12Elasticity

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ELASTICITY
▢ Types of Bodies:

Table 1: Rigid, Elastic and Plastic Bodies

Feature
Rigid body
Perfectly elastic body
Perfectly plastic body
Definition
External force produces no deformation
Completely regains original shape and size after removal of deforming force
Does not regain original shape and size after removal of deforming force
Examples
Diamond
Quartz, fibre, phosphor, bronze
Mud, wax, plastic
Most rigid
Diamond
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▢ Stress:
❖ Definition: Internal restoring force per unit area developed in a body due to external deforming force
❖ Nature: Tensor quantity
❖ Unit: \(N\ m^{-2}\)
❖ Dimension: \([ML^{-1}T^{-2}]\)

Table 1: Types of Stress

Type
Force direction
Effect
Formula
Normal stress
Perpendicular to surface
Change in length
\(\frac{F}{A}\)
Tangential / shearing stress
Parallel to surface
Change in shape
\(\frac{F}{A}\)
Volume stress
Normal force on total surface area
Change in volume
\(\frac{F}{A}\)
▢ Strain:
❖ Definition: Ratio of change in dimension to original dimension
❖ Nature: Dimensionless
❖ Unit: No unit

Table 1: Types of Strain

Type
Meaning
Formula
Longitudinal strain
Change in length per unit original length
\(\frac{\Delta l}{l}\)
Shearing strain
Angular deformation produced by tangential stress
\(\theta\)
Volumetric strain
Change in volume per unit original volume
\(\frac{\Delta V}{V}\)
▢ Hooke's Law:
❖ Statement 1: Within elastic limit, extension produced in wire is directly proportional to load applied
❖ Formula 1: \(e\propto F\)
❖ Spring Force: \(F=-ke\)
❖ Statement 2: Within elastic limit, stress is directly proportional to strain
❖ Formula 2: \(Stress\propto Strain\)
❖ Modulus of Elasticity: \(E=\frac{Stress}{Strain}=constant\)
❖ Negative Sign: Restoring force acts opposite to extension
▢ Modulus of Elasticity:

Table 1: Types of Elastic Modulus

Modulus
Related stress
Related strain
Formula
Property of
Young's modulus \((Y)\)
Normal stress
Longitudinal strain
\(Y=\frac{F/A}{\Delta l/l}=\frac{Fl}{A\Delta l}\)
Solids only
Modulus of rigidity \((\eta)\)
Shearing stress
Shearing strain
\(\eta=\frac{F/A}{\theta}=\frac{F}{A\theta}\)
Solids only
Bulk modulus \((K)\)
Normal / volume stress
Volumetric strain
\(K=\frac{F/A}{-\Delta V/V}\)
Solids, liquids and gases
▢ Young's Modulus:
❖ Formula: \(Y=\frac{F/A}{\Delta l/l}=\frac{Fl}{A\Delta l}\)
❖ Property: Property of solid material only
❖ Factors:
  • •
    Increases by mixing impurity in solid
  • •
    Decreases by increasing temperature
▢ Modulus of Rigidity:
❖ Formula: \(\eta=\frac{Shearing\ stress}{Shearing\ strain}=\frac{F}{A\theta}\)
❖ Property: Characteristic of solids only
❖ Reason: Liquids and gases do not have fixed shape
❖ For Liquid: \(\eta=0\)
▢ Bulk Modulus:
❖ Formula: \(K=\frac{Normal\ stress}{Volumetric\ strain}=\frac{F/A}{-\Delta V/V}\)
❖ Negative Sign: Volume decreases when force/pressure is applied
❖ Property: Solids, liquids and gases
❖ Magnitude: Maximum for solids, less for liquids, least for gases
❖ Types:

Table 1: Types of Bulk Modulus

Type
Formula
Isothermal bulk modulus
\(K_T=P\)
Adiabatic bulk modulus
\(K_A=\gamma P\)
Gamma
\(\gamma=\frac{C_p}{C_v}\)
❖ Compressibility:
◉ Definition: Reciprocal of bulk modulus
◉ Formula: \(C=\frac{1}{K}\)
▢ Poisson's Ratio:
❖ Definition: Ratio of lateral strain to longitudinal strain
❖ Formula: \(\sigma=\frac{Lateral\ strain}{Longitudinal\ strain}=\frac{-\Delta r/r}{\Delta l/l}=\frac{-\Delta r\cdot l}{\Delta l\cdot r}\)
❖ Negative Sign: Radius decreases when length increases

