physics-class-11-unit-10-chapter-04-mechanical-properties-of-fluids-4-lecture-4-5-vy-csifyfd4

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Bernoulli Equation </primary>
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- Assume steady or a streamline of an incompressible liquid.

- E= Kinetic energy +  Potential Energy

- $=\frac{1}{2} m v^2+m g h$

- Energy difference  between R1 and R2

- $E_1 - E_2 = (\frac{1}{2}m{v_1}^2 + mgh_1) - (\frac{1}{2}m{v_2}^2 + mgh_2)$

- $ W = E_1 - E_2 - (\frac{1}{2}m{v_1}^2 + mgh_1)- (\frac{1}{2}m{v_2}^2 + mgh_2)$

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<footer style = "font-size: 18px"> $\rightarrow$ Mechanical Properties of Fluids $\rightarrow$ <span style = "color:blue"> Bernoulli Equation </span> $\rightarrow$ Bernoulli Equation $\rightarrow$ Venturimeter </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Bernoulli Equation </primary>
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- Pressure at $O=(P + \Delta P)$, Force at$O=(P+\Delta P) A$

- Pressure at $O^{\prime}=? \quad$ ? Force at $O^{\prime}=P A$

- Difference in force between $O^{\prime}$ and $O=(P+\Delta P) A-P  A=(\Delta P) A$

- Work done, $\Delta W=(\Delta P) Al=(\Delta P) V$

- $ W = (P_2 - P_1)V = (\frac{1}{2}m{v_1}^2 + mgh_1)- (\frac{1}{2}m{v_2}^2 + mgh_2)$

- $P_1+\frac{1}{2} P v_1^2+P gh_1=P_2+\frac{1}{2} P v_2^2+Pg h_2$

- <span style="color:blue">$P +\frac{1}{2} P v^2+P gh$ =  constant</span>

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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-10-chapter-04-mechanical-properties-of-fluids-4-lecture-4_5-vy-csifyfd4-02.jpg)

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<footer style = "font-size: 18px">Mechanical Properties of Fluids $\rightarrow$ Bernoulli Equation $\rightarrow$ <span style = "color:blue"> Bernoulli Equation </span> $\rightarrow$ Venturimeter $\rightarrow$ Venturimeter </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Venturimeter </primary>
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- A venturimeter consists of: The inlet section venturimeter is of the same diameter as that type or the pipe line in which it is installed which is followed by the short convergent section with a converging cone angle of $15-20$ and its length parallel to the axis is approximately equal to $2.7(D - \left.D_T\right)$, where $D$ in the pipe diameter and $D_T$ is the throat diameter. The cylindrical throat - the section of constant cross-section with its length equal to diameter. The flow are is minimum at the throat. The commonly diameter of throat is the the pipe diameter.

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<footer style = "font-size: 18px">Bernoulli Equation $\rightarrow$ Bernoulli Equation $\rightarrow$ <span style = "color:blue"> Venturimeter </span> $\rightarrow$ Venturimeter $\rightarrow$ Surface Tension </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Venturimeter </primary>
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- Venturimeter is a device which measure the speed of a fluid within a pipe.

- $A v=\text { constant } $

- $A_2 v_2-A_1 v_1$

- Since $A_2>A_1$, $v_1>v_2$.

- $P_1<P_2 $

- $P_1+\frac{1}{2} P{v_1^2}=P_2+\frac{1}{2} {v_2 ^2}$

- $v_1$, can be determined.

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<footer style = "font-size: 18px">Bernoulli Equation $\rightarrow$ Venturimeter $\rightarrow$ <span style = "color:blue"> Venturimeter </span> $\rightarrow$ Surface Tension $\rightarrow$ Surface Tension </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Surface Tension </primary>
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- Liquid itself does not have shape and it takes the shape of the container.

- Surface of a liquid acts like a membrane under tension.

- Surface tension is defined as the force per unit length.
   
- $ S = \frac{F}{l}$

- Unit: $N/m$

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

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Surface Tension </primary>
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- The surface tension decreases with increasing temperature.

- Let's take a vessel containing water.

- At A, the water molecule experiences an attractive force due to all other water molecules, uniform in all directions.

- At B, there are no counterparts above, so there will be forces which will not be balanced by upward forces.

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<footer style = "font-size: 18px">Surface Tension $\rightarrow$ Surface Tension $\rightarrow$ <span style = "color:blue"> Surface Tension </span> $\rightarrow$ Surface Energy $\rightarrow$ Angle of Contact </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Surface Energy </primary>
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- Surface energy is the work done per unit area to create a new surface.

- $S=\frac{F}{L} $

- $W=F \Delta x $

- $=(S L)\Delta x $

- $=S \Delta A $

- $ S =\frac{W}{\Delta A}=J / m^2$

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<footer style = "font-size: 18px">Surface Tension $\rightarrow$ Surface Tension $\rightarrow$ <span style = "color:blue"> Surface Energy </span> $\rightarrow$ Angle of Contact $\rightarrow$ Angle of Contact </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Angle of Contact </primary>
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- S $\cos \theta$ = Vertical component of surface tension and supports weight

- $2 \pi r S \cos \theta=W $

- $ \Rightarrow \cos \theta=\frac{W}{2 \pi rs} $

- $\Rightarrow \theta=\cos ^{-1}\left(\frac{W}{2 \pi r}\right)$

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<footer style = "font-size: 18px">Surface Tension $\rightarrow$ Surface Energy $\rightarrow$ <span style = "color:blue"> Angle of Contact </span> $\rightarrow$ Angle of Contact $\rightarrow$ Capillarity </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Angle of Contact </primary>
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- A insect can walk on the surface of water. The base of the insect is approximately spherical within a radius $10^{-4}m$. The insect has 6 legs and a mass 0.002gm. Calculate the angle...that its legs makes with the vertical. Assume the temperature to the $20^\circ C$.

- $2 \pi S\cos \theta=W$

- $({-28})\left(10^{-4} \mathrm{m}\right)(0.072 N / \mathrm{m})=\frac{1}{6}\left(2 \times 10^{-6} \mathrm{kg}\right)\left(9.8 \mathrm{ms}^{-2}\right) $

- $\cos _\theta =\frac{0.33}{0.90} -0.37 $

- $\theta=\cos ^{-1}(0.37) \simeq 68.2^{\circ}$

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<footer style = "font-size: 18px">Surface Energy $\rightarrow$ Angle of Contact $\rightarrow$ <span style = "color:blue"> Angle of Contact </span> $\rightarrow$ Capillarity $\rightarrow$ Capillarity </footer>

### <Success style="font-size:25px"> Mechanical Properties Of Fluids 4 L-4 </Success>
### <primary style="font-size:24px"> Capillarity </primary>
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- Volume of a cylinder = $\pi r^2 h$

- Magnitude of the vertically upward force due to the surface tension

- $F = (S cos \theta) L = 2 \pi r S cos \theta$

- (Since $L = 2 \pi r$)

- $ 2 \pi r S cos \theta = mg$
- $= \rho Vg = \pi r^2h\rho g$

- $h = \frac{2Scos \theta}{\rho gr}$

- $\theta \simeq 0 , cos\theta \simeq 1$

- $h = \frac{2S}{\rho gr}$
	
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<footer style = "font-size: 18px">Angle of Contact $\rightarrow$ Capillarity $\rightarrow$ <span style = "color:blue"> Capillarity </span> $\rightarrow$ Thank You $\rightarrow$  </footer>