• Speed of light: $c = 3\times10^{8}m/s$
• Planck constant: $h = 6.63\times10^{-34}J~s$ or $1242~eV-nm$
• Gravitation constant: $G = 6.67\times10^{-11}m^{3}kg^{-1}s^{-2}$
• Boltzmann constant: $k = 1.38\times10^{-23}J/K$
• Molar gas constant: $R = 8.314~J/(mol~K)$
• Avogadro's number: $N_{A} = 6.023\times10^{23}mol^{-1}$
• Charge of electron: $e = 1.602\times10^{-19}C$
• Permittivity of vacuum: $\epsilon_{0} = 8.85\times10^{-12}F/m$
• Coulomb constant: $\frac{1}{4\pi\epsilon_{0}} = 9\times10^{9}Nm^{2}/C^{2}$
• Permeability of vacuum: $\mu_{0} = 4\pi\times10^{-7}N/A^{2}$

• Faraday constant: $F = 96485~C/mol$
• Mass of electron: $m_{e} = 9.1\times10^{-31}kg$
• Mass of proton: $m_{p} = 1.6726\times10^{-27}kg$
• Mass of neutron: $m_{n} = 1.6749\times10^{-27}kg$
• Atomic mass unit: $u = 1.66\times10^{-27}kg$ or $931.49~MeV/c^{2}$
• Stefan-Boltzmann constant: $\sigma = 5.67\times10^{-8}W/(m^{2}K^{4})$
• Rydberg constant: $R_{\infty} = 1.097\times10^{7}m^{-1}$
• Bohr magneton: $\mu_{B} = 9.27\times10^{-24}J/T$
• Bohr radius: $a_{0} = 0.529\times10^{-10}m$
• Standard atmosphere: $atm = 1.01325\times10^{5}Pa$
• Wien displacement constant: $b = 2.9\times10^{-3}m~K$

1.1: Notation & Vectors

• Vector Notation: $\vec{a}=a_{x}\hat{i}+a_{y}\hat{j}+a_{z}\hat{k}$
• Magnitude: $a=|\vec{a}|=\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}$
• Dot product: $\vec{a}\cdot\vec{b}=a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z}=ab~cos~\theta$
• Cross product: $\vec{a}\times\vec{b}=(a_{y}b_{z}-a_{z}b_{y})\hat{i}+(a_{z}b_{x}-a_{x}b_{z})\hat{j}+(a_{x}b_{y}-a_{y}b_{x})\hat{k}$
• Magnitude of cross product: $|\vec{a}\times\vec{b}|=ab~sin~\theta$

1.2: Kinematics

• Average Vel. and Accel.: $\vec{v}_{av}=\Delta\vec{r}/\Delta t$, $\vec{a}_{av}=\Delta\vec{v}/\Delta t$
• Instantaneous Vel. and Accel.: $\vec{v}_{inst}=d\vec{r}/dt$, $\vec{a}_{inst}=d\vec{v}/dt$
• Motion in a straight line with constant a: $v=u+at$, $s=ut+\frac{1}{2}at^{2}$, $v^{2}-u^{2}=2as$
• Relative Velocity: $\vec{v}_{A/B}=\vec{v}_{A}-\vec{v}_{B}$
• Projectile Eq: $x=ut~cos~\theta$, $y=ut~sin~\theta-\frac{1}{2}gt^{2}$, $y=x~tan~\theta-\frac{g}{2u^{2}cos^{2}\theta}x^{2}$
• Time, Range, Height: $T=\frac{2u~sin~\theta}{g}$, $R=\frac{u^{2}sin~2\theta}{g}$, $H=\frac{u^{2}sin^{2}\theta}{2g}$

