$T_i$: initial temperature

$T_f$: final temperature

$T_f-T_i=\Delta T$

$\Delta l_T=\alpha l_0(T_f-T_i) = \alpha l_0 \Delta T$

where: $\alpha$ : Linear coefficient of thermal expansion

Dimension: ${[L] } =\alpha[L][T]$

Unit: $\alpha =\frac{1}{[T]}=/^\circ C$

$\Delta l=\frac{F l_0}{A Y}$

$\Delta l \equiv \Delta l_T$

$\alpha l_0 \Delta T=(\frac{F}{A}) \frac{l_0}{Y}$

$\sigma=\alpha y \Delta T$

$\frac{F}{A}=\sigma$ Thermal stress

Pieces of rail track, each $10 {m}$ long are laid with a clearance of $5{mm}$ at a temperature $30^{\circ} {C}$

(i) At what temperature do the pieces just start touching?

(i) What is the thermal stress developed of these were no clearance?

$\alpha=18 \times 10^{-6} /{^\circ {c}},$

$Y=200 \times 10^6{N} / {m}^2$

$l_0=10 {m}$

$\Delta l_T =5 \mathrm{mm}=5 \times 10^{-3} \mathrm{m}$

$T_i =30^{\circ} \mathrm{C}$

$T_f =?$ and $\sigma=?$

Using, $\Delta l_T=\alpha l_0 \Delta T$

$\Delta T=\frac{\Delta l_T}{\alpha l_0}=\frac{5 \times 10^{-3}}{18 \times 10^{-6} \times 10^{\circ}} \simeq 28^{\circ}{C}$

$\Delta T=T_f-T_i=28^{\circ} {C} \Rightarrow I_f=30^{\circ}{C}+28^{\circ}{C}=58 ^{\circ}{C}$

$\sigma =\alpha Y \Delta T$

$=18 \times 10^{-6} \times 200 \times 10^6 \times 28$

$=1008 {N} /{m}^2$.

A bronze bar $5{m}$ long and a cross-sectional area of $200{m}^2$ is placed between two rigid walls.

At a temperature $-10^{\circ}{C}$, the gap between the bar and the right wall is $20{mm}$. Find the temperature at which the Compressive stress in the bar will be $30 \times 10^3 {N} / {n}^2$?

Given: $\alpha$ coefficient of thermal.

Expainsion $=12 \times 10^6 /^\circ{c}$

Young's modulus $=18 \times 10^6 {N} /{m}^2$

$\Delta x=\operatorname{strain} \times L_0=\frac{\operatorname{stress}}{Y} \times L_0$

$=\frac{30 \times 10^3{N} /{m}^2}{80 \times 10^6 {N} / {m}^2} \times 5{m}$ = $1.875 \times 10^{-3}{m}$

$L= 5 m+20 {m m+} {\Delta x}$

$L_0=5 m$

$L=L_0(1+\alpha(T_f-T_i))$

$\frac{L-L_0}{L_0}=\alpha(T_f+10^{\circ}{C})$

$\Rightarrow \frac{1875 \times 10^{-3}}{5}=12 \times 10^{-6}(T_f+10^{\circ}C)$ $\Rightarrow T_{f}=21.2 ^{\circ} {C}$

A cylindrical specimen of a certain alloy having $y=108 \times 10^6 {N} /{m}^2$ with a diameter $3.9{mm}$ experiences an elastic deformation when a tensile load of $2000{N}$ is applied calculate the maximum length of the specimen before deformation if the maximum allowed elongation is $0.42 {mm}$

$l_0$ = original diameter

$A_0$ = original area of cross section

$A_0=\pi\left(\frac{l_0}{2}\right)^2=\pi \frac{l_0^2}{4}$

$l_0=\frac {\Delta {l}\times {y}}{\sigma}$

$\sigma =\frac{F}{A_0}=\frac{0.4^2\times{10}^{-3}\times{10}^8\times{10}^6}{2000\times{4}}\times\pi(3.9 \times 10^{-3})^2 = 0.257 {m}$

Hooke's law: $F \propto x$

Force: $F=-k x$

$W=\int {|F|} d x =\int_0^S k x d x=\frac{1}{2} k s^2$

Potential Energy Stored: $U =\frac{1}{2} k s^2$

$F =G A \theta$

$d x =L d \theta$

$W=U =\int G A \theta L d \theta$

Potential Energy: $U =\frac{1}{2} G A L \theta^2$

Applications of elasticity on different components of human body.

Bones: stress break the bones.

Arteries and Veins: They carry blood, inner walls are elastic.

Focus: In a lifetime of 70 years, the human heart beats some 2.5 billion times.

This durable pump is the centerpiece of the cardiovascular system.

1) Hooke's law

2) Different kinds of elastic modulus

3) Stress versus strain curves.

4) Diffrence in elastic and plastic deformation

5) Properties of Matter

6) Elastic properties of human bady.

7) A number of example problem.

1) A matirial having. large $y$, if requires a large' force to produce a small elongation or compression.

2) Matirial which stretches/compression to a longer extent due to a given lord in termed an mare elastic.

3) Stress is not a vector quntity.