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\lhead{PHYS 1P23/1P93 Formula sheer}
\chead{}
\rhead{2016/2017}
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\title{Formula Sheet, Physics 1P23/1P93}
\author{Edward Sternin/Thad Harroun}
\date{Available at tests, midterms, and exams, 2015/16}

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\begin{document}

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{\footnotesize
\begin{landscape}
\begin{multicols}{3}

%==============================================================================
\textbf{Fundamental Constants}

\begin{itemize}
\addtolength{\itemsep}{-2pt}
\item[$g = $]
  9.80 m s$^{-2}$
\item[$c = $] 
  speed of light in vacuum $ = 3.00\times 10^8$ ms$^{-1}$
\item[$e = $]
  charge of electron  $ = 1.60\times10^{-19}$ C
\item[$m_e = $]
  mass of electron $ = 9.11\times10^{-31}$ kg % \\ $ = 0.511$ MeV/$c^2$
\item[$m_n = $]
  mass of neutron or proton $ = 1.67\times10^{-27}$ kg % \\ $ = 938$ MeV/$c^2$
\item[$N_A = $]
  Avogadro's number $ = 6.022 \times 10^{23}$ molecules/mol
\item[$k_B = $]
  Boltzmann's constant $ = 1.38 \times 10^{-23} {\mbox{J}}/{\mbox{K}}$
\item[$R = $]
  universal gas constant $ = 8.31 \frac{\mbox{J}}{\mbox{mol K}} = 0.0821 \frac{\mbox{l atm}}{\mbox{mol K}}$
\item[ $c_{\mbox{w}} = $] 
  specific heat of water $ = 1 {\mbox{ cal}}/({\mbox{g K}})$]
\item[ $1 {\mbox{ cal}} $] $= 4.186 {\mbox{ J}}$]
\item[ $\sigma = $] 
  Stefan-Boltzmann constant $ = 5.67\times 10^{-8}\>\frac{\mbox{W}}{\mbox{m}^2 \mbox{K}^4}$
\end{itemize}

%==============================================================================
\textbf{Energy}

kinetic energy $K=\frac{1}{2}mv^2$\\
graviational potential energy $U_g=mgy$\\
spring potential energy $U_s=\frac{1}{2}kx^2$\\
work done by a force $W=F\cdot d$\\
power $P=\frac{\Delta E}{\Delta t}$\\

%==============================================================================
\textbf{Heat and thermodynamics}

\[
 T_F = \frac{9}{5}\> \frac{^\circ \mbox{F}}{^\circ \mbox{C}}\> T_C + 32^\circ \mbox{F},  \qquad
 T_C = \frac{5}{9}\> \frac{^\circ \mbox{C}}{^\circ \mbox{F}}\> ( T_F - 32^\circ \mbox{F}),
\]
absolute $T = T_C + 273$\\

thermal expansion:\\
\begin{eqnarray*}
 \mbox{length (1D):} & & \Delta L = \alpha L_0 \Delta T\\
 \mbox{volume (3D):} & & \Delta V = \beta  V_0 \Delta T
\end{eqnarray*}

heat capacity $Q=C\Delta T$,\\
specific heat $Q=cm\Delta T$ 

{latent heat:} 
$Q = m L_f$ {\scriptsize (fusion)}, $Q = m L_v$ {\scriptsize (vaporization)} 

{heat conduction:}
$
\frac{\displaystyle \Delta Q}{\displaystyle \Delta t} = k A \frac{\displaystyle \Delta T}{\displaystyle L}
$

{radiation:} 
$ 
\frac{\displaystyle \Delta Q}{\displaystyle \Delta t} = e \sigma A \left(T^4 - T_{\rm surround}^4\right) 
$

{ideal gas law:} 
$
\frac{\displaystyle p V}{\displaystyle T} = k_B N = n R
$

{Maxwell distribution:}
$
  \frac{1}{2} m v_{rms}^2 = \frac{3}{2} k_B T\>,  \quad
  v_{rms} = \sqrt{\frac{\displaystyle 3 R T}{\displaystyle M}}
$