1.3: Newton's Laws and Friction

• Linear momentum: $\vec{p}=m\vec{v}$
• Newton's 1st law: inertial frame.
• Newton's 2nd law: $\vec{F}=\frac{d\vec{p}}{dt}$ or $\vec{F}=m\vec{a}$
• Newton's 3rd law: $\vec{F}_{AB}=-\vec{F_{BA}}$
• Frictional force: $f_{static,max}=\mu_{s}N$, $f_{kinetic}=\mu_{k}N$
• Banking angle: $\frac{v^{2}}{rg}=tan~\theta$, $\frac{v^{2}}{rg}=\frac{\mu+tan~\theta}{1-\mu~tan~\theta}$
• Centripetal force: $F_{c}=\frac{mv^{2}}{r}$, $a_{c}=\frac{v^{2}}{r}$
• Pseudo force: $\vec{F}_{pseudo}=-m\vec{a}_{0}$, $F_{centrifugal}=-\frac{mv^{2}}{r}$
• Minimum speed to complete vertical circle: $v_{min,bottom}=\sqrt{5gl}$, $v_{min,top}=\sqrt{gl}$
• Conical pendulum: $T=2\pi\sqrt{\frac{L~cos~\theta}{g}}$

1.4: Work, Power and Energy

• Work: $W=\vec{F}\cdot\vec{S}=FS~cos~\theta$, $W=\int\vec{F}\cdot d\vec{S}$
• Kinetic energy: $K=\frac{1}{2}mv^{2}=\frac{p^{2}}{2m}$
• Potential energy: $F=-\partial U/\partial x$ for conservative forces.
• Gravitational/Spring PE: $U_{gravitational}=mgh$, $U_{spring}=\frac{1}{2}kx^{2}$
• Work done by conservative forces is path independent: $\oint\vec{F}_{conservative}\cdot d\vec{r}=0$
• Work-energy theorem: $W=\Delta K$
• Mechanical energy: $E=U+K$ (Conserved if forces are conservative).
• Power: $P_{av}=\frac{\Delta W}{\Delta t}$, $P_{inst}=\vec{F}\cdot\vec{v}$

1.5: Centre of Mass and Collision

• Centre of mass: $x_{cm}=\frac{\sum x_{i}m_{i}}{\sum m_{i}}$, $x_{cm}=\frac{\int xdm}{\int dm}$
• CM configurations:
  1. $m_{1}$, $m_{2}$ separated by r: $\frac{m_{1}r}{m_{1}+m_{2}}$
  2. Triangle (CM = Centroid): $y_{c}=\frac{h}{3}$
  3. Semicircular ring: $y_{c}=\frac{2r}{\pi}$
  4. Semicircular disc: $y_{c}=\frac{4r}{3\pi}$
  5. Hemispherical shell: $y_{c}=\frac{r}{2}$
  6. Solid Hemisphere: $y_{c}=\frac{3r}{8}$
  7. Cone: height of CM from base is $h/4$ for solid cone, $h/3$ for hollow cone.
• Motion of the CM: $\vec{v}_{cm}=\frac{\sum m_{i}\vec{v}_{i}}{M},\vec{p}_{cm}=M\vec{v}_{cm},\vec{a}_{cm}=\frac{\vec{F}_{ext}}{M}$
• Impulse: $\vec{J}=\int\vec{F}dt=\Delta\vec{p}$
• Momentum conservation: $m_{1}v_{1}+m_{2}v_{2}=m_{1}v_{1}^{\prime}+m_{2}v_{2}^{\prime}$
• Elastic Collision: $\frac{1}{2}m_{1}{v_{1}}^{2}+\frac{1}{2}m_{2}{v_{2}}^{2}=\frac{1}{2}m_{1}{v_{1}^{\prime}}^{2}+\frac{1}{2}m_{2}{v_{2}^{\prime}}^{2}$
• Coefficient of restitution: $e=\frac{-(v_{1}^{\prime}-v_{2}^{\prime})}{v_{1}-v_{2}}$ (1 completely elastic, 0 completely in-elastic)
• If $v_{2}=0$ and $m_{1}\ll m_{2}$ then $v_{1}^{\prime}=-v_{1}$
• If $v_{2}=0$ and $m_{1}\gg m_{2}$ then $v_{2}^{\prime}=2v_{1}$
• Elastic collision with $m_{1}=m_{2}$: velocities exchange ($v_{1}^{\prime}=v_{2}$ and $v_{2}^{\prime}=v_{1}$)

