MP Board Class 11 Physics Revision Notes

Chapter 10: Mechanical Properties of Fluids (द्रवों के यांत्रिक गुण)

Density ($\rho$): Mass per unit volume. $$\rho = \frac{\text{Mass } (m)}{\text{Volume } (V)}$$
- SI Unit: $\text{kg/m}^3$ | Dimensional Formula: $[M^1 L^{-3} T^0]$
Relative Density / Specific Gravity (आपेक्षिक घनत्व): $$\text{Relative Density} = \frac{\text{Density of Substance}}{\text{Density of Water at } 4^\circ\text{C}}$$ (Note: It is a unitless and dimensionless quantity).
Fluid Pressure ($P$): Normal force applied per unit area. $$P = \frac{\text{Force } (F)}{\text{Area } (A)}$$
- SI Unit: $\text{N/m}^2$ or Pascal ($\text{Pa}$)
- 1 Atmospheric Pressure ($1\text{ atm}$): $1.013 \times 10^5\text{ Pa} = 760\text{ mm of Hg}$
Hydrostatic Pressure (Variation with Depth): Pressure at depth $h$ below the free surface of a liquid at rest: $$P = P_0 + \rho g h$$
- Where $P_0$ = Atmospheric Pressure, $\rho$ = Liquid density, $g$ = Acceleration due to gravity.
- Gauge Pressure ($P_{\text{gauge}}$): $P - P_0 = \rho g h$
- Absolute Pressure ($P_{\text{abs}}$): $P_0 + \rho g h$

Pascal's Law: Pressure applied to an enclosed, incompressible fluid is transmitted equally and undiminished to every portion of the fluid and the walls of the vessel. $$\frac{F_1}{A_1} = \frac{F_2}{A_2}$$
- Applications: Hydraulic Lift, Hydraulic Brakes.
Archimedes' Principle (आर्किमिडीज का सिद्धांत): When a body is partially or fully immersed in a fluid, it experiences an upward force (Upthrust/Buoyant Force $F_B$) equal to the weight of the fluid displaced by it. $$F_B = V_{\text{in}} \cdot \rho_{\text{fluid}} \cdot g$$
- Where $V_{\text{in}}$ = Volume of the immersed body.

Streamline Flow (धारारेखी प्रवाह): Liquid particles follow a smooth path; velocity at any point remains constant with time.
Turbulent Flow (विक्षुब्ध प्रवाह): Irregular flow above a critical velocity where liquid motion becomes chaotic.
Equation of Continuity (संतत्य समीकरण): Based on the Law of Conservation of Mass. For an incompressible, non-viscous fluid flowing through a pipe of varying cross-section: $$A_1 v_1 = A_2 v_2 \implies A \cdot v = \text{Constant}$$
- $A$ = Area of cross-section, $v$ = Velocity of flow.
- Mass flow rate: $\frac{dm}{dt} = \rho A v$
Reynolds Number ($R_e$): A dimensionless number that determines the nature of liquid flow. $$R_e = \frac{\rho v d}{\eta}$$
- $R_e < 1000$: Streamline / Laminar Flow
- $R_e > 2000$: Turbulent Flow
- $1000 < R_e < 2000$: Unstable Flow

Bernoulli's Principle: For an ideal (incompressible and non-viscous) fluid in a streamlined flow, the total energy per unit mass (or volume) remains constant along a streamline.
- Conservation Law: Based on Conservation of Energy.
$$P + \frac{1}{2} \rho v^2 + \rho g h = \text{Constant}$$
- Pressure Head: $\frac{P}{\rho g}$
- Velocity Head: $\frac{v^2}{2g}$
- Gravitational/Elevation Head: $h$

Torricelli’s Law (Speed of Efflux): Speed of liquid coming out of an orifice at depth $h$: $$v = \sqrt{2gh}$$
Venturimeter: Device used to measure the rate of flow of liquid through a pipe. $$Q = A_1 A_2 \sqrt{\frac{2gh}{A_1^2 - A_2^2}}$$
Dynamic Lift: Dynamic lift on an airplane wing (aerofoil), Magnuseffect in a spinning ball.