Table 1: Poisson's Ratio Points

Point
Value / Formula
Theoretical range
\(-1\ to\ +\frac{1}{2}\)
Practical range
\(0\ to\ +\frac{1}{2}\)
If change in volume is zero
\(\sigma=0.5\)
Fractional change in volume
\(\frac{\Delta V}{V}=(1-2\sigma)\frac{\Delta L}{L}\)
▢ Elastic Energy Stored:
❖ Definition: Work done in deforming a body stored as elastic potential energy

Table 1: Elastic Energy Formulae

Quantity
Formula
Elastic P.E. in stretched wire
\(U=\frac{1}{2}\times Stretching\ force\times Extension\)
Elastic P.E.
\(U=\frac{1}{2}Fe\)
Using spring constant
\(U=\frac{1}{2}ke^2\)
Energy density
\(\frac{U}{V}=\frac{1}{2}\times Stress\times Strain\)
▢ Molecular Theory of Elasticity:

Table 1: Molecular Theory Points

Point
Description
Mutual potential energy
\(U\)
Intermolecular distance
\(r\)
Force between molecules
\(F=-\frac{dU}{dr}\)
Equilibrium distance
\(r_0\)
At \(r=r_0\)
Potential energy minimum; attractive and repulsive forces balance; \(F=0\)
For \(r>r_0\)
Attractive force restores molecules to equilibrium
For \(r
Repulsive force restores molecules to equilibrium
Molecular motion
Molecules oscillate about equilibrium / mean position
Near \(r_0\)
\(F-r\) graph approximately straight line
Hooke's law explanation
Extension proportional to applied force near equilibrium
Force constant
\(K=-\frac{dF}{dr}\)
Meaning of \(K\)
Negative gradient of tangent to force-distance curve at \(r=r_0\)
Breaking point separation
\(r'\)
At breaking point
Restoring force decreases with increasing separation
Condition for \(r'\)
\(\frac{dF}{dr}=0\)
Graph relation
\(r'\) corresponds to point of inflexion of \(U-r\) graph
Breaking strain
\(\frac{r-r_0}{r_0}\)
▢ High-Yield Recall:

Table 1: Elasticity One-Liners

Fact
Answer
Most rigid body
Diamond
Perfectly elastic example
Quartz
Perfectly plastic example
Mud
Stress
Force per unit area
Stress nature
Tensor quantity
Strain
Change in dimension / original dimension
Strain unit
No unit
Hooke's law
Stress \(\propto\) strain within elastic limit
Young's modulus
\(Y=\frac{Fl}{A\Delta l}\)
Young's modulus property
Solids only
Modulus of rigidity
\(\eta=\frac{F}{A\theta}\)
Rigidity modulus of liquid
Zero
Bulk modulus
\(K=\frac{F/A}{-\Delta V/V}\)
Bulk modulus property
Solids, liquids and gases
Isothermal bulk modulus
\(K_T=P\)
Adiabatic bulk modulus
\(K_A=\gamma P\)
Compressibility
\(C=\frac{1}{K}\)
Poisson's ratio
\(\sigma=\frac{Lateral\ strain}{Longitudinal\ strain}\)
Practical Poisson's ratio
\(0\ to\ \frac{1}{2}\)
Zero volume change Poisson's ratio
\(0.5\)
Fractional volume change
\(\frac{\Delta V}{V}=(1-2\sigma)\frac{\Delta L}{L}\)
Elastic P.E.
\(U=\frac{1}{2}Fe=\frac{1}{2}ke^2\)
Energy density
\(\frac{U}{V}=\frac{1}{2}\times Stress\times Strain\)
Molecular force
\(F=-\frac{dU}{dr}\)
At equilibrium separation
Potential energy minimum and force zero
Breaking strain
\(\frac{r-r_0}{r_0}\)
Q1.
A metal block is stretched along its length, reducing cross-sectional area. What happens to Young's modulus and modulus of rigidity?
📅CEE MBBS 2025
Q2.
Young's modulus for perfectly rigid body is:
📅CEE Nursing 2024
Q3.
The area of cross-section of a steel wire (Y = 2 × 10¹⁰ N/m²) is 100 m². The force required to double its length is:
📅CEE MBBS 2024a
Q4.
If the force constant of a spring is 100 N/m and the potential energy of the spring is 0.5 J, then the extension produced is:
📅CEE MBBS 2024a
Q5.
With rise in temperature, Young's modulus of elasticity:
📅CEE MBBS 2023
Q6.
Young's modulus for a perfectly elastic body is:
📅CEE MBBS 2022
Q7.
Length of a wire increases by 2% when a mass is hung. The linear strain is:
📅CEE MBBS 2022
Q8.
Modulus of rigidity of rigid body is:
📅CEE BDS 2023
Q9.
A steel wire of length 1 m and cross section area = 2 cm² is stretched by 1 mm by a certain force. What is the energy stored in the wire? \(Y=2\times10^{11}N/m^2\)
📅CEE BDS 2021
Q10.
Modulus of rigidity of rigid body is:
📅CEE BDS 2021
Q11.
What is the SI (MKS) unit of Young's modulus?
📅IOM 2011
Q12.
The coefficient of restitution for a perfectly elastic collision is:
📅IOM 2009
Q13.
The breaking stress of a wire of radius \(3\,\mathrm{mm}\) is \(F\). What is the breaking stress of a wire of the same material and radius \(6\,\mathrm{mm}\)?
📅MOE 2013, 2012
Q14.
The initial slopes of the stress–strain graphs for rubber and copper are \(\tan\theta_1\) and \(\tan\theta_2\), respectively. Which relation is correct?
📅MOE 2010
Q15.
The elastic energy stored per unit volume in a wire having strain \(S\) and Young’s modulus \(Y\) is:
📅MOE 2014
Q16.
If both the length and applied load of a wire are doubled without exceeding its elastic limit, the ratio of stress to strain:
📅KU 2014, 2011
Q17.
Which statement correctly describes Hooke's law?
📅KU 2014
Q18.
The modulus of rigidity is the ratio of:
📅KU 2013
Q19.
A wire experiences tensile stress \(P\) and has Young’s modulus \(Y\). Its elastic energy density is:
📅IE 2013
Q20.
A wire has Poisson’s ratio \(0.20\) and longitudinal tensile strain \(2\times10^{-3}\). Its approximate percentage change in volume is:
📅IE 2013
Q21.
If the tensile stress is \(S\) and Young’s modulus is \(Y\), the strain-energy density is:
📅BP 2010
Q22.
A nylon rope of diameter \(3\,\mathrm{cm}\) breaks under \(1.5\times10^5\,\mathrm N\). What is the breaking load of a rope of the same material and diameter \(1.5\,\mathrm{cm}\)?
📅BP 2009
Q23.
Two springs have constants \(k_A=2k_B\). When the same force stretches each spring, spring A stores energy \(E\). The energy in spring B is:
📅BP 2012
Q24.
Find the Young’s modulus of a wire with \(L=300\,\mathrm{cm}\), \(A=0.003\,\mathrm{cm^2}\), extension \(0.5\,\mathrm{cm}\), and applied force \(10^7\,\mathrm{dyne}\).
📅KU 2012
Q25.
Using Hooke’s law formally, find the force needed to double the length of a wire of area \(100\,\mathrm{m^2}\) and \(Y=2\times10^{10}\,\mathrm{N/m^2}\).
📅KU 2010
Q26.
For the same tensile stress, steel and rubber differ in that steel undergoes:
📅Bangladesh 2009
Q27.
Under the purely linear-elastic approximation, what stress would produce a longitudinal strain of \(1\) in a wire of Young’s modulus \(Y\)?
📅IOM 1996
Q28.