1.6: Rigid Body Dynamics

• Angular velocity/accel: $\omega=\frac{d\theta}{dt}, \vec{v}=\vec{\omega}\times\vec{r}, \alpha=\frac{d\omega}{dt}, \vec{a}=\vec{\alpha}\times\vec{r}$
• Rotation with const $\alpha$: $\omega=\omega_{0}+\alpha t$, $\theta=\omega t+\frac{1}{2}\alpha t^{2}, \omega^{2}-{\omega_{0}}^{2}=2\alpha\theta$
• Moment of Inertia: $I=\sum_{i}m_{i}{r_{i}}^{2}$, $I=\int r^{2}dm$
• MI values: ring ($mr^{2}$), disk ($\frac{1}{2}mr^{2}$), shell ($\frac{2}{3}mr^{2}$), solid sphere ($\frac{2}{5}mr^{2}$), rod ($\frac{1}{12}ml^{2}$), hollow cylinder ($mr^{2}$), solid cylinder ($\frac{1}{2}mr^{2}$), rectangle ($\frac{m(a^{2}+b^{2})}{12}$)
• Theorem of Parallel Axes: $I_{||}=I_{cm}+md^{2}$
• Theorem of Perp. Axes: $I_{z}=I_{x}+I_{y}$
• Radius of Gyration: $k=\sqrt{I/m}$
• Angular Momentum: $\vec{L}=\vec{r}\times\vec{p}$, $\vec{L}=I\vec{\omega}$
• Torque: $\vec{\tau}=\vec{r}\times\vec{F}$, $\vec{\tau}=\frac{d\vec{L}}{dt}$, $\tau=I\alpha$
• Conservation of $\vec{L}$: $\vec{\tau}_{ext}=0\Rightarrow\vec{L}=const$
• Equilibrium condition: $\Sigma\vec{F}=\vec{0}$, $\sum\vec{\tau}=\vec{0}$
• Kinetic Energy: $K_{rot}=\frac{1}{2}I\omega^{2}$, Total $K=\frac{1}{2}m{v_{cm}}^{2}+\frac{1}{2}I_{cm}\omega^{2}$
• Dynamics: $\vec{\tau}_{cm}=I_{cm}\vec{\alpha}$, $\vec{F}_{ext}=m\vec{a}_{cm}$, $\vec{p}_{cm}=m\vec{v}_{cm}$
• Angular momentum of combined motion: $L=I_{cm}\vec{\omega}+\vec{r}_{cm}\times m\vec{v}_{cm}$

1.7: Gravitation

• Gravitational force: $F=G\frac{m_{1}m_{2}}{r^{2}}$
• Potential energy: $U=-\frac{GMm}{r}$
• Gravitational acceleration: $g=\frac{GM}{R^{2}}$
• Variation of g with depth: $g_{inside}\approx g(1-\frac{h}{R})$
• Variation of g with height: $g_{outside}\approx g(1-\frac{2h}{R})$
• Effect of non-spherical earth shape: g at pole > g at equator ($\because R_{e}-R_{P}\approx21~km$)
• Effect of earth rotation on apparent weight: $mg_{\theta}^{\prime}=mg-m\omega^{2}R~cos^{2}\theta$
• Orbital velocity of satellite: $v_{o}=\sqrt{\frac{GM}{R}}$
• Escape velocity: $v_{e}=\sqrt{\frac{2GM}{R}}$
• Kepler's laws:
  1. First: Elliptical orbit with sun at one of the focus.
  2. Second: Areal velocity is constant. ($dL/dt=0$).
  3. Third: $T^{2}\propto a^{3}$. In circular orbit $T^{2}=\frac{4\pi^{2}}{GM}a^{3}$.

