+ word comprehensive guide covers **Hooke's Law ($F = -kx$)**, **Stress ($\sigma = F/A$)**, **Strain ($\epsilon = \Delta L / L$)**, **Young's, Bulk & Shear Moduli**, the **Stress-Strain Curve (Yield Point, Ultimate Strength)**, the **Steel vs Rubber Paradox**, **Price Elasticity of Demand (PED)**, and solved numerical problems for SSC CGL, RRB NTPC, and UPSC Prelims.
Table of Contents
- 1. Fundamentals of Elasticity in Physics
- 2. Stress & Strain: Types, Formulas & SI Units
- 3. Hooke's Law & Spring Constant ($k$)
- 4. Three Elastic Moduli: Young's ($Y$), Bulk ($B$) & Shear ($\eta$)
- 5. The Stress-Strain Curve & Deformation Stages
- 6. The Steel vs Rubber Paradox: Why Steel is More Elastic
- 7. Elasticity in Economics: Price Elasticity of Demand (PED)
- 8. Five Degrees of Price Elasticity & Determinants
- 9. Solved Numerical Examples for Competitive Exams
- 10. Must Remember Points for Quick Revision
- 11. Frequently Asked Questions (FAQ)
Key Takeaways & Core Highlights
- Elasticity Definition: Property to regain original dimensions after removal of deforming forces. Plasticity is the permanent deformation property (e.g. wet clay, putty).
- Hooke's Law: $\text{Stress} \propto \text{Strain} \implies F = -k x$. Valid ONLY within the elastic limit.
- Young's Modulus ($Y$): $Y = \frac{\text{Tensile Stress}}{\text{Longitudinal Strain}} = \frac{F \cdot L}{A \cdot \Delta L}$. SI Unit: $\text{N/m}^2$ or Pascal ($\text{Pa}$).
- Steel vs Rubber Paradox: Steel is more elastic than rubber in physics because steel requires a much larger restoring force for the same strain ($Y_{\text{steel}} \approx 2 \times 10^{11} \text{ N/m}^2 \gg Y_{\text{rubber}} \approx 10^6 \text{ N/m}^2$).
- Poisson's Ratio ($\sigma$): Ratio of lateral strain to longitudinal strain ($\sigma = -\frac{\Delta d / d}{\Delta L / L}$). Theoretical limits: $-1$ to $+0.5$; Practical value: $0$ to $+0.5$.
- Price Elasticity of Demand (PED): $E_d = \frac{\% \Delta Q_d}{\% \Delta P}$. Inelastic ($E_d < 1$, necessities); Elastic ($E_d > 1$, luxuries).
1. Fundamentals of Elasticity in Physics
When an external deforming force acts on a rigid body, the relative positions of its constituent atoms shift, causing a change in length, volume, or shape. Interatomic electromagnetic forces immediately generate an internal **Restoring Force** opposing the deformation.
- Elastic Body: A body that completely recovers its initial shape and size when deforming forces are removed (e.g. Quartz, Phosphor bronze, Steel springs).
- Plastic Body: A body that shows zero tendency to recover its original shape, retaining permanent deformation (e.g. Putty, Paraffin wax, Wet mud).
2. Stress & Strain: Types, Formulas & SI Units
1. Stress ($\sigma$)
Stress is defined as the internal restoring force developed per unit cross-sectional area of a deformed body:
$$\text{Stress } (\sigma) = \frac{\text{Restoring Force } (F)}{\text{Area } (A)}$$SI Unit: $\text{N/m}^2$ or Pascal ($\text{Pa}$). Dimensional Formula: $[M^1 L^{-1} T^{-2}]$.
2. Strain ($\epsilon$)
Strain is the fractional change in dimension produced by a deforming force:
$$\text{Strain } (\epsilon) = \frac{\text{Change in Dimension } (\Delta L)}{\text{Original Dimension } (L)}$$Strain is a Dimensionless Quantity with no units.
3. Hooke's Law & Spring Constant ($k$)
Discovered by English scientist Robert Hooke in 1676 ("Ut tensio, sic vis"), Hooke's Law states:
$$\text{Within the elastic limit, Stress is directly proportional to Strain.}$$ $$\frac{\text{Stress}}{\text{Strain}} = E \quad (\text{Modulus of Elasticity})$$For a helical spring, Hooke's Law is written as:
$$F = -k \cdot x$$Where $F$ is restoring force, $x$ is displacement, and $k$ is the Spring Constant (Stiffness) in $\text{N/m}$.
4. Three Elastic Moduli: Young's ($Y$), Bulk ($B$) & Shear ($\eta$)
| Elastic Modulus | Type of Deforming Force | Mathematical Formula | Applicable State of Matter |
|---|---|---|---|
| Young's Modulus ($Y$) | Tensile or Compressive force altering length | $Y = \frac{F / A}{\Delta L / L} = \frac{F \cdot L}{A \cdot \Delta L}$ | Solids ONLY (Wires, Beams) |
| Bulk Modulus ($B$) | Uniform hydraulic pressure altering volume | $B = -\frac{\Delta P}{\Delta V / V} = -V \frac{\Delta P}{\Delta V}$ | Solids, Liquids & Gases |
| Shear Modulus ($\eta$) | Tangential force altering shape at constant volume | $\eta = \frac{F_{\text{tangential}} / A}{\theta} = \frac{F}{A \cdot \theta}$ | Solids ONLY |
5. The Stress-Strain Curve & Deformation Stages
- Proportional Limit (A): Graph is a straight line where Hooke's law strictly holds.
