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# Chapter 27: Equations and Fundamental Concepts

Difficulty Level: At Grade Created by: CK-12

Simple Harmonic Motion and Wave Motion

TTsp=1/f=2πmk\begin{align*}T & = 1 / f \\ T_{sp} & = 2 \pi \sqrt{\frac{m} {k}}\end{align*}

vTP=λf=2πLg\begin{align*}v & = \lambda f \\ T_P & = 2 \pi \sqrt{\frac{L} {g}}\end{align*}

fnfnfbeat=nv2L=nv4L=|f1f2|nodes at both ends(n is odd)node at one end\begin{align*}f_n & = \frac{nv} {2L} && \text{nodes at both ends} \\ f_n & = \frac{nv} {4L}&& \text{(n is odd)} \text{node at one end} \\ f_{beat} & = |f_1 - f_2|\end{align*}

vsound=343m/s\begin{align*}v_{sound} = 343 \;\mathrm{m/s}\end{align*} (in air at 20C\begin{align*}20 \;\mathrm{C}\end{align*})

A note: 440Hz\begin{align*}440 \;\mathrm{Hz}\end{align*}

C note: 524Hz\begin{align*}524 \;\mathrm{Hz}\end{align*}

D note: 588Hz\begin{align*}588 \;\mathrm{Hz}\end{align*}

E note: 660Hz\begin{align*}660 \;\mathrm{Hz}\end{align*}

G note: 784Hz\begin{align*}784 \;\mathrm{Hz}\end{align*}

Fluids and Thermodynamics

3/2 kTPPP+(ρgh)+(12ρv2)ϕC=<12 mv2>avg=F/A=P0+ρgh=0=Av=K+273.15\begin{align*}3/2 \ kT & = <\frac{1} {2}\ \text{mv}^2>_{avg} \\ P & = F / A \\ P & = P_0 + \rho gh \\ \triangle P + \triangle (\rho gh) + \triangle \left (\frac{1} {2} \rho v^2 \right ) & = 0 \\ \phi & = A \cdot v \\ ^\circ C & = ^\circ K + 273.15\end{align*}

PVFbuoyQinWkη=NkT=nRT=(ρwaterVdisplaced)g=W+U+Qout=P V=12ρv2;u=ρgh=W/Qin;ηCarnot=1(Tlow/Thigh)\begin{align*}PV & = NkT = nRT \\ F_{buoy} & = - (\rho_{water}V_{displaced})g \\ Q_{in} &= W + \triangle U + Q_{out} \\ W & = P\triangle\ \text{V} \\ k & = \frac{1} {2} \rho v^2; u = \rho gh \\ \eta & = W/Q_{in}; \eta_{Carnot} = 1 - (T_{low} / T_{high})\end{align*}

kρairRρwaterPATMOSPHNavo=1.381×1023 J/K=1.29 kg/m3=8.315 J/mol-K=1000 kg/m3=101,000 N/M2=6.022×1023 mol1\begin{align*}k & = 1.381 \times 10^{-23}\ \text{J/K} \\ \rho_{air} & = 1.29\ \text{kg/m}^3\\ R & = 8.315\ \text{J/mol-K} \\ \rho_{water} & = 1000\ \text{kg/m}^3 \\ P_{ATMOSPH} & = 101,000\ \text{N/M}^2 \\ N_{avo} & = 6.022 \times 10^{23}\ \text{mol}^{-1}\end{align*}

Properties of Fundamental Particles

mprotonqelectronrhydrogen atom=1.6726×1027 kg=qproton=1.602×1019 C0.529×1010 m\begin{align*}m_{proton} & = 1.6726 \times 10^{-27}\ \text{kg} \\ q_{electron} & = - q_{proton} = -1.602 \times 10^{-19}\ \text{C} \\ r_{hydrogen \ atom} & \approx 0.529 \times 10^{-10}\ \text{m}\end{align*}

melectron1 amu E=9.109×1031 kg=1.6605×1027 kg=931.5 Mev/c2= mc2\begin{align*}m_{electron} & = 9.109 \times 10^{-31}\ \text{kg} \\ 1\ \text{amu} & = 1.6605 \times 10^{-27}\ \text{kg} = 931.5\ \text{Mev/c}^2 \\ \triangle\ \text{E} & = \triangle\ \text{mc}^2\end{align*}

mneutron=1.6749×1027 kg\begin{align*}m_{neutron} = 1.6749 \times 10^{-27}\ \text{kg}\end{align*}

