physics-class-11-unit-09-chapter-02-mechanical-properties-of-solids-2-lecture-2-3-cvhtr1nrgj0

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Young's Modulus </primary>
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- $Y = /\left(\frac{(F / A)}{(\Delta l/l_0)}\right)=\frac{\text { Stress }}{\text { Strain }} $

- $[Y]=\left[\frac{M L T^{-2}}{L^2} \times 1\right] $

- Unit of Y $=\frac{N}{m^2}$.

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<footer style = "font-size: 18px"> $\rightarrow$ Mechanical Properties of Solids $\rightarrow$ <span style = "color:blue"> Young's Modulus </span> $\rightarrow$ Mechanical Properties of Solids $\rightarrow$ Experimental Determination of Young's Modulus </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Mechanical Properties of Solids </primary>
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| Material $\left(\mathrm{N} / \mathrm{m}^2\right)$ | Tensile Strength $\left(\mathrm{N} / \mathrm{m}^2\right)$ | Compressive Strength| Sheer Strength $\left(\mathrm{N} / \mathrm{m}^2\right)$ |
| -------- | -------- | -------- | -------- |
| Iron    |  $170 \times 10^6$  | $550 \times 10^6$    | $170 \times 10^6$    |
| Steel   | $500 \times 10^6$      | $500 \times 10^6$   | $250 \times 10^6$    |
| Brick     | $10^6$      | $35 \times 10^6$  |  -    |
| Concrete    | $2 \times 10^6$    | $20 \times 10^6$  | $2 \times 10^6$     |
| Aluminium   | $200 \times 10^6$      | $200 \times 10^6$  | $200 \times 10^6$     |
| Wood (pine)  | $40 \times 10^6$      | $35 \times 10^{6}$  | $5\times10^6$     |

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<footer style = "font-size: 18px">Mechanical Properties of Solids $\rightarrow$ Young's Modulus $\rightarrow$ <span style = "color:blue"> Mechanical Properties of Solids </span> $\rightarrow$ Experimental Determination of Young's Modulus $\rightarrow$ Experimental Determination of Young's Modulus </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Experimental Determination of Young's Modulus </primary>
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* Radius of both wires $=Y_0$
* Initial length$=L_0$
* Elongation due to weights $=\Delta L$
* Mass causing elongation $=M$

- $Y=(\frac{M g}{\pi r_0^2}) / (\frac{\Delta L}{L_0}) $

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<footer style = "font-size: 18px">Young's Modulus $\rightarrow$ Mechanical Properties of Solids $\rightarrow$ <span style = "color:blue"> Experimental Determination of Young's Modulus </span> $\rightarrow$ Experimental Determination of Young's Modulus $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Experimental Determination of Young's Modulus </primary>
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* Load causes an elongation.

* Reading is noted of the difference between the two vernier scales when there are equal weights as compared to when there are unequal weights.

* Difference between readings is taken as the elongation.

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<footer style = "font-size: 18px">Mechanical Properties of Solids $\rightarrow$ Experimental Determination of Young's Modulus $\rightarrow$ <span style = "color:blue"> Experimental Determination of Young's Modulus </span> $\rightarrow$ Examples $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- Complete Bulk modulus of water from the data.

- Initial volume $V_i=100.0 \mathrm{L}$

- Pressure increase, $\Delta P=100.0 \mathrm{atm} $

- $1 atm = 1.013 \times 10^5 \mathrm{Pa} $

- Final volume, $V_f=100.5 \mathrm{L} $

- $B=\Delta P / \frac{\Delta V}{V_i} \quad \Delta V=V_f-V_i=0.5 \mathrm{L} $

- =$\frac{100 \times 1.013 \times 10^5 \times 100 \mathrm{L}}{0.5 \mathrm{L}}=2.026 \times 10^9 \mathrm{Pa} =2.026 \times 10^9 \mathrm{N} /\mathrm{m}^2 $

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

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- Determine the volume contraction of solid copper cube 10cm on each edge when subjected to a hydraulic pressure of $7.0\times {10^6} Pa $.

