Elasticity Study Notes

States of Matter
- The states of matter in bulk can be conveniently divided into solids and fluids.
- Solids: Tend to be rigid and maintain their shape.
- Fluids: Do not maintain their shape but flow.
- Liquids: Flow until they occupy the lowest possible regions in their container.
- Gases: Expand to fill their container regardless of its shape.
Density
- An important property of a substance is the ratio of its mass to its volume, known as its density (ρ):
  $\rho = \frac{m}{V}$
- The ratio of the density of a substance to that of water is called specific gravity or relative density, which is a dimensionless quantity.
- If the volume of a substance equals the volume of water, relative density can be defined as:
  $\text{Relative Density} = \frac{\text{mass of a substance}}{\text{mass of an equal volume of water}}$

Density of the Earth
- The radius of the Earth is $6.37 \times 10^6 \text{ m}$ and its mass is $5.98 \times 10^{24} \text{ kg}$ .
- Density calculation is required.
Mass of Air in a Room
- A room is 5 m long, 4 m wide, and 3 m high. The density of air is $1.3 \text{ kg/m}^3$ .
- Volume $(V) = l \times b \times h = 5 \text{ m} \times 4 \text{ m} \times 3 \text{ m} = 60 \text{ m}^3$
- $m = \text{Density} \times V = (1.3 \text{ kg/m}^3)(60 \text{ m}^3) = 78 \text{ kg}$
Density of Oil in a Flask
- A 100 cm³ flask weighs 23.00 g when empty and 97.84 g when full of oil.
- $\text{Mass of oil} = (97.84 \text{ g}) - (23.00 \text{ g}) = 74.84 \text{ g}$
- Volume of oil is $100 \text{ cm}^3$ , therefore:
  $\text{Density} = \frac{74.84 \text{ g}}{100 \text{ cm}^3} = 0.784 \text{ g/cm}^3 = 748 \text{kg/m}^3$

When an external force is applied to a solid, it produces a change in shape, size, or volume of the solid known as deformation.
Elasticity: Ability of a solid (material) to regain its original shape and size after the deforming force is removed.
- Elastic Deformation: Temporary deformation that recovers when the force is removed.
- Plastic Deformation: Permanent deformation after the force is removed. It can be further divided into:
- Dislocation
- Fracture
Definition of Terms
- Deformation: Change in shape and size of a material due to an external force.
- Plasticity: Condition where material remains permanently deformed post-removal of force.
- Common properties of materials include:
- Hardness: Resists cutting or penetration.
- Brittleness: Allow little bending or deformation.
- Malleability: Can be hammered or rolled into shapes.
- Ductility: Can be permanently bent.
- Elasticity: Enables return to original shape.
- Density: Mass per unit volume.
- Fusibility: Ability to become liquid via heat.
- Conductivity: Carries heat and electricity.

Hooke’s Law: $F = -kx$
- This law states that the restoring force ( $F$ ) from a spring is proportional to the displacement ( $x$ ) from its equilibrium position.
- The spring constant $k$ characterizes the elasticity of the spring.
  $k = \frac{\Delta F}{\Delta x}$

Stress: Refers to the force applied to deform a material per unit area, defined as: $\text{Stress} = \frac{F}{A}$ where $F$ represents the applied force and $A$ is the area.
- Units: Pascals (Pa), N/m²
- Examples include change in length per unit length and change in volume per unit volume.
Strain: The relative change in dimensions (length or volume) due to applied stress.
- Examples include change in length per unit length and change in volume per unit volume.

Tensile Stress: Arises when forces are directed away from each other (lengthening).
Compressive Stress: Arises when forces are directed toward each other (shortening).

The elastic limit is the maximum stress without permanent deformation.
Ultimate Strength: Greatest stress experienced without breaking.
Shear Modulus: Not applicable to fluids (gases and liquids) as they do not possess shear modulus like solids.

Elastic deformation (strain) is directly proportional to the applied force (stress) when the elastic limit is not exceeded:
$\text{Modulus of Elasticity} = \frac{\text{Stress}}{\text{Strain}}$
Commonly measured modulus types include:
- Young’s Modulus (Y): Measures longitudinal elasticity.
- Bulk Modulus (K): Measures volumetric elasticity.
- Shear Modulus (G): Measures shear elasticity.

Example 1: A steel wire 10 m long and 2 mm in diameter is stressed by a weight of 200 N, causing a strain of 3.08 mm.

Calculate the stress:
- $A = \frac{\pi(0.002 m)^2}{4} \implies A = 3.14 \times 10^{-6} m^2$
- $\text{Stress} = \frac{F}{A} = \frac{200 N}{3.14 \times 10^{-6} m^2} = 6.37 \times 10^7 Pa$

Example 2: Calculate Young's Modulus. If Stress is $6.37 \times 10^7 Pa$ and Strain is $3.08 \times 10^{-4}$ :

$E = \frac{\text{Stress}}{\text{Strain}} = \frac{6.37 \times 10^7 Pa}{3.08 \times 10^{-4}} = 207 \times 10^9 Pa$ .

Example 3: A block of unknown material with dimensions 60 mm x 60 mm x 20 mm is under a shearing force of 0.245 N with displacement of 5 mm.

Calculate the shear modulus:
- $\text{Shear stress} = \frac{0.245 N}{\text{Area}} = \frac{0.245 N}{60\text{ mm} \times 20\text{ mm}} = 2450 N/m^2$ .

When materials are stretched or compressed, they store energy.
Work done on the elastic material is stored as potential energy.
The energy stored can be described as:
- Energy per unit volume for tensile strain:
  $E = \frac{1}{2} \frac{F x}{V} = \frac{1}{2} kx^2$
For shear strain:
$E = \frac{1}{3} \frac{F \Delta x}{A}$

Poisson's ratio relates to elastic moduli (B, S, Y).
Defined as:
$\sigma = \frac{3B - 2S}{6B + 2S}$
$Y = 2S(1 + \sigma)$
$Y = 3B(1 - 2\sigma)$

Example: To find the pressure required to reduce the volume of a 1 kg water from 100 to 99 L where bulk modulus for water is $2 x 10^{11} N/m^2$ . The change in volume can be calculated.
Example: A solid brass sphere: Calculate changes in volume subjected to different pressures, utilizing the Bulk modulus of seawater.
Example: A slingshot potential energy: Calculate energy stored when stretched and use it to find speed at which stone leaves the slingshot.
Example: The force needed to punch a hole in steel, given shear stress.

The concepts of elasticity, stress and strain, and energy storage are fundamental in understanding material properties and their applications in various fields including engineering, construction, and materials science.