Radioactivity, Nuclear Physics, and Quantum Mechanics

(x)(p)λN1 ev(energy of a photon)h/4π=h/p=N0 (12) t/tH1240 nm\begin{align*}(\triangle x)(\triangle p) & \approx h/4\pi \\ \lambda & = h/p \\ N & = \mathrm{N0} \ \left (\frac{1} {2} \right ) \ t/t_H \\ 1\ \text{ev} & \rightarrow 1240\ \text{nm} \\ \text{(energy of a photon)}\end{align*}

(E)(t)EphotonKmaxh/4π=hf=pc=qV=hf+ϕ\begin{align*}(\triangle E)(\triangle t) & \approx h/4\pi \\ E_{photon} & = hf = pc \\ Kmax & = qV = hf + \phi\end{align*}

\begin{align*}h & = 6.626 \times 10^{-34}\ \text{J.s} \\ ^AZ & = \text{element \ Z}\ \text{with A nucleons} \\ ^{14}C: t_H & = 5,730\ \text{years} (\text{half life} = t_h)\\ ^{239}Pu: t_H & = 24,119\ \text{years} \\ E_o & = - 13.605\ \text{ev} \text{(Hydrogen ground state)}\end{align*}

Light

\begin{align*}\lambda_{blue} & \approx 450\ \text{nm} \\ \lambda_{green} & \approx 500\ \text{nm} \\ \lambda_{red} & \approx 600\ \text{nm} \end{align*}

\begin{align*}n_i \sin(\theta_i) & = n_r \sin(\theta_r) \\ c & = 2.998 \times 10^8\ \text{m/s} \\ m\lambda & = \text{dsin}(\theta)\end{align*}

\begin{align*}n_{air} & \approx n_{vacuum} = 1.00 \\ n_{water} & = 1.33 \\ n & = c/v_{material}\end{align*}

primary: Red, Green, Blue

secondary: Magenta, Cyan, Yellow

\begin{align*} \frac{1} {f} = \frac{1} {d_o} + \frac{1} {di} && M = h_i/h_o = d_i/d_o\end{align*}

Electricity and Magnetism

\begin{align*}F_E & = k \ q_1q_2 / r^2 \\ E & = F_E / q \\ E & = - \triangle V / \triangle x\end{align*}

\begin{align*}F_B & = qv \times B =\ \text{qvB} \sin(\theta) \\ B_{wire} & = \mu_o I / 2\pi r \\ F_{wire} & = \ell(I \times B) = \ell\ \text{IB} \sin(\theta)\end{align*}

(direction: RHR)

(direction: RHR)

(direction: RHR)

\begin{align*}k & = 8.992 \times 10^9\ \text{N} \cdot\ \text{m}^2/C^2 \\ \mu_o & = 4\pi \times 10^{-7}\ \text{T} \cdot\ \text{m/A} \\ \phi & = \text{BA} \cos(\theta)\end{align*}

\begin{align*}U_{el} = q\triangle V\end{align*}

Point charges: \begin{align*}E(r) = k q / r^2\end{align*} and \begin{align*}V(r) = k q / r\end{align*}

\begin{align*}(k = 1/4\pi \varepsilon_o\end{align*} where \begin{align*}\varepsilon_o = 8.854 \times 10^{-12} \ C^2/\;\mathrm{N} \cdot\ \text{m}^2)\end{align*}

\begin{align*}V = -\triangle \phi / \triangle t =\ \text{Blv}\end{align*}

Electric Circuits

\begin{align*}\triangle V & = IR \\ I & = \triangle q/ \triangle t = \triangle V/R \\ \tau & = RC\end{align*}

\begin{align*}P & = \triangle E / \triangle t = I\triangle V = I^2R = V^2/R \\ R & = \rho l /A \\ V & = -L(\triangle I/\triangle t)\end{align*}

\begin{align*}Q & = C \triangle V \\ C_{parallel \ plate} & = k \varepsilon A/d \\ C_{parallel} & = C_1 + C_2 + \ldots\end{align*}

\begin{align*}R_{series} & = R_1 + R_2 + \ldots \\ 1 / R_{parallel} & = (1/R_1) + (1/R_2) + \ldots \\ 1 / C_{series} & = (1/C_1) + (1/C_2) + \ldots\end{align*}