{thermal energy of an ideal gas:} 
$
  \Delta E_{\rm th} = \frac{3}{2} N k_B T = \frac{3}{2} n RT
$

{heat capacity of an ideal gas (monatomic):} \\
$
  \mbox{ } C_V=\frac{3}{2}R=12.5 \mbox{ J/mol}\cdot\mbox{K, @ const } V
$\\
$ 
  \mbox{ } C_p=\frac{5}{2}R=20.8 \mbox{ J/mol}\cdot\mbox{K, @ const } p
$

{first law of thermodynamics:} $\Delta U = +Q -W$ 

$W=p \Delta V$ (isobaric), 
$W=nRT\ln \frac{\displaystyle V_f}{\displaystyle V_i}$ (isothermal)

{heat engine efficiency:} $e = \frac{\displaystyle W}{\displaystyle Q_h} =
	1 - \frac{\displaystyle Q_c}{\displaystyle Q_h} = 
	1 - \frac{\displaystyle T_c}{\displaystyle T_h}$

{heat pump CoP cooling:} $e = \frac{\displaystyle Q_c}{\displaystyle W_{in}} = 
\frac{\displaystyle T_c}{\displaystyle T_h-T_c}$

{entropy:} $\Delta S = \frac{\displaystyle Q}{\displaystyle T}$ (reversible process)\\
$\Delta S=mc\ln\left(T_f/T_i\right)+nC\ln\left(V_f/V_i\right)$\\
{second law of thermodynamics:} $\Delta S_{\rm total} \geq 0$

%==============================================================================
\textbf{Mechanical properties of matter}

{stress $\propto$ strain}

\begin{eqnarray*}
  F & = & {Y} \> {{\Delta L}\over L_0} \> A, \qquad {Y} = \mbox{Young's modulus} \\
  F & = & {S} \> {{\Delta X}\over L_0} \> A, \qquad {S} = \mbox{shear modulus} \\
  \frac{F}{A} & = & - {B} \> {{\Delta V}\over V_0} \> , \qquad  {B} = \mbox{bulk modulus}
\end{eqnarray*}

{pressure} $p=F/A$

\[
   1\>\mbox{atm} = 1.013\times10^5\>\mbox{Pa} =760\>\mbox{torr} = 760\>\mbox{mm~Hg}
\]

{mass density = mass per unit volume,} $\rho=m/V$ 

{specific gravity of X} $= \rho_{\mbox{\tiny X}}/\rho_{\mbox{\tiny H$_2$O}}$,
$\rho_{\mbox{\tiny H$_2$O}}=1000\,$kg/m$^3$

%==============================================================================
\vspace{1em}
\textbf{Fluids}

hydrostatic pressure: 
$
  p= p_0 + \rho g h
$

{equation of continuity (conservation of mass)}
\[
  { {\Delta m}\over{\Delta t} } = \rho A v = \mbox{const}
\]

{Bernoulli's equation for an ideal fluid}
\[
  p + {1\over2}\rho v^2 + \rho g y = \mbox{const}
\]

{Poiseuille'e equation for a viscous flow}
\[
  Q = { {\pi R^4 \Delta p}\over {8\eta L} }\>, \qquad \eta = \mbox{viscosity}
\]

{buoyancy}
\[
F_B=\rho_fV_fg
\] 

%==============================================================================
\vspace{1em}
\textbf{Waves and Sound}

simple harmonic oscillator, $F=-kx$, $U=\frac{1}{2}kx^2$
\begin{eqnarray*}
  & f={1}/{T} \quad \omega={2\pi}/{T}=2\pi f = \sqrt{{k}/{m}} & \\
  & \left\{ \begin{array}{l}
  x = A \cos\phi = A \cos \omega t,    \\
  v = -A \omega \sin\omega t            \\
  a = -A \omega^2 \cos\omega t
  \end{array} \right. &
\end{eqnarray*}

pendulum $\omega=\sqrt{{g}/{L}}$

traveling wave 
\[
  y=A\sin (  \frac{2\pi}{T}t \mp \frac{2\pi}{\lambda}x + \phi_0 ) , \quad v=\lambda/T
\]

