2.1 Pressure in Liquids
Pressure in liquids is caused by the weight of the liquid column acting on a surface area at a given depth.
1. Formula for Liquid Pressure
The pressure at depth $h$ in a liquid of density $\rho$ is given by:
$$P = h \rho g$$
- $P$ = Pressure ($\text{Pa}$ or $\text{N m}^{-2}$)
- $h$ = Depth of liquid column ($\text{m}$)
- $\rho$ = Density of liquid ($\text{kg m}^{-3}$)
- $g$ = Gravitational acceleration ($\text{m s}^{-2}$)
2. Factors Affecting Liquid Pressure
- Depth ($h$): Liquid pressure increases linearly with depth. Points at the same horizontal level experience identical pressure.
- Density of Liquid ($\rho$): Higher density liquid exerts greater pressure at the same depth.
- Independent Factors: Pressure in a liquid does not depend on the shape or surface area of the container.
3. Total Pressure at Depth
When atmospheric pressure ($P_{\text{atm}}$) is considered, the total (absolute) pressure at depth $h$ is:
$$P_{\text{total}} = P_{\text{atm}} + h \rho g$$
4. Applications of Liquid Pressure
- Dams: Dam walls are built thicker at the base to withstand higher liquid pressure at greater depths.
- Submarines: Built with reinforced thick steel hulls to withstand extreme pressure at deep levels.
- Intravenous (IV) Drips: Placed at a elevated height so the liquid pressure exceeds blood pressure, allowing fluid flow into the vein.
2.2 Atmospheric Pressure
Atmospheric Pressure ($P_{\text{atm}}$) is the force per unit area exerted by the weight of the air column surrounding the Earth's surface.
1. Value of Standard Atmospheric Pressure
- $1\text{ atm} = 760\text{ mmHg} = 76\text{ cmHg}$
- $1\text{ atm} \approx 10.3\text{ m H}_2\text{O}$
- $1\text{ atm} = 1.013 \times 10^5\text{ Pa} \approx 10^5\text{ N m}^{-2}$
2. Measuring Atmospheric Pressure
- Simple Mercury Barometer: Uses a vertical glass tube inverted over a dish of mercury. Atmospheric pressure balances a $76\text{ cm}$ mercury column ($P_{\text{atm}} = h \rho g$).
- Fortin Barometer: High-precision laboratory barometer using mercury.
- Aneroid Barometer: Portable mechanical barometer containing an evacuated metal capsule that expands or contracts with changing air pressure (used as altimeters in aircraft).
3. Instruments for Measuring Gas Pressure
- Manometer: U-shaped tube containing liquid.
- If liquid levels are equal: Gas pressure $P_{\text{gas}} = P_{\text{atm}}$.
- If gas side is lower: $P_{\text{gas}} = P_{\text{atm}} + h \rho g$.
- If open side is lower: $P_{\text{gas}} = P_{\text{atm}} - h \rho g$.
- Bourdon Gauge: Mechanical instrument containing a coiled C-shaped tube that uncurls when gas pressure increases inside.
2.3 Pascal's Principle
Pascal's Principle states that pressure applied to an enclosed fluid is transmitted equally and undiminished to every portion of the fluid and to the walls of the container.
1. Hydraulic System Equation
A small input force ($F_1$) on a small piston ($A_1$) creates a pressure $P$, which produces a larger output force ($F_2$) on a larger piston ($A_2$):
$$P = \frac{F_1}{A_1} = \frac{F_2}{A_2} \implies F_2 = F_1 \left(\frac{A_2}{A_1}\right)$$
2. Conservation of Energy in Hydraulic Systems
Assuming no fluid friction, work done by the input piston equals work done on the output piston:
$$W_{\text{in}} = W_{\text{out}} \implies F_1 d_1 = F_2 d_2$$
Where $d_1$ and $d_2$ are the displacement distances of the input and output pistons, respectively.
3. Applications of Pascal's Principle
- Hydraulic Jack: Used in vehicle repair shops to lift heavy loads with minimal effort.
- Hydraulic Brakes: Force on a brake pedal is transmitted via hydraulic fluid to press brake shoes against discs on all four wheels simultaneously.
2.4 Archimedes' Principle
Archimedes' Principle states that an object fully or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
1. Formula for Buoyant Force ($F_B$)
$$F_B = \rho V g$$
- $\rho$ = Density of the fluid ($\text{kg m}^{-3}$)
- $V$ = Volume of fluid displaced = Volume of submerged portion of the object ($\text{m}^3$)
- $g$ = Gravitational acceleration ($\text{m s}^{-2}$)
2. Principle of Flotation
- An object floats when Buoyant Force = Total Weight of Object ($F_B = W$).
- An object sinks when Weight exceeds maximum Buoyant Force ($W > F_B$).
- Density condition for floating: Average density of object ($\rho_{\text{object}}$) $\le$ Density of fluid ($\rho_{\text{fluid}}$).
3. Applications of Archimedes' Principle
- Submarines: Flood or purge ballast tanks with water to adjust overall density, allowing sinking, floating, or hovering underwater.
- Hydrometers: Used to measure liquid density based on depth of flotation in a test liquid.
- Hot Air Balloons: Control altitude by heating air inside the balloon envelope (reducing air density inside) to generate sufficient upward buoyant force.
- Plimsoll Line on Ships: Safety lines painted on ship hulls indicating maximum safe loading depths in various ocean water densities.
2.5 Bernoulli's Principle
Bernoulli's Principle states that as the velocity of a moving fluid increases, the pressure within the fluid decreases, and vice versa.
1. Fluid Flow through a Venturi Tube
- A narrow constriction in a pipe causes fluid flow speed to increase.
- Higher speed at constriction leads to a localized region of lower pressure.
- Pressure differences can be observed via vertical manometer tubes attached to narrow and wide segments.
2. Aerofoil and Lift Generation
- Air travels faster over the curved top surface of an aerofoil wing $\implies$ Low Pressure region.
- Air travels slower underneath the flatter lower surface $\implies$ High Pressure region.
- The resulting pressure difference ($\Delta P = P_{\text{bottom}} - P_{\text{top}}$) produces an upward Lift Force:
$$F_{\text{lift}} = (P_{\text{bottom}} - P_{\text{top}}) \times A$$
3. Applications of Bernoulli's Principle
- Bunsen Burner: High-velocity gas jet lowers internal pressure, drawing outside air through the air hole for complete combustion.
- Insecticide Sprayer: Air pushed rapidly past a nozzle creates low pressure, drawing liquid insecticide up the vertical feed tube.
- Carburettor: Air entering a Venturi choke creates low pressure to siphon fuel into the air intake stream.