Name Symbols Unit Typical examples
Voltage Source \begin{align*}\triangle V\end{align*} volt \begin{align*}(V)\end{align*} \begin{align*}9 \;\mathrm{V}\end{align*} (cell phone charger); \begin{align*}12 \;\mathrm{V}\end{align*} (car); \begin{align*}120 \;\mathrm{VAC}\end{align*} (U.S. wall outlet)
Resistor \begin{align*}R\end{align*} Ohm \begin{align*}(\Omega)\end{align*} \begin{align*}144 \Omega (100 \;\mathrm{w}, 120v \;\mathrm{bulb});\end{align*} \begin{align*}1 k\Omega \;\mathrm{(wet \ skin)}\end{align*}
Capacitor \begin{align*}C\end{align*} Farad \begin{align*}(F)\end{align*} RAM in a computer, \begin{align*}700 \;\mathrm{MF}\end{align*} (Earth)
Inductor \begin{align*}L\end{align*} Henry \begin{align*}(H)\end{align*} \begin{align*}7 \;\mathrm{H}\end{align*} (guitar pickup)
Diode by type none light-emitting diode (LED); solar panel
Transistor by type none Computer processors

Equation Sheet

Mathematics

\begin{align*}\sin(\theta) & = b/c \rightarrow b = c \cdot \sin(\theta) \\ \cos(\theta) & = a/c \rightarrow a = c \cdot \cos(\theta) \\ \tan(\theta) & = b/a \rightarrow b = a \cdot \tan(\theta) \\ c^2 & = a^2 + b^2\end{align*}

\begin{align*}180^\circ & = \pi\ \text{radians}\\ C_{circle} & = 2\pi R \\ A_{circle} & = \pi R^2 \\ V_{sphere} & = (4/3)\pi R^3 \\ V_{cylinder} & = \pi R^2h\end{align*}

If \begin{align*}X\end{align*} is any unit, then \begin{align*}\ldots\end{align*}

\begin{align*}1\ \text{mX} & = 0.001\ \text{X} = 10^{-3}\ \text{X} \\ 1 \mu \mathrm{X} & = 0.000 001\ \text{X} = 10^{-6}\ \text{X} \\ 1\ \text{nX} & = 0.000 000 001\ \text{X} = 10^{-9}\ \text{X}\end{align*}

\begin{align*}1\ \text{kX} & = 1 000\ \text{X} = 10^3\ \text{X} \\ 1\ \text{MX} & = 1 000 000\ \text{X} = 10^6\ \text{X} \\ 1\ \text{GX} & = 1 000 000 000\ \text{X} = 10^9\ \text{X}\end{align*}

If \begin{align*}ax^2 + bx + c = 0\end{align*}, then \begin{align*}\ldots\end{align*}

\begin{align*}x = \frac{-b \pm \sqrt{b^2 - 4ac}} {2a}\end{align*}

% difference = |(measured – accepted) / accepted | \begin{align*}\times 100\end{align*}%

vector dot product: \begin{align*}a \cdot b = ab \ \cos\theta\end{align*} (product is a scalar)---\begin{align*}\theta\end{align*} is angle between vectors

vector cross product: \begin{align*}a \times b = ab \ \sin\theta\end{align*} (direction is given by RHR)

Kinematics Under Constant Acceleration

\begin{align*}\triangle x & = x_{final} - x_{initial} \\ \triangle\ \text{(anything)} & = \text{final value} - \text{initial value} \\ v_{avg} & = \triangle x / \triangle t \\ a_{avg} & = \triangle v / \triangle t\end{align*}

\begin{align*}x(t) & = x_0 + v_0t + \frac{1} {2} a_x t^2 \\ v(t) & = v_0 + at \\ v^2 & = v_0^2 + 2a(\triangle x)\end{align*}

\begin{align*}(x = x_0\ \text{and} \ v = v_0\ \text{at} \ t = 0)\end{align*}

\begin{align*}g & = 9.81\ \text{m/s}^2 \approx 10\ \text{m/s}^ 2 \\ 1\ \text{km} & = 1000\ \text{m} \\ 1\ \text{meter} & = 3.28\ \text{ft} \\ 1\ \text{mile} & = 1.61\ \text{km}\end{align*}

