physics-class-11-unit-09-chapter-01-mechanical-properties-of-solids-1-lecture-1-3-3oryx8mnans

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Rigid Bodies </primary>
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- **Rigid Bodies** are defined as objects whose internal particle distances remain constant during motion, yet this does not preclude bending, stretching, extension, or other forms of deformation.

- **Example:** Steel rods deform under extremely high force.

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<footer style = "font-size: 18px"> $\rightarrow$ Mechanical Properties Of Solids $\rightarrow$ <span style = "color:blue"> Rigid Bodies </span> $\rightarrow$ Elastic Materials $\rightarrow$ Non-elastic Materials </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Elastic Materials </primary>
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- **Bungee jump** : The activity of jumping off a very high bridge or similar structure, with a long elastic rope tied to your legs, so that the rope pulls you back before you hit the ground:

- Springs are examples of elastic materials.

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<footer style = "font-size: 18px">Mechanical Properties Of Solids $\rightarrow$ Rigid Bodies $\rightarrow$ <span style = "color:blue"> Elastic Materials </span> $\rightarrow$ Non-elastic Materials $\rightarrow$ Young's Modulus </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Non-elastic Materials </primary>
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- Non-elastic materials do not return to their original shape and size after force is removed.

- Non-elastic materials include clay or wheat dough.
-These are referred to as plastic materials.
-The internal structure of materials determines whether they can regain their original shape.

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<footer style = "font-size: 18px">Rigid Bodies $\rightarrow$ Elastic Materials $\rightarrow$ <span style = "color:blue"> Non-elastic Materials </span> $\rightarrow$ Young's Modulus $\rightarrow$ Young's Modulus </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Young's Modulus </primary>
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- **Two rods** : $l_0$ = initial length , $\Delta$ l: extension

- Case 1. Same initial length, but different extension same area of cross section.

- Case 2. Same initial length, but different cross section area A and same extension.

- Case 3. Different initial length, but same extension same area of cross section.

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<footer style = "font-size: 18px">Elastic Materials $\rightarrow$ Non-elastic Materials $\rightarrow$ <span style = "color:blue"> Young's Modulus </span> $\rightarrow$ Young's Modulus $\rightarrow$ Young's Modulus </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Young's Modulus </primary>
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- $F  \propto \Delta l$

- $F \propto \frac{1}{L_0}$

- $F \propto A$

- $F \propto  \frac{\Delta l}{L_0} . A$

- $F = Y (\frac{\Delta l}{L_0}) A$

- $(\frac{\Delta l}{L_0})$: frational change in length.

- Y: Young's modulus

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<footer style = "font-size: 18px">Non-elastic Materials $\rightarrow$ Young's Modulus $\rightarrow$ <span style = "color:blue"> Young's Modulus </span> $\rightarrow$ Young's Modulus $\rightarrow$ Young's Modulus </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Young's Modulus </primary>
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- A spring is tied at one end.
 
- It has an unstretched length L<sub>0</sub>.

- The spring is acted upon by a force  F.

- It undergoes an expansion denoted by ΔL.

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<footer style = "font-size: 18px">Young's Modulus $\rightarrow$ Young's Modulus $\rightarrow$ <span style = "color:blue"> Young's Modulus </span> $\rightarrow$ Young's Modulus $\rightarrow$ Young's Modulus </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Young's Modulus </primary>
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- For Spring:

- $\Delta l \propto F$

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

- Now, $\Delta l \propto l_0$

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

- Young's Modulus = $\frac{\text{tensile stress}}{\text{tensile strain}}$

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Young's Modulus </primary>
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| Material |value Y |
| -------- | -------- |
|Aluminium|$6.9 \times 10^{10} N/m^2$|
|Copper|$1.1 \times 10^{11} N/m^2$|
|Steel|$2.0 \times 10^{11} N/m^2$|
|Nylon|$3.7 \times 10^{9} N/m^2$|
|Brass|$9 \times 10^{9} N/m^2$|

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Example </primary>
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- For example: Circus

- A circus act features one performer supporting six others.

- Each of the six performers weighs 50 kg.

- The main performer bears their combined weight.

