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Hooke's Law and Elasticity

Hooke's law states that within the elastic limit, the stress developed in a body is directly proportional to the strain produced. For a simple spring, it establishes a linear relation between restoring force and displacement from equilibrium.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of stress, strain, elastic limit, Young's modulus, bulk modulus, modulus of rigidity, and spring constant under simple one-dimensional loading.

PGT extension

Energy stored in stretched strings and springs, tensor treatment of elasticity for complex deformations, interrelations between elastic constants, and dynamic behavior involving oscillations under central/restoring forces.

Prerequisites

Newton's Laws of Motion
Work, Energy, and Power
Simple Harmonic Motion

Fundamentals of Hooke's Law

States that extension (or compression) x is directly proportional to the applied load F within the proportional limit.
Mathematically expressed as $F = -kx$, where k is the stiffness constant or spring constant.
The negative sign indicates that the restoring force acts in the direction opposite to the displacement.

Stress-Strain Relationship

Normal stress (sigma) is defined as restoring force per unit area, measured in N/m² or Pascal.
Longitudinal strain (epsilon) is the fractional change in length (Δl / l).
Within elastic limits, Hooke's law is written as Stres$s = E \times Strain$, where E is the relevant modulus of elasticity.

Elastic Moduli

Young's modulus (Y) relates longitudinal stress to longitudinal strain for solid wires and rods.
Bulk modulus (B) relates hydrostatic pressure to volumetric strain, applicable to fluids and solids alike.
Modulus of rigidity (η) relates shearing stress to shearing strain, governing shape deformation.

Energy Stored in an Elastic Medium

Work done in stretching an elastic spring is stored as potential energy within the material.
Elastic potential energy density is proportional to the square of the strain multiplied by the elastic modulus.
Total potential energy $U = 1/2 kx^{2}$ for a linearly elastic spring.

Derivations

Potential Energy of a Stretched Spring

Conditions: Spring obeys Hooke's law up to extension x

Consider an external force $F_ext = kx required$ to stretch a spring slowly by an infinitesimal displacement dx.
The elemental work done dW is given by d$W = F_{ext} dx = kx dx$.
Integrate the work done from zero extension to a final extension x: $W = \int_{0}^{x} kx dx = \frac{1}{2}kx^2$.
By conservation of energy, this work appears as elastic potential energy U.

Result: $U = \frac{1}{2}kx^2$

Work Done in Stretching a Wire

Conditions: Wire of initial length L and cross-sectional area A stretched by length ΔL

From Hooke's law, tension T in the wire for an elongation l is given by T = \frac{YA}{L} l.
The small work done to further elongate the wire by dl is dW = T dl = \frac{YA}{L} l dl.
Integrating from l = 0 to $l = \Delta L gives W = \int_{0}^{\Delta L} \frac{YA}{L} l dl = \frac{YA}{2L} (\Delta L)^2$.
Expressing in terms of stress and strain: W = \frac{1}{2} \times (Stress) \times (Strain) \times (Volume).

Result: $W = \frac{1}{2} \frac{YA (\Delta L)^2}{L}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Hooke's Law (Linear Form)	$$F = -kx$$	F = Restoring force, k = Spring constant, x = Displacement N, N/m, m	Within elastic limit
Young's Modulus	$$Y = \frac{FL}{A \Delta L}$$	F = Applied force, L = Original length, $A = Cross-sectional area$, ΔL = Change in length N/m² or Pa	Tensile or compressive stress
Bulk Modulus	$$B = -V \frac{\Delta P}{\Delta V}$$	V = Original volume, ΔP = Change in pressure, ΔV = Change in volume N/m² or Pa	Uniform volumetric compression
Modulus of Rigidity	$$\eta = \frac{F/A}{\theta}$$	F/A = Shearing stress, $\theta = Angle of shear$ N/m² or Pa	Shape deformation without volume change
Elastic Potential Energy (Spring)	$$U = \frac{1}{2} kx^2$$	k = Spring constant, x = Extension Joule (J)	Conservative elastic deformation
Potential Energy Density	$$u = \frac{1}{2} (Stress) \times (Strain)$$	u = Energy per unit volume J/m³	Linear elastic regime
Compressibility	$$K = \frac{1}{B}$$	K = Compressibility, B = Bulk modulus m²/N or Pa&sup{-1}	Isothermal or adiabatic volumetric changes
Speed of Transverse Wave in Stretched String	$$v = \sqrt{\frac{T}{\mu}}$$	T = Tension (elastic force), $\mu = Mass per unit length$ m/s	Uniform linear mass density

