PHY222 Midterm 2: Fluids, Thermodynamics, Waves, and Light – Equation Sheet Study Guide

Fluids

Basic Fluid Properties and Pressure

Fluids are substances that flow and take the shape of their container. Key properties include density, pressure, and buoyancy.

• Density (\( \rho \)): Mass per unit volume. $\rho = \frac{m}{V}$

• Pressure (\( P \)): Force per unit area, perpendicular to the surface. $P = \frac{dF_\perp}{dA}$

• Hydrostatic Pressure: Pressure difference due to fluid column height. $P_2 - P_1 = -\rho g (y_2 - y_1)$ $P = P_0 + \rho g h$

• Buoyant Force (\( F_B \)): Upward force on a submerged object. $F_B = m_{disp} g = \rho_F V_{disp} g$

Fluid Flow and Continuity

• Continuity Equation: Conservation of mass in fluid flow. $\frac{\Delta m}{\Delta t} = \rho A v$ For incompressible fluids: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$ $\frac{\Delta V}{\Delta t} = A v$ (volume flow rate) $A_1 v_1 = A_2 v_2$

• Bernoulli's Equation: Conservation of energy for flowing fluids. $P_1 + \rho g y_1 + \frac{1}{2} \rho v_1^2 = P_2 + \rho g y_2 + \frac{1}{2} \rho v_2^2$

• Viscous Flow (Poiseuille's Law): $F_v = \eta A \frac{v}{\ell}$ $\frac{\Delta V}{\Delta t} = \frac{\pi R^4 (P_1 - P_2)}{8 \eta \ell}$

Example: Calculating the pressure at a certain depth in water using $P = P_0 + \rho g h$.

Temperature, Thermal Expansion, and the Ideal Gas Law

Temperature Scales and Thermal Expansion

Temperature can be measured in Celsius, Fahrenheit, or Kelvin. Materials expand or contract with temperature changes.

• Temperature Conversions: $T_C = \frac{5}{9}(T_F - 32^\circ)$ $T_K = T_C + 273.15$

• Linear Expansion: $\Delta \ell = \alpha \ell_0 \Delta T$ $\ell = \ell_0 (1 + \alpha \Delta T)$

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

• Thermal Stress: $\frac{F}{A} = \alpha E \Delta T$

Ideal Gas Law

• Equation of State: $PV = nRT = NkT$ $k = \frac{R}{N_A}$

• Combined Gas Law: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

Example: Calculating the final volume of a gas after a temperature and pressure change using the combined gas law.

Kinetic Theory of Gases

Distribution of Molecular Speeds

The kinetic theory describes the motion of gas molecules and relates microscopic motion to macroscopic properties.

• Maxwell-Boltzmann Distribution: $f(v) = 4 \pi N \left( \frac{m}{2 \pi k T} \right)^{3/2} v^2 e^{-mv^2/2kT}$

• Average Speed: $\bar{v} = \frac{\sum_i v_i}{N} = \frac{1}{N} \int_0^\infty v f(v) dv = \sqrt{\frac{8kT}{\pi m}}$

• Root Mean Square (rms) Speed: $v_{rms} = \sqrt{\bar{v^2}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

• Most Probable Speed: $v_p = \sqrt{\frac{2kT}{m}}$

• Kinetic Energy: $K = \frac{1}{2} m v^2 = \frac{3}{2} k T$

• Pressure from Molecular Motion: $P = \frac{1}{3} \frac{N m \bar{v^2}}{V}$

Example: Calculating the rms speed of oxygen molecules at room temperature.

First Law of Thermodynamics

Internal Energy, Heat, and Work

The first law relates changes in internal energy to heat and work.

• Internal Energy of an Ideal Gas: $E_{int} = \frac{3}{2} N k T = \frac{3}{2} n R T$

• First Law: $\Delta E_{int} = Q - W$

• Heat Capacities: $\Delta E_{int} = n C_V \Delta T$ $Q = n C_V \Delta T$ (constant volume) $Q = n C_P \Delta T$ (constant pressure) $C_P = C_V + R$ $\gamma = \frac{C_P}{C_V}$

• Work Done by a Gas: $W = \int_{V_1}^{V_2} P dV$

• Latent Heat: $Q = \pm m L$

• Adiabatic Processes: $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$ $p_1 V_1^{\gamma} = p_2 V_2^{\gamma}$ $W = \frac{P_1 V_1^{\gamma} (V_2^{1-\gamma} - V_1^{1-\gamma})}{1-\gamma}$

• Heat Transfer: $Q = mc \Delta T$ $\frac{Q}{t} = kA \frac{T_1 - T_2}{\ell}$ $\frac{dQ}{dt} = -kA \frac{dT}{dx}$ $\frac{Q}{t} = Ae \sigma (T_1^4 - T_2^4)$

Example: Calculating the work done by a gas during an isothermal expansion.

Second Law of Thermodynamics

Heat Engines, Refrigerators, and Entropy

The second law introduces the concepts of efficiency, entropy, and the direction of heat flow.