Viscosity: The property of a fluid by virtue of which it opposes relative motion between its different layers.
Newton's Law of Viscosity: $$F = -\eta A \frac{dv}{dx}$$
- $\eta$ = Coefficient of viscosity
- $\frac{dv}{dx}$ = Velocity gradient
- SI Unit of $\eta$: $\text{N}\cdot\text{s/m}^2$ or $\text{Pa}\cdot\text{s}$ or Poiseuille (Pl)
- CGS Unit: Poise ($1\text{ Pl} = 10\text{ Poise}$)
Stoke's Law: Viscous drag force on a spherical body of radius $r$ moving with velocity $v$ through a fluid of viscosity $\eta$: $$F = 6 \pi \eta r v$$
Terminal Velocity ($v_t$): Maximum constant velocity acquired by a spherical body falling through a viscous medium. $$v_t = \frac{2}{9} \frac{r^2 (\rho - \sigma) g}{\eta}$$
- $\rho$ = Density of the sphere
- $\sigma$ = Density of the fluid

Surface Tension ($T$ or $S$): Force per unit length acting on an imaginary line drawn on the liquid surface. $$T = \frac{F}{L}$$
- SI Unit: $\text{N/m}$ or $\text{J/m}^2$ | Dimensional Formula: $[M^1 L^0 T^{-2}]$
- Effect of Temperature: Surface tension decreases with an increase in temperature.
Surface Energy ($W$): Work done to increase the surface area of a liquid film by $\Delta A$. $$W = T \cdot \Delta A$$
Excess Pressure inside Droplets and Bubbles:
- Inside a liquid drop (1 surface): $$P_{\text{excess}} = \frac{2T}{R}$$
- Inside a liquid bubble inside liquid: $$P_{\text{excess}} = \frac{2T}{R}$$
- Inside a soap bubble in air (2 surfaces): $$P_{\text{excess}} = \frac{4T}{R}$$
Angle of Contact ($\theta$):
- $\theta < 90^\circ$ (Acute): Liquid wets the container wall (e.g., Water in glass), concave meniscus.
- $\theta > 90^\circ$ (Obtuse): Liquid does not wet the wall (e.g., Mercury in glass), convex meniscus.
- $\theta = 0^\circ$: Pure water and clean glass.
Ascent Formula (Capillary Rise / Height of Liquid Column): $$h = \frac{2 T \cos\theta}{r \rho g}$$
- $r$ = Radius of the capillary tube
- Zurin's Law: $h \propto \frac{1}{r} \implies h \cdot r = \text{Constant}$

Physical Quantity	Symbol	Formula	SI Unit
Pressure	$P$	$P = F / A$	$\text{Pa}$ or $\text{N/m}^2$
Gauge Pressure	$P_g$	$P_g = \rho g h$	$\text{Pa}$
Equation of Continuity	-	$A_1 v_1 = A_2 v_2$	$\text{m}^3/\text{s}$ (Flow rate)
Bernoulli's Equation	-	$P + \frac{1}{2}\rho v^2 + \rho g h = \text{C}$	$\text{J/m}^3$ or $\text{Pa}$
Terminal Velocity	$v_t$	$\frac{2 r^2 (\rho - \sigma) g}{9 \eta}$	$\text{m/s}$
Surface Tension	$T$	$T = F / L$	$\text{N/m}$
Capillary Rise	$h$	$\frac{2 T \cos\theta}{r \rho g}$	$\text{m}$

Derivations to Focus On:
- Bernoulli's Theorem and its derivation.
- Ascent Formula ($h = \frac{2 T \cos\theta}{r \rho g}$) for capillary tube.
- Terminal Velocity expression using Stoke's Law.
Conceptual Questions:
- Why do two boats moving parallel close to each other pull together? (Bernoulli's Theorem)
- Why action of spray pump/atomizer works? (Bernoulli's Principle)
- Effect of impurities and temperature on surface tension.