Graphs and experiments

Stress-Strain Curve for a Ductile Metal

Draw: A graph with Strain on the x-axis and Stress on the y-axis showing proportional limit, yield point, ultimate tensile strength, and fracture point.

Interpret: Region up to proportional limit follows Hooke's law. Beyond yield point, permanent plastic deformation occurs.

Exam use: Direct questions identifying yield strength, breaking stress, and regions of permanent set.

Load vs. Elongation Graph for a Spring

Draw: A straight line passing through the origin with Load (F) on y-axis and Elongation (x) on x-axis.

Interpret: The slope of the straight line directly yields the spring constant k.

Exam use: Numerical data interpretation to compute stored potential energy or stiffness.

Searle's Apparatus Experiment Setup

Draw: Two parallel wires (experimental and reference) suspended from a rigid support, carrying a spirit level and micrometer screw gauge.

Interpret: Used to determine Young's modulus of a wire by precisely measuring elongation against added mass while compensating for thermal expansion.

Exam use: Questions on least count, error propagation, and formula identification for Y.

Solved numericals

TGT core

A spring of force constant 200 N/m is compressed by 5 cm. Calculate the elastic potential energy stored in the spring.

Identify given values: $k = 200 N/m$, x = 5 cm = 0.05 m.
Apply the formula for potential energy: $U = 1/2 kx^{2}$.
Substitute values: $U = 0.5 \times 200 \times (0.05)^{2}$.
Calculate result: $U = 100 \times 0.0025 = 0.25 J$.

Answer: 0.25 J

PGT extension

A steel wire of length 2 m and cross-sectional area 1 mm² is suspended vertically. When a mass of 10 kg is attached to its lower end, find the elongation produced. (Given $Y_steel = 2 \times 10^{11} N/m^{2}$, $g = 9.8 m/s^{2})$

Identify parameters: L = 2 m, $A = 1 mm^{2} = 10^{-6} m^{2}$, M = 10 kg, $g = 9.8 m/s^{2}$, $Y = 2 \times 10^{11} N/m^{2}$.
Force $F = Mg = 10 \times 9.8 = 98 N$.
Rearrange Young's modulus formula for elongation: Δ$L = FL / (YA)$.
Substitute values: Δ$L = (98 \times 2) / (2 \times 10^{11} \times 10^{-6})$.
Calculate: Δ$L = 196 / (2 \times 10^5) = 9.8 \times 10^{-4} m = 0.98 mm$.

Answer: 9.8 × 10^{-4} m (0.98 mm)

PGT extension

Two identical springs of spring constant k are connected in parallel and then in series. Find the ratio of their effective spring constants (k_{parallel} / k_{series}).

For springs in parallel: $k_{parallel} = k + k = 2k$.
For springs in series: 1 / $k_{series} = 1/k + 1/k = 2/k$, so $k_{series} = k / 2$.
Take the ratio: k_{parallel} / $k_{series} = 2k / (k / 2) = 4$.

Answer: 4

Assumptions and limitations

Hooke's law is an empirical approximation valid only within the proportional (elastic) limit of a material.
Assumes homogeneous, isotropic materials when defining macroscopic elastic constants like Young's and Bulk moduli.
Fails under large deformations where plastic flow, permanent set, or material rupture occurs.