- Consider : g = 10 ${m s^{-2}}$

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<footer style = "font-size: 18px">Young's Modulus $\rightarrow$ Young's Modulus $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Example </primary>
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- 6 performers and each weight$ \rightarrow 50 \{kg}$

- $\text { Total weight }  =300 \{kg} $

- $\text { Total force }  =300 \times 10 = 3000 \{N}$

- We know, femur bone = 0.5 $\{m}$ long.

- =$10^{-3} \{m}^2 \text { (cross-sectional area) }$
 
- $Y=10^{10} \{N} / \{m}^2 $
 
- $ \Delta l = \frac{(\frac{3000}{2}) (0.5 \{m})} {(10^{10} \{N}  \{m}^2) \times 10^{-3} \{m}^2} \simeq 10^{-6} \{m}$

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<footer style = "font-size: 18px">Young's Modulus $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $Y=2 \times 10^{11} \{N} / \{m}^2 $

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

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

- $\Delta l=\frac{F \times l}{A y} \simeq 2.5 \{mm}$

- For Nylon ( Y = 50 times lower than steel) $\Delta l |_{Nylon} = 50 \times 2.5 \{mm}$

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/890201fe-c49e-4826-00f4-7968ec629900/public)

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Shear Stress $\rightarrow$ Shear Stress </footer>

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### <primary style="font-size:24px"> Shear Stress </primary>
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- Example: thick book

- We apply a force to the book's top surface with cross-sectional area 'A.'

- This force causes an extension, $\Delta x$, from the book's original length, L.

- Deformation is known as <span style="color:blue">shear</span>, and the force per unit area is called <span style="color:blue">shearing strength</span>.

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/16e444fe-d8d0-4e51-932f-6b382d6fbb00/public)

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Shear Stress </span> $\rightarrow$ Shear Stress $\rightarrow$ Shearing Strain </footer>

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### <primary style="font-size:24px"> Shear Stress </primary>
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- $F \propto \Delta x$

- $F \propto \frac{1}{L_0}$

- $F \propto A$

- $F \propto (\frac{\Delta x}{L_0}) A$

- $[\frac{F}{A} = G (\frac{\Delta x}{L_0})]$

- Where G is shear modulus

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<footer style = "font-size: 18px">Example $\rightarrow$ Shear Stress $\rightarrow$ <span style = "color:blue"> Shear Stress </span> $\rightarrow$ Shearing Strain $\rightarrow$ Volumne Deformation </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Shearing Strain </primary>
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- Shearing Strain = $\frac{\Delta x}{L_0} = \tan\theta \simeq \sin\theta \sim \theta $

- $\tan \theta  = \frac{\sin \theta }{\cos \theta }$

- Angle is small.

- $\sin \theta \rightarrow \theta $

- $\cos \theta \rightarrow 1 $

- $F \propto (\frac{\Delta x}{L_0}) A$

- $\frac{F}{A} = G (\frac{\Delta x}{L_0})$

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9956d9b2-6cde-43c6-f6fb-06725b112600/public)

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<footer style = "font-size: 18px">Shear Stress $\rightarrow$ Shear Stress $\rightarrow$ <span style = "color:blue"> Shearing Strain </span> $\rightarrow$ Volumne Deformation $\rightarrow$ Hooke's Law </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 1 L-1 </Success>
### <primary style="font-size:24px"> Volumne Deformation </primary>
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- $\frac{F}{A}$ =  Pressure due to the liquid

- $\Delta P \propto \frac {\Delta V }{V_0}$

- $\Delta P = -B \frac {\Delta V }{V_0}$

- <span style="color:blue">B: Bulk Modulus</span>

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<footer style = "font-size: 18px">Shear Stress $\rightarrow$ Shearing Strain $\rightarrow$ <span style = "color:blue"> Volumne Deformation </span> $\rightarrow$ Hooke's Law $\rightarrow$  </footer>

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### <primary style="font-size:24px"> Hooke's Law </primary>
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- $\frac{F}{A} = Y(\frac{\Delta}{L_0})$

- $\frac{F}{A} = G(\frac{\Delta x}{L_0})$

- $\Delta P = B(\frac{\Delta V }{V_0})$

- Hooke's Law: Stress $\propto$ Strain.

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<footer style = "font-size: 18px">Shearing Strain $\rightarrow$ Volumne Deformation $\rightarrow$ <span style = "color:blue"> Hooke's Law </span> $\rightarrow$  $\rightarrow$ Thank You </footer>