Find the strain energy per unit volume for \(Y=2\times10^{11}\,\mathrm{Pa}\) and strain \(0.05\) (linear-elastic approximation).
📅MOE 2065
Q29.
A steel wire has \(L=2.5\,\mathrm m\), \(A=0.8\times10^{-6}\,\mathrm{m^2}\), \(Y=2\times10^{11}\,\mathrm{Pa}\), and is stretched by \(8\,\mathrm N\). Find its extension.
📅MOE 2062
Q30.
For tensile stress \(S\) and Young’s modulus \(Y\), the elastic energy per unit volume is:
📅MOE 2010
Q31.
Work done in extending an elastic wire of length \(L\), cross-sectional area \(A\), and Young’s modulus \(Y\) by \(x\) is:
📅IE-08
Q32.
An elastic string stores energy \(U\) at an extension of \(2\,\mathrm{cm}\). What energy does it store at \(10\,\mathrm{cm}\), assuming Hooke’s law holds?
📅IE-08
Q33.
For seawater of surface density \(\rho_0\) and bulk modulus \(B\), the approximate increase in density at depth \(h\) is:
📅BPKIHS-09
Q34.
Which material has the highest Young’s modulus among the following?
📅BPKIHS-05
Q35.
A vertical wire of density \(3\times10^3\,\mathrm{kg/m^3}\) and breaking stress \(10^7\,\mathrm{N/m^2}\) hangs under its own weight. What is the maximum length before it breaks?
Q36.
Four wires of the same material experience the same axial force. Which has the greatest extension?
Q37.
A wire of length \(1\,\mathrm m\) and radius \(4\,\mathrm{mm}\) is twisted through \(30^\circ\). Find the surface shear angle (small-angle approximation).
📅BPKIHS
Q38.
An elastic spring has natural length \(L\). In uniform rotation, its extensions are \(1\,\mathrm{cm}\) and \(5\,\mathrm{cm}\) when the angular speed is doubled, with \(kx=m\omega^2(L+x)\). Find \(L\).
Q39.
An elastic wire is extended by \(\ell\) under a final load \(Mg\). The work done in stretching it gradually from zero load is:
Q40.
At depth \(h\) in water of density \(\rho\) and bulk modulus \(B\), the additional elastic energy density due to hydrostatic pressure is approximately:
Q41.
Brass and steel wires have equal lengths and undergo equal extension under equal loads. If \(Y_S=2Y_B\), their radii satisfy:
Q42.
The force constant of an elastic wire of length \(L\), area \(A\), and Young’s modulus \(Y\) is:
Q43.
For a beam of fixed dimensions under the same transverse load, the central deflection is proportional to:
Q44.
A wire has longitudinal tensile strain \(10^{-3}\) and Poisson’s ratio \(0.2\). Find its approximate percentage volume change.
Q45.
An air bubble rises from depth \(h\) in a lake and its radius increases \(n\) times. Atmospheric pressure is equivalent to \(H\,\mathrm{mm}\) of mercury. Assuming constant temperature, express \(h\) in millimetres of water.
Q46.
The bulk modulus of water is \(10^9\,\mathrm{Pa}\). What depth causes a \(0.2\%\) density increase if \(\rho=10^3\,\mathrm{kg/m^3}\)?
Q47.
Water of initial volume \(1\,\mathrm L\) is subjected to an additional pressure of \(2\times10^7\,\mathrm{Pa}\). If its compressibility is \(5\times10^{-10}\,\mathrm{Pa^{-1}}\), find the decrease in volume.
Q48.
Estimate the interatomic force constant in iron if \(Y=2\times10^{11}\,\mathrm{Pa}\) and the atomic spacing is \(3\times10^{-10}\,\mathrm m\).
Q49.
According to Hooke’s law, as stress increases within the proportional limit, the stress-to-strain ratio:
Q50.
A uniform wire initially breaks under a load of \(80\,\mathrm{kgf}\). If it is cut into two equal lengths, the breaking load of either half is:
Q51.
If the breaking force of a wire of radius \(r\) is \(F\), what is the breaking force of a similar wire of radius \(2r\)?
Q52.
Two solid rods A and B of the same material and length are twisted by equal torques. Their radii are \(r_1\) and \(r_2\). Find \(\theta_A/\theta_B\).
Q53.
A metal bar is heated from \(0^\circ\mathrm C\) to \(100^\circ\mathrm C\) while its expansion is completely prevented. For fixed \(Y\), area and temperature rise, the developed force varies with linear expansion coefficient \(\alpha\) as:
Q54.
The length of a spring is \(a\) under tension \(4\,\mathrm N\) and \(b\) under tension \(5\,\mathrm N\). What is its length under \(9\,\mathrm N\)?
Q55.
Which type of stress primarily changes the shape of a body without changing its volume?
📅IOM 2015
Q56.
Young’s modulus is defined as:
📅IOM 2015
Q57.
For most metals, Young’s modulus generally _____ as temperature increases.
📅IOM 2016
Q58.
Two wires have length ratio \(1:2\), but the same material, radius, and applied tensile force. Their longitudinal strains are in the ratio:
📅KU 2017