Common misconceptions

Common error: Hooke's law is a universal law applicable to all materials under any load.

Correct: Hooke's law holds true only up to the proportional limit for elastic deformations.

Beyond the elastic limit, stress is no longer proportional to strain, and permanent deformation ensues.

Common error: Cutting a spring in half halves its spring constant.

Correct: Cutting a spring into half doubles its spring constant.

Spring constant is inversely proportional to its length (k ∝ 1/L); shorter springs are stiffer.

Common error: Modulus of elasticity has dimensions of force.

Correct: Modulus of elasticity has dimensions of pressure or stress (ML^{-1}T^{-2}).

It is the ratio of stress to dimensionless strain.

Quick revision

Hooke's law: $F = -kx or Stress = E \times Strain$.
Young's modulus governs longitudinal strain, Bulk modulus governs volumetric strain, and Rigidity modulus governs shear.
Elastic potential energy stored in a spring is $U = 1/2 kx^{2}$.
Spring constant in series combination: 1/$k_{eq} = \sum(1/k_i); in parallel: k_{eq} = \sum(k_i)$.

Must remember

Hooke's law states that extension or compression is directly proportional to the applied load within the proportional limit, expressed mathematically as $F = -kx$.
The negative sign in $F = -kx$ indicates that the restoring force acts in the direction opposite to the displacement.
Young's modulus ($Y$) relates longitudinal stress to longitudinal strain for solid wires and rods: $Y = \frac{FL}{A \Delta L}$.
Bulk modulus ($B$) relates hydrostatic pressure to volumetric strain: $B = -V \frac{\Delta P}{\Delta V}$, and compressibility is $K = \frac{1}{B}$.
Modulus of rigidity ($\eta$) governs shape deformation without volume change: $\eta = \frac{F/A}{\theta}$.
Elastic potential energy stored in a linearly elastic spring is given by $U = \frac{1}{2} kx^2$, and potential energy density is $u = \frac{1}{2} (Stress) \times (Strain)$.
Cutting a spring of length $L$ and spring constant $k$ into two equal halves doubles its spring constant to $2k$, since $k \propto \frac{1}{L}$.
The dimensions of any modulus of elasticity are $[ML^{-1}T^{-2}]$, which is identical to the dimensions of stress and pressure.

Glossary

Hooke's LawThe linear relationship between stress and strain within the proportional limit of an elastic material, $F = -kx$.

Young's Modulus (Y)The elastic modulus governing longitudinal tensile or compressive stress and strain in solid materials.

Bulk Modulus (B)The elastic modulus governing volumetric compression under uniform hydrostatic pressure.

Modulus of Rigidity (\eta)The elastic modulus governing shearing stress and shape deformation without volume change.

Spring Constant (k)A measure of the stiffness of a spring, representing the restoring force per unit displacement ($N/m$).

Compressibility (K)The reciprocal of the bulk modulus, representing the fractional change in volume per unit increase in pressure.

Comparison of Elastic Moduli

Modulus	Symbol	Deformation Type	Governing Equation
Young's Modulus	$Y$	Longitudinal (Length change)	$Y = \frac{FL}{A \Delta L}$
Bulk Modulus	$B$	Volumetric (Volume change)	$B = -V \frac{\Delta P}{\Delta V}$
Modulus of Rigidity	$\eta$	Shear (Shape/angle change)	$\eta = \frac{F/A}{\theta}$

Practice Quiz

Choose an option to reveal the explanation.

1. What does the negative sign in Hooke's law expression $F = -kx$ signify?

Explanation: The negative sign indicates that the restoring force is directed opposite to the displacement vector, returning the system to equilibrium.

2. If a spring of spring constant $k$ is cut into two equal halves, what is the spring constant of each new piece?

Explanation: Spring constant is inversely proportional to length ($k \propto 1/L$). Halving the length doubles the spring constant to $2k$.

