Study Guide: Fluids, Vibrations & Waves, Sound, and Kinetic Theory (Chapters 10–13)

Fluids and Fluid Mechanics

Phases of Matter and Fluids

The three common phases of matter are solid, liquid, and gas. Liquids and gases are collectively called fluids because they have the ability to flow.

• Solids: Fixed shape and volume.

• Liquids: Fixed volume, variable shape.

• Gases: Variable shape and volume.

• Fluids: Substances that flow (liquids and gases).

Density and Specific Gravity

Density is defined as mass per unit volume:

• Formula: $\rho = \frac{m}{V}$

• Units: kg/m3

• Specific Gravity (SG): Ratio of the density of a material to the density of water at 4°C.

Pressure in Fluids

Pressure is the force applied per unit area:

• Formula: $P = \frac{F}{A}$

• Units: N/m2 (Pascal, Pa)

• Pressure at Depth: $P = \rho g h$

• Atmospheric Pressure: Standard value is $1.013 \times 10^5$ N/m2

• Gauge Pressure: Difference between absolute pressure and atmospheric pressure.

Pascal’s Principle and Hydraulic Systems

Pascal’s principle states that an external pressure applied to a confined fluid is transmitted undiminished throughout the fluid.

• Hydraulic Lift: $\frac{F_{out}}{A_{out}} = \frac{F_{in}}{A_{in}}$

• Used in car lifts and other machinery.

Measuring Pressure

• Manometer: Measures pressure in a fluid.

• Barometer: Measures atmospheric pressure.

Archimedes’ Principle and Buoyancy

An object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced.

• Buoyant Force: $F_B = m_F g = \rho_F V_{displ} g$

• Explains why objects float or sink.

• Example: A boat floats because the buoyant force equals its weight.

Vibrations and Waves

Simple Harmonic Motion (SHM)

SHM occurs when the restoring force is proportional to the negative displacement from equilibrium.

• Restoring Force: $F = -k x$

• Amplitude: Maximum displacement from equilibrium.

• Period (T): Time for one complete cycle.

• Frequency (f): Number of cycles per second; $f = \frac{1}{T}$

• Period for Mass-Spring: $T = 2\pi \sqrt{\frac{m}{k}}$

• Energy in SHM: $E = \frac{1}{2} m v^2 + \frac{1}{2} k x^2$

Pendulums

• Simple Pendulum Period: $T = 2\pi \sqrt{\frac{\ell}{g}}$ (for small amplitudes)

• Assumes negligible friction.

Damped and Resonant Oscillations

• Damping: Friction causes amplitude to decrease over time; energy transforms to heat.

• Resonance: Large amplitude occurs when driving frequency matches natural frequency.

Waves: Types and Properties

• Wave: Disturbance that travels through a medium, carrying energy.

• Transverse Wave: Oscillations perpendicular to direction of travel (e.g., waves on a string).

• Longitudinal Wave: Oscillations parallel to direction of travel (e.g., sound).

• Wavelength (λ): Distance between successive crests.

• Wave Speed: $v = \lambda f$

• Amplitude: Maximum height or depth from equilibrium.

Wave Intensity and Reflection

• Intensity: Energy per unit time per unit area (W/m2).

• Intensity for Point Source: $I \propto \frac{1}{r^2}$

• Law of Reflection: Angle of reflection equals angle of incidence.

Interference and Standing Waves

• Superposition Principle: Resultant displacement is sum of individual displacements.

• Constructive Interference: Waves add to increase amplitude.

• Destructive Interference: Waves subtract to decrease amplitude.

• Standing Waves: Produced by interference of waves traveling in opposite directions.

• Nodes: Points of zero amplitude.

• Antinodes: Points of maximum amplitude.

Sound

Nature and Speed of Sound

Sound is a longitudinal wave that travels in air and other materials. Its speed increases with temperature.

• Speed of Sound in Air (20°C): 343 m/s

Pitch, Frequency, and Audible Range

• Pitch: Determined by frequency; higher frequency means higher pitch.

• Audible Range: 20 Hz to 20,000 Hz for humans.

Sound Intensity and Decibel Scale

• Intensity: Related to amplitude squared.

• Sound Level (β): Measured in decibels (dB).

• Formula: $\beta = 10 \log \left( \frac{I}{I_0} \right)$

• Reference Intensity (I0): $10^{-12}$ W/m2

Musical Instruments and Harmonics

• Standing Waves: Produced in strings and air columns.

• Fundamental Frequency: Lowest frequency, wavelength $\lambda_1 = 2\ell$

• Harmonics: Higher frequencies, whole-number multiples of fundamental.

Temperature and Kinetic Theory

Atomic Theory of Matter

All matter is composed of atoms, typically about $10^{-10}$ m in diameter.

• Atomic Mass Unit (u): Based on C-12 isotope (12.0000 u).

States of Matter and Temperature

• Solids, Liquids, Gases: Differ by strength of intermolecular forces and average speed of particles.

• Temperature: Measure of hotness; measured in °C, °F, and K.

• Standard Points: Freezing (0°C, 32°F, 273.15 K), Boiling (100°C, 212°F, 373.15 K).

• Kelvin-Celsius Relation: $T(K) = T(°C) + 273.15$

Thermal Equilibrium

• Objects in contact reach same temperature over time.

Thermal Expansion

• Linear Expansion: $\Delta \ell = \alpha \ell_0 \Delta T$

• Volume Expansion: $\Delta V = \beta V_0 \Delta T$

• Coefficient of Volume Expansion: $\beta \approx 3\alpha$ for solids.

• Water Exception: Volume decreases from 0°C to 4°C as temperature increases.

Ideal Gas Law

• Equation: $PV = nRT$

• R: 8.314 J/(mol·K) (SI units), 0.08214 L·atm/(mol·K) (other units)

• One Mole: Mass in grams equals atomic/molecular mass.

• Avogadro’s Number: $N_A = 6.02 \times 10^{23}$

• Alternate Form: $PV = NkT$

• Boltzmann’s Constant: $k = \frac{R}{N_A} = 1.38 \times 10^{-23}$ J/K

Kinetic Theory of Gases

• Gas molecules move rapidly and randomly.

• Average Kinetic Energy: $KE_{avg} = \frac{1}{2} m v^2_{avg} = \frac{3}{2} k T$

• Wide distribution of molecular speeds at any moment.

Summary Table: Key Formulas and Concepts

Concept	Formula	Units
Density	$\rho = \frac{m}{V}$	kg/m3
Pressure	$P = \frac{F}{A}$	Pa (N/m2)
Pressure at Depth	$P = \rho g h$	Pa
Buoyant Force	$F_B = \rho_F V_{displ} g$	N
SHM Force	$F = -k x$	N
SHM Period (Spring)	$T = 2\pi \sqrt{\frac{m}{k}}$	s
SHM Period (Pendulum)	$T = 2\pi \sqrt{\frac{\ell}{g}}$	s
Wave Speed	$v = \lambda f$	m/s
Sound Level	$\beta = 10 \log \left( \frac{I}{I_0} \right)$	dB
Linear Expansion	$\Delta \ell = \alpha \ell_0 \Delta T$	m
Volume Expansion	$\Delta V = \beta V_0 \Delta T$	m3
Ideal Gas Law	$PV = nRT$	Pa·m3
Kinetic Energy (Gas)	$KE_{avg} = \frac{3}{2} k T$	J