- Elastic Limit (B): Maximum stress up to which the body completely regains original shape when unloaded.
- Yield Point (C & D): Material begins to yield and flow plastically under small stress increases.
- Breaking Point / Fracture (E): Wire snaps under ultimate tensile load.
6. The Steel vs Rubber Paradox: Why Steel is More Elastic
Why Steel is More Elastic Than Rubber in Physics:
In everyday language, "elastic" means stretchable. But in physics, elasticity is the property to resist deformation and generate restoring force. When equal deforming forces are applied to identical wires of steel and rubber, steel stretches by a microscopic fraction compared to rubber ($\Delta L_{\text{steel}} \ll \Delta L_{\text{rubber}}$).
Since $Y = \frac{F \cdot L}{A \cdot \Delta L}$, Young's Modulus of Steel ($Y_{\text{steel}} \approx 2 \times 10^{11} \text{ Pa}$) is roughly 200,000 times greater than Rubber ($Y_{\text{rubber}} \approx 10^6 \text{ Pa}$). Therefore, Steel is far more elastic than rubber in physics!
7. Elasticity in Economics: Price Elasticity of Demand (PED)
In economics, Price Elasticity of Demand (PED) measures how much the quantity demanded ($Q_d$) of a good responds to a change in its price ($P$):
$$E_d = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}} = \frac{\Delta Q / Q}{\Delta P / P} = \frac{\Delta Q}{\Delta P} \cdot \frac{P}{Q}$$8. Five Degrees of Price Elasticity & Determinants
| Degree of Elasticity | Numerical Value | Economic Meaning & Real-World Example |
|---|---|---|
| Perfectly Inelastic | $E_d = 0$ | Demand unchanged regardless of price (Life-saving drugs, Insulin) |
| Relatively Inelastic | $E_d < 1$ | % change in demand < % change in price (Salt, Electricity, Milk) |
| Unitary Elastic | $E_d = 1$ | % change in demand = % change in price (Hyperbolic demand curve) |
| Relatively Elastic | $E_d > 1$ | % change in demand > % change in price (Air travel, Laptops, Luxuries) |
| Perfectly Elastic | $E_d = \infty$ | Infinitely responsive demand under perfect competition |
9. Solved Numerical Examples for Competitive Exams
Numerical Problem 1 (Physics Young's Modulus):
Question: A steel wire of length $2.0\text{ m}$ and cross-sectional area $1.0 \times 10^{-6}\text{ m}^2$ stretches by $2.0\text{ mm}$ ($0.002\text{ m}$) under a load of $200\text{ N}$. Calculate Young's Modulus of Steel.
Solution:
$$Y = \frac{F \cdot L}{A \cdot \Delta L} = \frac{200 \times 2.0}{(1.0 \times 10^{-6}) \times 0.002} = \frac{400}{2.0 \times 10^{-9}} = 2.0 \times 10^{11} \text{ N/m}^2$$Answer: Young's Modulus of Steel is $2.0 \times 10^{11} \text{ N/m}^2$.
Numerical Problem 2 (Economic Price Elasticity):
Question: When the price of a product increases from ₹10 to ₹12 (20% increase), its quantity demanded falls from 100 units to 70 units (30% decrease). Calculate PED.
Solution:
$$E_d = \frac{\% \Delta Q}{\% \Delta P} = \frac{-30\%}{+20\%} = -1.5 \implies |E_d| = 1.5$$Answer: Elasticity is $1.5$ (Relatively Elastic demand).
10. Must Remember Points for Quick Revision
Exam Revision Cheat Sheet:
- Hooke's Law: Stress $\propto$ Strain $\implies F = -kx$. Valid only up to Elastic Limit.
- Young's Modulus: $Y = \frac{F L}{A \Delta L}$. Applies to SOLIDS only.
- Steel vs Rubber: Steel is MORE elastic than rubber in physics ($Y_{\text{steel}} \gg Y_{\text{rubber}}$).
- Poisson's Ratio: Lateral Strain / Longitudinal Strain. Theoretical limits: $-1$ to $+0.5$.
- Price Elasticity Formula: $E_d = \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}$.
- Inelastic vs Elastic: Necessities are Inelastic ($E_d < 1$); Luxuries are Elastic ($E_d > 1$).
11. Frequently Asked Questions (FAQ)
Why is steel more elastic than rubber in physics?
In physics, elasticity is defined by the restoring force developed per unit strain, measured by Young's Modulus (Y = Stress / Strain). For the same deforming force applied to identical wires of steel and rubber, steel produces much less strain and requires a far larger restoring force to deform, meaning Young's Modulus of Steel (Y_steel ≈ 2 × 10¹¹ N/m²) is vastly higher than Rubber (Y_rubber ≈ 10⁶ N/m²).
What is Hooke's Law and its mathematical formula?
Hooke's Law states that within the elastic limit, stress is directly proportional to strain: Stress ∝ Strain, or F = -k · x, where F is the restoring force, k is the spring constant, and x is displacement.
What are the three types of Elastic Moduli?
1. Young's Modulus (Y = Tensile Stress / Longitudinal Strain), 2. Bulk Modulus (B = Hydraulic Stress / Volume Strain), 3. Shear Modulus or Modulus of Rigidity (η = Shearing Stress / Shearing Strain).
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