• Efficiency of a Heat Engine: $e = \frac{W}{Q_H} = 1 - \left| \frac{Q_L}{Q_H} \right|$ $e_{ideal} = 1 - \frac{T_L}{T_H}$

• Coefficient of Performance (COP): $COP = \frac{|Q_L|}{|W|} = \frac{|Q_L|}{|Q_H| - |Q_L|}$ $COP_{ideal} = \frac{T_L}{T_H - T_L}$

• Entropy (S): $\Delta S_{rev} = \int_a^b \frac{dQ}{T}$ $\Delta S_{isotherm} = \frac{Q}{T}$ $\Delta S = \Delta S_{syst} + \Delta S_{env} > 0$ $S = k \ln W$

Example: Calculating the change in entropy for a reversible isothermal process.

Wave Motion

Wave Properties and Equations

Waves transfer energy through oscillations. Key properties include speed, frequency, and wavelength.

• Wave Speed: $v = \lambda f = \frac{\omega}{k}$

• Frequency and Period: $f = \frac{\omega}{2\pi} = \frac{1}{T}$

• Wavelength: $\lambda = \frac{2\pi}{k}$

• Wave Function: $D(x, t) = A \sin(kx \pm \omega t)$

• Wave Equation: $\frac{\partial^2 D(x, t)}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 D(x, t)}{\partial t^2}$

• Speed on a String: $v = \sqrt{\frac{F_T}{\mu}}$

• Speed in Bulk Media: $v = \sqrt{\frac{B}{\rho}}$ (bulk modulus) $v = \sqrt{\frac{E}{\rho}}$ (Young's modulus)

• Energy in a Wave: $E = 2 \pi^2 \rho S v t f^2 A^2$ $\bar{P} = \frac{E}{t}$ $I = \frac{\bar{P}}{S} = \frac{\bar{P}}{4 \pi r^2}$ $\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}$

• Standing Waves: $DSW(x, t) = 2A \sin k_n x \cos \omega_n t$ $\lambda_n = \frac{2\ell}{n}$ $f_n = \frac{n v}{2\ell}$ $k_n = \frac{n \pi}{\ell}$ $\omega_n = \frac{n \pi v}{\ell}$ (n = 1, 2, 3...)

Example: Calculating the fundamental frequency of a string fixed at both ends.

Sound

Sound Waves and Properties

Sound is a longitudinal wave in a medium. Its properties include pressure variation, intensity, and resonance.

• Pressure Variation: $\Delta P(x, t) = -B \frac{\partial D(x, t)}{\partial x} = -\Delta P_M \cos(kx - \omega t)$ $\Delta P_M = B A k$

• Intensity and Decibel Level: $\beta (dB) = 10 \log \frac{I}{I_0}$ $I = \frac{(\Delta P_M)^2}{2 v \rho}$

• Interference (Path Difference): $|AC - BC| = m \lambda$ (constructive, m = 0, 1, ...) $|AC - BC| = (m + 1/2) \lambda$ (destructive, m = 0, 1, ...)

• Resonance in Tubes: $f_n = \frac{n v}{2\ell}$ (open tube, n = 1, 2, 3...) $f_n = \frac{n v}{4\ell}$ (closed tube, n = 1, 3, 5...)

• Beats: $f_{beat} = |f_1 - f_2|$ $D_{beat}(x, t) = 2A \cos\left(2\pi \frac{f_{beat}}{2} t\right) \sin\left(2\pi \frac{f_1 + f_2}{2} t\right)$

• Doppler Effect: $f_L = \frac{v \pm v_L}{v \mp v_S} f_S$

Example: Calculating the beat frequency when two tuning forks of slightly different frequencies are sounded together.

The Wave Nature of Light; Interference

Refraction, Polarization, and Interference

Light exhibits wave properties such as refraction, polarization, and interference.

• Index of Refraction: $n = \frac{c}{v}$ $\lambda_n = \frac{\lambda}{n}$

• Snell's Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

• Polarization: $I_1 = \frac{I_0}{2}$ (initially unpolarized light) $I_0 \cos^2 \theta$ (initially polarized light) $\tan \theta_p = \frac{n_2}{n_1}$

• Double-Slit Interference: $\delta = \frac{2\pi}{\lambda} d \sin \theta$ $I = I_0 \cos^2 \frac{\delta}{2} = I_0 \cos^2 \left( \frac{\pi d \sin \theta}{\lambda} \right)$ $y_m = \ell m \frac{\lambda}{d}$ $d \sin \theta = m \lambda$ (constructive, m = 0, \pm1, ...) $d \sin \theta = (m + 1/2) \lambda$ (destructive, m = 0, \pm1, ...)

• Thin Film Interference: $\phi = 0$ (if $n_1 > n_2$), $\phi = \pi$ (if $n_1 < n_2$) $2t - |\phi_2 - \phi_1| \frac{\lambda_n}{2\pi} = m \lambda_n$ (constructive, m = 0, 1, ...) $2t - |\phi_2 - \phi_1| \frac{\lambda_n}{2\pi} = (m - 1/2) \lambda_n$ (destructive, m = 1, 2, ...)

Example: Calculating the position of bright fringes in a double-slit experiment.

Additional info: This study guide expands on the equation sheet by providing definitions, context, and examples for each formula, making it suitable for exam preparation in a college-level physics course.