12Elasticity

📚
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.
Which of the following unit of Young's modulus is in MKS system:
📅IOM 2011
Q2.
The elasticity of highly elastic body is:
📅IOM 2009
Q3.
Breaking stress (in N/m²) of a wire of radius 3 mm is F. The breaking stress of the same material of radius 6 mm will be:
📅MOE 2013, 2012
Q4.
The stress-strain graph for copper and rubber shows slopes tanθ₁ (rubber) and tanθ₂ (copper). Then:
📅MOE 2010
Q5.
The energy density of a wire of strain S and Young's modulus Y is:
📅MOE 2014
Q6.
When length and load are doubled for a wire, the ratio of stress to strain:
📅KU 2014, 2011
Q7.
Which statement is correct about elasticity?
📅KU 2014
Q8.
Modulus of rigidity is:
📅KU 2013
Q9.
Energy stored per unit volume under pressure P (Young's modulus Y):
📅IE 2013
Q10.
For Poisson's ratio 0.20 and longitudinal strain 2×10⁻³, % volume change is:
📅IE 2013
Q11.
Energy stored per unit volume (stress S, Young's modulus Y):
📅BP 2010
Q12.
Breaking strength of nylon rope (1.5cm dia) if 3cm rope breaks at 1.5×10⁵ N:
📅BP 2009
Q13.
Energy stored in spring B (K_A=2K_B) when spring A stores E:
📅BP 2012
Q14.
Young's modulus for wire (300cm, 0.003cm², ΔL=0.5cm, F=10⁷ dyne):
📅KU 2012
Q15.
Force to double length of material (Y=2×10¹⁰ N/m², A=100 m²):
📅KU 2010
Q16.
Steel vs. rubber elasticity means for same stress:
📅Bangladesh 2009
Q17.
Stress required to double wire length:
📅IOM 1996
Q18.
Energy stored per unit volume (Y=2×10¹¹ N/m², strain=0.05):
📅MOE 2065
Q19.
Extension of steel wire (L=2.5m, A=0.8×10⁻⁶ m², Y=2×10¹¹ Pa, F=8N):
📅MOE 2062
Q20.
Elastic energy per unit volume (stress S, Y):
📅MOE 2010
Q21.
Work done stretching wire (L, A, Y) by x:
📅IE-08
Q22.
PE when string stretched from 2cm (U) to 10cm:
📅IE-08
Q23.
Change in seawater density (ρ₀, B) at depth h:
📅BPKIHS-09
Q24.
Substance with highest elasticity:
📅BPKIHS-05
Q25.
Length of wire (breaking stress=10⁷ N/m², density=3×10³ kg/m³) breaking under own weight:
Q26.
Wire elongating most under same load:
Q27.
Angle of shear at surface for twisted wire (30° twist, L=1m, r=4mm):
📅BPKIHS
Q28.
Original spring length if elongation changes from 1cm to 5cm when ω doubled:
Q29.
Work done stretching wire by l under weight Mg:
Q30.
Elastic energy per unit volume in water (B, ρ, depth h):
Q31.
Relation between R_B (brass) and R_S (steel) for same ΔL under same F (Y_steel=2Y_brass):
Q32.
Spring constant K for metal wire (L, A, Y):
Q33.
Depression at center of loaded beam is proportional to:
Q34.
% volume change for longitudinal strain 10⁻³ (Poisson's ratio=0.2):
Q35.
Depth of lake if bubble radius increases n times (atm pressure = H mmHg):
Q36.
Depth for 0.2% density increase in rubber ball (B=10⁹ N/m², ρ=10³ kg/m³):
Q37.
Decrease in 1L water volume under 2×10⁷ N/m² (compressibility=5×10⁻¹⁰ m²/N):
Q38.
Interatomic force constant for iron (Y=2×10¹¹ N/m², spacing=3×10⁻¹⁰ m):
Q39.
Hooke's Law: ratio of stress to strain when stress increases:
Q40.
Breaking load for wire divided into two identical parts:
Q41.
Breaking force for wire radius 2r vs. radius r:
Q42.
Ratio of twist angles for rods A and B (radii r₁ and r₂) under same torque:
Q43.
Force developed in bar heated 0°C→100°C while constrained:
Q44.
Length of spring when tension is 9N (4N→a, 5N→b):
Q45.
Stress that changes object shape:
📅IOM 2015
Q46.
Young's modulus is:
📅IOM 2015
Q47.
Effect of temperature on Young's modulus:
📅IOM 2016
Q48.
Strain ratio for wires (length ratio 1:2, same radius/material, same force):
📅KU 2017