- Given: $B_{cu} =140 \times 10^9 \mathrm{N} / \mathrm{m}^2 $

- $B = \frac{\Delta P}{( \Delta V / \mathrm{V_i})}$

- $V_i =(10 \mathrm{cm})^3 $=$(0.1 \mathrm{m})^3=0.001 \mathrm{m}^3$

- $\Delta V=\frac{\Delta P}{B} \times V_i =\frac{7 \times 10^6}{140 \times 10^9} \times 0.001 \mathrm{m}^3 $

- $=0.5 \times 10^{-6} \mathrm{m}^3$

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

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- Two wires each of diameter 0.25 cm, one made of steel and the other of brass as shown below. The unloaded length of the steel wire is 1.5m and that of the brass wire is 1.0m. Compute the elongations of the steel and brass wires.

- Given: $Y_{\text {steel }}=20 \times 10^{10} \mathrm{N} / \mathrm{m}^2 \text { and } Y_{\text {brass}}=9.0 \times 10^{10} \mathrm{N} / \mathrm{m}^2$

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

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- $d=0.25 \mathrm{cm}=25 \times 10^{-4} \mathrm{m} $

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

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

- <span style="color:green">**Steel**</span>

- $\Delta l_{\text {steel}} =\frac{100 \times 1.5 \times 4}{\pi(25)^2 \times 10^{-8} \times 20 \times 10^{10}}m $

- $=1 \times 10^{-4} \mathrm{m} $

- <span style="color:green">**Brass**</span>

- $\Delta l_{\text {brass}} =\frac{60 \times 1 \times 4}{\pi(25)^2 \times 10^{-8} \times 9 \times 10^{10}}=1.35 \times 10^{-4} \mathrm{m}$

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

### <Success style="font-size:25px"> Mechanical Properties Of Solids 2 L-2 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- A cylindrical specimen of a titanium alloy having an elastic modulus of 108GPa and an original diameter of 3.9mm will experience only elastic diformation when a tensile load of 2000v is applied. Computed maximum length of the specimen before deformation of the maximum allowable elongation is 0.42mm.

- The initial area of the cylinder is $A_0=\pi\left(\frac{l_0}{2}\right)^2 \quad d_0=3.9 \mathrm{mm}$

- Original length, $L_0  = \frac{\Delta l \times Y}{F/A_0}$

- $ = \frac {(0.4 \times 10^{-3}) \times (108 \times 10^6 N/m^2) \pi (3.9 \times 10^{-3})^2}{4 \times 2000 N}$

- $=0.257 m=257 \mathrm{mm}$

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<footer style = "font-size: 18px">Examples $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Properties of Matter $\rightarrow$ Toughness </footer>

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### <primary style="font-size:24px"> Brittleness and Hardness </primary>
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- <span style="color:green">**Brittleness**</span>

- Breaks being subjected to stress without undergoing significant deformation under strain.

- <span style="color:green">**Hardness**</span>

- Measure of how resistant a material in to a permanent shape change when subjected to an applied force.

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<footer style = "font-size: 18px">Properties of Matter $\rightarrow$ Toughness $\rightarrow$ <span style = "color:blue"> Brittleness and Hardness </span> $\rightarrow$ Resilience $\rightarrow$ Stiffness </footer>

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### <primary style="font-size:24px"> Resilience </primary>
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- <span style="color:green">**Resilience**</span>

- Resilience is the capacity of a material to absorb energy when it is deformed elastically and then the energy in released upon unloading.

- It is obtained from the area of the stress v/s strain graph.

- $U_{\text {res }}=\frac{1}{2} \int_0^{(\Delta x) d} \sigma d x=\frac{1}{2} \frac{F}{A} \int_0^{(\Delta x) d} x $

- $U_{\text {res }}=\frac{1}{2} \frac{F}{A}(\Delta x) d$

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<footer style = "font-size: 18px">Toughness $\rightarrow$ Brittleness and Hardness $\rightarrow$ <span style = "color:blue"> Resilience </span> $\rightarrow$ Stiffness $\rightarrow$ Thank You </footer>