Newtonian Physics and Centripetal Motion

\begin{align*}a & = F_{net} / m F_g = mg \\ F_{net} & = \Sigma F_{all \ forces} = ma\end{align*}

\begin{align*}f_k = \mu_kF_N F_{sp} & = -k(\triangle x) \\ f_ s \le \ \mu_sF_N F_G & = Gm_1m_2 / r^2 \\ F_C & = mv^2 / r\end{align*}

\begin{align*}G & = 6.672 \times 10^{-11}\ \text{N} \cdot\ \text{m}^2/kg^2 \\ 1\ \text{kg} & = 1000\ \text{g} = 2.2\ \text{lbs} \\ 1\ \text{N} & = 1\ \text{kg} \cdot\ \text{m/s}^2\end{align*}

Momentum and Energy Conservation

\begin{align*}\Sigma p_{initial} & = \Sigma p_{final} \\ E_{initial} & = E_{final} \\ E & = K + U + W\end{align*}

\begin{align*}p & = mv \\ K & = 1/2\ \text{mv}^2\end{align*}

\begin{align*}F_{avg} & = \triangle p / \triangle t \\ U_g & = mgh \\ U_{sp} & = \frac{1} {2} k(\triangle x)^2 \\ U_g & = -Gm_1m_2/ r\end{align*}

\begin{align*}W & = F \cdot \triangle x \\ P & = \triangle W / \triangle t \\ P & = F \cdot v\end{align*}

\begin{align*}1\ \text{J} & = 1\ \text{N} \cdot\ \text{m} \\ 1\ \text{W} & = 1\ \text{J/s} \\ 1\ \text{food \ Calorie} & = 4180\ \text{J} \\ 1\ \text{ev} & = 1.602 \times 10^{-19}\ \text{J} \\ 1\ \text{kwh} & = 3.600 \times 10^6\ \text{J}\end{align*}

Rotational Motion

\begin{align*}d & = r\theta \\ v & = r\omega \\ a & = r\alpha \\ \omega & = 2\pi / T\end{align*}

\begin{align*}\theta(t) & = \theta_0 + \omega_0t + \frac{1} {2} \alpha t^2 \\ \omega(t) & = \omega_0 + \alpha t \\ \omega^ 2 & = \omega_0^2 + 2\alpha(\triangle \theta) \\ a_C & = -r\omega^2\end{align*}

\begin{align*}\tau = I \alpha \\ L = r \times p = I \omega \\ \tau = r \times F = \triangle L / \triangle t\end{align*}

\begin{align*}K = 1/2 I\omega^2\end{align*}

\begin{align*}I_{\text{ring about cm}} & = \text{MR}^2 \\ I_{\text{disk about cm}} & = \frac{1} {2} \text{MR}^2 \\ I_{\text{rod about end}} & = (1/3)\text{ML}^2 \\ I_{\text{solid sphere about cm}} & = (2/5)\text{MR}^2\end{align*}

Astronomy

\begin{align*}P_\ast & = 4 \times 10^{26}\ \text{W} \\ M_\ast & = 1.99 \times 10^{30}\ \text{kg} \\ R_\ast & = 6.96 \times 10^8\ \text{m}\end{align*}

\begin{align*}1\ \text{light-year} (ly) & = 9.45 \times 10^{15}\ \text{m} \\ M_{Earth} & = 5.97 \times 10^{24}\ \text{kg} \\ R_{Earth} & = 6.38 \times 10^6\ \text{m}\end{align*}

\begin{align*}\;\mathrm{Earth-Sun \ distance} & = 1.496 \times 10^{11}\ \text{m} \\ M_{Moon} & = 7.35 \times 10^{22}\ \text{kg} \\ R_{Moon} & = 1.74 \times 10^6\ \text{m} \\ \mathrm{Earth-Moon \ distance} & = 3.84 \times 10^8\ \text{m}\end{align*}

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