{waves on a string under tension $F$}
\[
  v_{\mbox{\scriptsize string}} = \sqrt{\frac{F}{m/L}}
\]

%%$$
%%  v_{\mbox{\scriptsize solid}} = \sqrt{\frac{\cal Y}{\rho}} \quad
%%  v_{\mbox{\scriptsize liquid}} = \sqrt{\frac{\cal B}{\rho}}  \quad
%%  v_{\mbox{\scriptsize gas}} = \sqrt{\frac{\gamma k_B T }{m}}
%%$$

{standing waves (nodes at both ends, string length $L$)}
\[
f_n=\frac{v}{2L}n \qquad n=1\mbox{(fundamental)};2,3,...\mbox{(harmonics)}
\]

{sound intensity}
\[
  \beta = 10 \> \log \frac{I}{I_0}, \mbox { dB} \qquad
  I_0=1\times 10 ^{-12} \mbox{ W/m}^2
\]

{point source}
\[
  I = \frac{P}{A} = \frac{P}{4\pi r^2}
\]

{Doppler (upper sign = approach, lower = recede)}
\[
  f = f_s \left( \frac{1\pm \frac{\textstyle v_o}{\textstyle v}}{1\mp \frac{\textstyle v_s}{\textstyle v}} \right)
  \quad \mbox{\scriptsize s=source, o=observer}
\]

%==============================================================================
\vspace{1em}
\textbf{Light and Optics}

diffraction $\sin\theta= m\lambda/W$

{two-slit interference fringes ($m=0,1,2,\dots$)}
\[]
  \sin\theta= m \frac{\lambda}{d} \> \mbox{\scriptsize (bright)} \quad
  \sin\theta= \left( m + \frac{1}{2} \right) \frac{\lambda}{d} \> \mbox{\scriptsize (dark)}
\]

{Bragg peaks (X-ray diffraction, atom spacing $d$)}
\[
  \sin\theta= \frac{m}{2} \frac{\lambda}{d} \qquad m=1,2,\dots
\]

{diffraction-limited resolving power (first dark fringe, aperture size $d$)}
\[
  \theta_{min} = \frac{\lambda}{d} \> \mbox{\scriptsize (slit)} \qquad
  \theta_{min} = 1.22 \frac{\lambda}{d} \> \mbox{\scriptsize (circular)}
\]

reflection $\theta_i=\theta_r$, refraction $n_1 \sin\theta_1 = n_2 \sin\theta_2$

{index of refraction of a medium $m$}
\[
  n = \frac{c}{v}=\frac{\lambda}{\lambda_m}
  \quad \mbox{\scriptsize {\it e.g}. $n_{air} \approx 1 \quad n_{water} \approx 1.33$}
\]

{total internal reflection}
$
  \theta_{\mbox{\scriptsize critical}} = \arcsin \frac{\displaystyle n_2}{\displaystyle n_1}
$

{polarization by reflection}
$
  \theta_{\mbox{\scriptsize Brewster}} = \arctan \frac{\displaystyle n_2}{\displaystyle n_1}
$

spherical mirrors $f=\pm \frac{1}{2} R$

{magnification}
$
  m = \frac{\displaystyle h_i}{\displaystyle h_o} = - \frac{\displaystyle d_i}{\displaystyle d_o}
$

{mirror/lens equation}
$
  \frac{\displaystyle 1}{\displaystyle d_o} + \frac{\displaystyle 1}{\displaystyle d_i} = \frac{\displaystyle 1}{\displaystyle f}
$

Rayleigh criterion $\theta_{\mbox{\scriptsize min}} \approx 1.22 \lambda/D$
\end{multicols}
\end{landscape}
}
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