physics-class-11-unit-09-chapter-03-mechanical-properties-of-solids-3-lecture-3-3-pp3jjpe5v2e

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Thermal Expansion </primary>
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- $T_i$:  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 $

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<footer style = "font-size: 18px"> $\rightarrow$ Mechanical Properties of Solids $\rightarrow$ <span style = "color:blue"> Thermal Expansion </span> $\rightarrow$ Thermal Expansion $\rightarrow$ Problem </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Thermal Expansion </primary>
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- $\Delta l=\frac{F l_0}{A Y}$
 
- $ \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

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<footer style = "font-size: 18px">Mechanical Properties of Solids $\rightarrow$ Thermal Expansion $\rightarrow$ <span style = "color:blue"> Thermal Expansion </span> $\rightarrow$ Problem $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- 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 $\sigma=?$

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<footer style = "font-size: 18px">Thermal Expansion $\rightarrow$ Thermal Expansion $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Solution $\rightarrow$ Problem </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- 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 $.
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<footer style = "font-size: 18px">Thermal Expansion $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- 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 $=18 \times 10^6 {N} /{m}^2$

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<footer style = "font-size: 18px">Problem $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Solution $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $\Delta x=\operatorname{strain} \times L_0=\frac{\operatorname{stress}}{Y} \times L_0$

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

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

- $ 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})$

- $\Rightarrow \frac{1875 \times 10^{-3}}{5}=12 \times 10^{-6}(T_f+10^{\circ}C)$ $\Rightarrow T_{f}=21.2 ^{\circ} {C}$
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<footer style = "font-size: 18px">Solution $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Examples $\rightarrow$ Potential Energy </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- 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} $

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

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<footer style = "font-size: 18px">Problem $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Potential Energy $\rightarrow$ Applications of Elasticity </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Potential Energy </primary>
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- 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$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Potential Energy </span> $\rightarrow$ Applications of Elasticity $\rightarrow$ Blood Pressure </footer>

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### <primary style="font-size:24px"> Blood Circulation </primary>
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- <span style="color:green"> Brings oxygen, nutrients, and hormones to cells.</span>
- Removes waste products from cells and excess heat.
	
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<footer style = "font-size: 18px">Applications of Elasticity $\rightarrow$ Blood Pressure $\rightarrow$ <span style = "color:blue"> Blood Circulation </span> $\rightarrow$ The Heart $\rightarrow$ Summary </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> The Heart </primary>
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- <span style="color:blue">**Focus:**</span> 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.

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ae870c1d-7588-4726-0a29-9047dad16800/public)

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<footer style = "font-size: 18px">Blood Pressure $\rightarrow$ Blood Circulation $\rightarrow$ <span style = "color:blue"> The Heart </span> $\rightarrow$ Summary $\rightarrow$ Points to Ponder </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 3 L-3 </Success>
### <primary style="font-size:24px"> Points to Ponder </primary>
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- 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.
	
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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-09-chapter-03-mechanical-properties-of-solids-3-lecture-3_3-pp3jjpe5v2e-601(r).jpg)

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<footer style = "font-size: 18px">The Heart $\rightarrow$ Summary $\rightarrow$ <span style = "color:blue"> Points to Ponder </span> $\rightarrow$ Thank You $\rightarrow$  </footer>