3. What are the dimensional formula and SI units of the modulus of elasticity?

Explanation: Modulus of elasticity is stress divided by strain (dimensionless). Thus its dimensions are $[ML^{-1}T^{-2}]$ and its unit is Pascal or $N/m^2$.

4. Which elastic modulus is applicable to fluids (liquids and gases)?

Explanation: Fluids do not sustain shearing stress or tensile stress permanently, but they resist uniform compression, making the bulk modulus applicable.

5. The elastic potential energy stored in a stretched spring of constant $k$ for an extension $x$ is given by:

Explanation: Integrating the restorative force $F = kx$ with respect to displacement gives the work done, $U = \frac{1}{2} kx^2$.

6. What is compressibility defined as in relation to elastic moduli?

Explanation: Compressibility $K$ is defined as the inverse of the bulk modulus ($K = 1/B$). It measures how easily a substance can be compressed.

7. Within what limit does Hooke's law hold true for typical structural materials?

Explanation: Hooke's law is valid strictly within the proportional limit, where stress is linearly proportional to strain.

8. How is elastic potential energy density $u$ expressed in terms of stress and strain within the linear elastic regime?

Explanation: Potential energy density (energy per unit volume) is given by the area under the stress-strain curve, which evaluates to $\frac{1}{2} \times \text{Stress} \times \text{Strain}$.

9. The speed of a transverse wave in a stretched string depends on tension $T$ and mass per unit length $\mu$ as:

Explanation: Dimensional analysis and wave equation derivation for stretched strings yield $v = \sqrt{T/\mu}$.

10. Modulus of rigidity ($\eta$) is defined as the ratio of:

Explanation: Modulus of rigidity relates shearing stress to shearing strain ($\eta = \frac{F/A}{\theta}$).

11. If two identical springs of constant $k$ are connected in series, the effective spring constant $k_{eff}$ is:

Explanation: For springs in series, $\frac{1}{k_{eff}} = \frac{1}{k} + \frac{1}{k}$, which yields $k_{eff} = k/2$.

12. If two identical springs of constant $k$ are connected in parallel, the effective spring constant $k_{eff}$ is:

Explanation: For springs in parallel, their stiffnesses add up directly, giving $k_{eff} = k + k = 2k$.

13. Which of the following statements regarding permanent deformation is correct?

Explanation: Exceeding the elastic limit causes permanent (plastic) deformation, meaning the material does not return to its original dimensions when unloaded.

14. What happens to the longitudinal strain of a wire if its length is doubled under the same tensile force, assuming elastic behavior?

Explanation: Longitudinal strain is $\Delta L / L$. If original length $L$ and extension $\Delta L$ both scale proportionally under identical stress, fractional strain remains constant for a given material and stress.

15. What is the primary difference between longitudinal stress and shearing stress?

Explanation: Normal/longitudinal stress acts perpendicularly to the surface area, whereas shearing stress acts tangentially or parallel to the surface.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A wire of length $L$ and cross-sectional area $A$ is stretched by a force $F$. If Young's modulus of the material is $Y$, the work done in stretching the wire by an amount $\Delta L$ is:

Explanation: Work done $W = \frac{1}{2} \times \text{Force} \times \text{Extension} = \frac{1}{2} \left(\frac{Y A \Delta L}{L}\right) \Delta L = \frac{YA (\Delta L)^2}{2L}$.

2. The bulk modulus of an ideal gas at constant temperature (isothermal process) is numerically equal to:

Explanation: For an isothermal process, $PV = \text{constant}$, differentiating gives $P \Delta V + V \Delta P = 0 \implies -V \frac{\Delta P}{\Delta V} = P$, so $B = P$.

