It is known that density $\mathrm{\rho }$ of air decreases with height $y$ as $\mathrm{\rho }=\mathrm{\rho }₀{\mathrm{e}}^{-\mathrm{y}/\mathrm{y}₀}$. Where ${\rho }_0= 1.25 \mathrm{kg} {\mathrm{m}}^{-3}$ is the density at sea level, and ${\mathrm{y}}_0$ is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of $g$ remains constant.

 

(i) It is defined as the force acting perpendicular to the surface of an object per unit surface area. Pressure, $P= \frac{F}{A}$

(ii) Pressure exerted by a fluid of density ρ at a depth $h$, $P= h\rho g$.

(iii) Absolute pressure at depth h below the free surface of a liquid of density ρ, $P= P_0+hpg$. where, P0 = atmospheric pressure.

(iv) Gauge Pressure $= (P-P_0)= h\rho g$

2. Units of pressure:

(i) $1 \mathrm{bar}= {10}^5 \mathrm{Pa}$

(ii) $1 \mathrm{torr} = 1\mathrm{mm} \mathrm{of} \mathrm{Hg}= 133.3\mathrm{ }\mathrm{Pa}$

(iii) $1 \mathrm{atm}= \mathrm{76 }\mathrm{cm} \mathrm{of} \mathrm{Hg}= 1.{\mathrm{013\times 10}}^5\mathrm{ }\mathrm{Pa}.$

3. Pascal's law:

It states that the pressure applied to an enclosed liquid is transmitted undiminished to every portion of the liquid and the wall of the container.

4. Archimedes’ principle:

It states that when a body is immersed partially or entirely in a fluid, it loses its weight, which is equal to the weight of the fluid displaced by the body.

5. Law of floatation:

A body floats in a liquid, if weight of the liquid displaced by the immersed portion of the body is equal to the weight of the body.

6. Types of flow:

(i) Streamline flow: 

The flow of the fluid is said to be Streamline, if at any given point, the velocity of each passing fluid particle remains constant in time.

(ii) Turbulent flow:

The flow of fluid in which velocity of all particles crossing a given point is not same and the motion of the fluid becomes irregular is called turbulent flow.

7. Critical velocity:

The maximum velocity of the liquid or fluid up to which its flow is streamlined is called critical velocity. The flow of fluid becomes turbulent if velocity of flow is more than the critical velocity

8. Equation of continuity:

For an ideal fluid flowing in a pipe, $A_1{v}_1= A_2{v}_2$ or $Av=$ constant.

9. Bernoulli's Theorem:

(i) It states that for an ideal fluid having streamline flow, the sum of pressure energy per unit volume, K.E. per unit volume and P.E. energy per unit volume is constant.

(ii) $\frac{P}{\rho }+\frac{1}{2}{v}^2+gh=$ constant

10. Viscosity:

(i) The property of a liquid (or fluid) by virtue of which an opposing force comes into play between different layers of the liquid whenever there is a relative motion between these layers is called viscosity.

(ii) Viscous drag or force between two layers of the fluid, $F= -\eta A\frac{dv}{dx}$

(iii) SI unit of coefficient viscosity is decapoise. $1\mathbf{ }\mathrm{decapoise}= 10\mathbf{ }\mathrm{poise}$.

(iv) Viscosity of liquid decreases with the increase in temperature, whereas viscosity of gases increases with the increase in temperature.

11. Stoke’s law:

(i) According to Stoke, viscous drag acting on a spherical body of radius r moving with velocity v in a fluid of coefficient of viscosity η is given by, $F= 6 \pi \eta r v$

(ii) Terminal velocity of the same spherical body, $v_T= \frac{2r^2(\rho -\sigma )g}{9\eta }$

12. Flow in a capillary tube:

Volume of liquid flowing per second through a capillary tube of length $l$ and radius $r$ is given by $V= \frac{\Delta P}{\frac{8\eta l}{\pi r^4}}$, where $V$ is the volume/second

13. Surface tension:

(i) Surface tension, undefined Force.

(ii) Excess of pressure inside air bubble in a liquid, undefined where undefined is radius of air bubble.

(iii) Excess pressure inside a soap bubble, undefined where undefined is radius of soap bubble.

(iv) Excess pressure inside a liquid drop, undefined where $R$ is radius of liquid drop.

(v) Surface energy undefined is defined as the potential energy of the molecules at the free surface of the liquid. Surface energy per unit area magnitude is equal to surface tension.