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# Chapter 27: Equations and Fundamental Concepts

Difficulty Level: At Grade Created by: CK-12

Simple Harmonic Motion and Wave Motion

$T & = 1 / f \\T_{sp} & = 2 \pi \sqrt{\frac{m} {k}}$

$v & = \lambda f \\T_P & = 2 \pi \sqrt{\frac{L} {g}}$

$f_n & = \frac{nv} {2L} && \text{nodes at both ends} \\f_n & = \frac{nv} {4L}&& \text{(n is odd)} \text{node at one end} \\f_{beat} & = |f_1 - f_2|$

$v_{sound} = 343 \;\mathrm{m/s}$ (in air at $20 \;\mathrm{C}$)

A note: $440 \;\mathrm{Hz}$

C note: $524 \;\mathrm{Hz}$

D note: $588 \;\mathrm{Hz}$

E note: $660 \;\mathrm{Hz}$

G note: $784 \;\mathrm{Hz}$

Fluids and Thermodynamics

$3/2 \ kT & = <\frac{1} {2}\ \text{mv}^2>_{avg} \\P & = F / A \\P & = P_0 + \rho gh \\\triangle P + \triangle (\rho gh) + \triangle \left (\frac{1} {2} \rho v^2 \right ) & = 0 \\\phi & = A \cdot v \\^\circ C & = ^\circ K + 273.15$

$PV & = NkT = nRT \\F_{buoy} & = - (\rho_{water}V_{displaced})g \\Q_{in} &= W + \triangle U + Q_{out} \\W & = P\triangle\ \text{V} \\k & = \frac{1} {2} \rho v^2; u = \rho gh \\\eta & = W/Q_{in}; \eta_{Carnot} = 1 - (T_{low} / T_{high})$

$k & = 1.381 \times 10^{-23}\ \text{J/K} \\\rho_{air} & = 1.29\ \text{kg/m}^3\\R & = 8.315\ \text{J/mol-K} \\\rho_{water} & = 1000\ \text{kg/m}^3 \\P_{ATMOSPH} & = 101,000\ \text{N/M}^2 \\N_{avo} & = 6.022 \times 10^{23}\ \text{mol}^{-1}$

Properties of Fundamental Particles

$m_{proton} & = 1.6726 \times 10^{-27}\ \text{kg} \\q_{electron} & = - q_{proton} = -1.602 \times 10^{-19}\ \text{C} \\r_{hydrogen \ atom} & \approx 0.529 \times 10^{-10}\ \text{m}$

$m_{electron} & = 9.109 \times 10^{-31}\ \text{kg} \\1\ \text{amu} & = 1.6605 \times 10^{-27}\ \text{kg} = 931.5\ \text{Mev/c}^2 \\\triangle\ \text{E} & = \triangle\ \text{mc}^2$

$m_{neutron} = 1.6749 \times 10^{-27}\ \text{kg}$

Radioactivity, Nuclear Physics, and Quantum Mechanics

$(\triangle x)(\triangle p) & \approx h/4\pi \\\lambda & = h/p \\N & = \mathrm{N0} \ \left (\frac{1} {2} \right ) \ t/t_H \\1\ \text{ev} & \rightarrow 1240\ \text{nm} \\\text{(energy of a photon)}$

$(\triangle E)(\triangle t) & \approx h/4\pi \\E_{photon} & = hf = pc \\Kmax & = qV = hf + \phi$

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

Light

$\lambda_{blue} & \approx 450\ \text{nm} \\\lambda_{green} & \approx 500\ \text{nm} \\\lambda_{red} & \approx 600\ \text{nm}$

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

$n_{air} & \approx n_{vacuum} = 1.00 \\n_{water} & = 1.33 \\n & = c/v_{material}$

primary: Red, Green, Blue

secondary: Magenta, Cyan, Yellow

$\frac{1} {f} = \frac{1} {d_o} + \frac{1} {di} && M = h_i/h_o = d_i/d_o$

Electricity and Magnetism

$F_E & = k \ q_1q_2 / r^2 \\E & = F_E / q \\E & = - \triangle V / \triangle x$

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

(direction: RHR)

(direction: RHR)

(direction: RHR)

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

$U_{el} = q\triangle V$

Point charges: $E(r) = k q / r^2$ and $V(r) = k q / r$

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

$V = -\triangle \phi / \triangle t =\ \text{Blv}$

Electric Circuits

$\triangle V & = IR \\I & = \triangle q/ \triangle t = \triangle V/R \\\tau & = RC$

$P & = \triangle E / \triangle t = I\triangle V = I^2R = V^2/R \\R & = \rho l /A \\V & = -L(\triangle I/\triangle t)$

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

$R_{series} & = R_1 + R_2 + \ldots \\1 / R_{parallel} & = (1/R_1) + (1/R_2) + \ldots \\1 / C_{series} & = (1/C_1) + (1/C_2) + \ldots$

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

Equation Sheet

Mathematics

$\sin(\theta) & = b/c \rightarrow b = c \cdot \sin(\theta) \\\cos(\theta) & = a/c \rightarrow a = c \cdot \cos(\theta) \\\tan(\theta) & = b/a \rightarrow b = a \cdot \tan(\theta) \\c^2 & = a^2 + b^2$

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

If $X$ is any unit, then $\ldots$

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

$1\ \text{kX} & = 1 000\ \text{X} = 10^3\ \text{X} \\1\ \text{MX} & = 1 000 000\ \text{X} = 10^6\ \text{X} \\1\ \text{GX} & = 1 000 000 000\ \text{X} = 10^9\ \text{X}$

If $ax^2 + bx + c = 0$, then $\ldots$

$x = \frac{-b \pm \sqrt{b^2 - 4ac}} {2a}$

% difference = |(measured – accepted) / accepted | $\times 100$%

vector dot product: $a \cdot b = ab \ \cos\theta$ (product is a scalar)---$\theta$ is angle between vectors

vector cross product: $a \times b = ab \ \sin\theta$ (direction is given by RHR)

Kinematics Under Constant Acceleration

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

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

$(x = x_0\ \text{and} \ v = v_0\ \text{at} \ t = 0)$

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

Newtonian Physics and Centripetal Motion

$a & = F_{net} / m F_g = mg \\F_{net} & = \Sigma F_{all \ forces} = ma$

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

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

Momentum and Energy Conservation

$\Sigma p_{initial} & = \Sigma p_{final} \\E_{initial} & = E_{final} \\E & = K + U + W$

$p & = mv \\K & = 1/2\ \text{mv}^2$

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

$W & = F \cdot \triangle x \\P & = \triangle W / \triangle t \\P & = F \cdot v$

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

Rotational Motion

$d & = r\theta \\v & = r\omega \\a & = r\alpha \\\omega & = 2\pi / T$

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

$\tau = I \alpha \\L = r \times p = I \omega \\\tau = r \times F = \triangle L / \triangle t$

$K = 1/2 I\omega^2$

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

Astronomy

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

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

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

## Date Created:

Feb 23, 2012

Sep 15, 2014
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