3. Two wires A and B are made of the same material and have equal lengths. The radius of wire A is twice that of wire B. If they are stretched by the same load, the ratio of their elongations ($\Delta L_A / \Delta L_B$) is:

Explanation: Since $\Delta L = \frac{FL}{A Y}$ and $A = \pi r^2$, elongation is inversely proportional to $r^2$. Thus $\frac{\Delta L_A}{\Delta L_B} = \frac{r_B^2}{r_A^2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.

4. A spring with spring constant $k$ is stretched by distance $x$. The potential energy stored is $U$. If it is further stretched by an additional distance $x$, the total potential energy stored becomes:

Explanation: Initial extension is $x$ ($U = \frac{1}{2}kx^2$). Total extension becomes $2x$. Total potential energy $U_{total} = \frac{1}{2}k(2x)^2 = 4 \left(\frac{1}{2}kx^2\right) = 4U$.

5. Which of the following physical quantities has dimensions different from the other three?

Explanation: Stress, Young's modulus, and Bulk modulus all have dimensions of pressure $[ML^{-1}T^{-2}]$. Strain is a dimensionless quantity.

6. The fractional change in volume of a glass slab under a uniform hydrostatic pressure $\Delta P$ is given by (where $B$ is bulk modulus):

Explanation: Bulk modulus is $B = -V \frac{\Delta P}{\Delta V}$, which rearranges to fractional volume change $\frac{\Delta V}{V} = -\frac{\Delta P}{B}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the main physical significance of the elastic limit in material science?

Explanation: The elastic limit is the boundary up to which a material recovers its original dimensions completely upon load removal; beyond it, permanent deformation sets in.

2. Calculate the force required to stretch a spring of spring constant $200\text{ N/m}$ by $5\text{ cm}$ within its elastic limit.

Explanation: Using $F = kx$, where $k = 200\text{ N/m}$ and $x = 0.05\text{ m}$, we get $F = 200 \times 0.05 = 10\text{ N}$.

3. For an adiabatic process in an ideal gas, what is its bulk modulus $B_{ad}$ equal to?

Explanation: For an adiabatic process, $PV^\gamma = \text{constant}$, differentiating yields $B_{ad} = \gamma P$, where $\gamma$ is the adiabatic index.

4. If a solid metal cube experiences hydraulic compression on all faces, which elastic modulus is directly involved?

Explanation: Hydraulic compression involves uniform volume change under hydrostatic pressure, which is governed by the bulk modulus.

5. What is the elastic potential energy stored in a spring of constant $100\text{ N/m}$ when extended by $0.2\text{ m}$?

Explanation: Using $U = \frac{1}{2} kx^2 = \frac{1}{2} (100) (0.2)^2 = 50 \times 0.04 = 2\text{ J}$.

6. Why does a spring become stiffer when it is cut into shorter pieces?

Explanation: Each segment experiences a larger fraction of stress for a given applied load because fewer bonds share the stretch, resulting in a higher spring constant ($k \propto 1/L$).

7. Which of the following correctly pairs an elastic modulus with its primary deformation type?

Explanation: Modulus of rigidity governs shearing deformation (change in shape without change in volume).

8. A uniform wire of length $L$ and radius $r$ is subjected to tensile force $F$. The extension produced is $\Delta L$. If the radius of the wire is halved ($r/2$) while keeping everything else same, the extension becomes:

Explanation: Since $\Delta L = \frac{FL}{\pi r^2 Y}$, extension is inversely proportional to $r^2$. Halving the radius increases the extension by a factor of $2^2 = 4$.

9. What is the dimensional formula for compressibility ($K$)?

Explanation: Compressibility is $K = 1/B$. Since bulk modulus $B$ has dimensions $[ML^{-1}T^{-2}]$, compressibility has dimensions $[M^{-1}LT^2]$ (inverse of pressure).

10. In the context of the stress-strain curve, what represents the yield point?

Explanation: The yield point marks the transition from elastic behavior to plastic behavior, where permanent deformation occurs upon load removal.