1.8: Simple Harmonic Motion

• Hooke's law: $F=-kx$ (for small elongation x)
• Acceleration: $a=\frac{d^{2}x}{dt^{2}}=-\frac{k}{m}x=-\omega^{2}x$
• Time period: $T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}}$
• Displacement: $x=A~sin(\omega t+\phi)$
• Velocity: $v=A\omega~cos(\omega t+\phi)=\pm\omega\sqrt{A^{2}-x^{2}}$
• Potential energy: $U=\frac{1}{2}kx^{2}$
• Kinetic energy $K=\frac{1}{2}mv^{2}$
• Total energy: $E=U+K=\frac{1}{2}m\omega^{2}A^{2}$
• Simple pendulum: $T=2\pi\sqrt{\frac{l}{g}}$
• Physical Pendulum: $T=2\pi\sqrt{\frac{I}{mgl}}$
• Torsional Pendulum: $T=2\pi\sqrt{\frac{I}{k}}$
• Superposition of two SHM's: $x_{1}=A_{1}sin~\omega t$, $x_{2}=A_{2}sin(\omega t+\delta)$ $\Rightarrow x=x_{1}+x_{2}=A~sin(\omega t+\epsilon)$
• $A=\sqrt{{A_{1}}^{2}+{A_{2}}^{2}+2A_{1}A_{2}cos~\delta}$
• $tan~\epsilon=\frac{A_{2}sin~\delta}{A_{1}+A_{2}cos~\delta}$
• Springs in series: $\frac{1}{k_{eq}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}$
• Springs in parallel: $k_{eq}=k_{1}+k_{2}$

1.9: Properties of Matter

• Modulus of rigidity: $Y=\frac{F/A}{\Delta l/l}$, $B=-V\frac{\Delta P}{\Delta V}$, $\eta=\frac{F}{A\theta}$
• Compressibility: $K=\frac{1}{B}=-\frac{1}{V}\frac{dV}{dP}$
• Poisson's ratio: $J=\frac{lateral~strain}{longitudinal~strain}=\frac{\Delta D/D}{\Delta l/l}$
• Elastic energy: $U=\frac{1}{2} \times stress \times strain \times volume$
• Surface tension: $S=F/l$
• Surface energy: $U=SA$
• Excess pressure in bubble: $\Delta p_{air}=2S/R$, $\Delta p_{soap}=4S/R$
• Capillary rise: $h=\frac{2S~cos~\theta}{r\rho g}$
• Hydrostatic pressure: $p=\rho gh$
• Buoyant force: $F_{B}=\rho Vg=$ Weight of displaced liquid
• Equation of continuity: $A_{1}v_{1}=A_{2}v_{2}$
• Bernoulli's equation: $p+\frac{1}{2}\rho v^{2}+\rho gh= constant$
• Torricelli's theorem: $v_{effux}=\sqrt{2gh}$
• Viscous force: $F=-\eta A\frac{dv}{dx}$
• Stoke's law: $F=6\pi\eta rv$
• Poiseuilli's equation: $\frac{Volume~flow}{time}=\frac{\pi pr^{4}}{8\eta l}$
• Terminal velocity: $v_{t}=\frac{2r^{2}(\rho-\sigma)g}{9\eta}$

2.1: Waves Motion

• General equation of wave: $\frac{\partial^{2}y}{\partial x^{2}}=\frac{1}{v^{2}}\frac{\partial^{2}y}{\partial t^{2}}$
• Notation: Amplitude A, Frequency $\nu$, Wavelength $\lambda$, Period T, Angular Frequency $\omega$, Wave Number k
• $T=\frac{1}{\nu}=\frac{2\pi}{\omega}$, $v=\nu\lambda$, $k=\frac{2\pi}{\lambda}$
• Progressive wave travelling with speed v: $y=f(t-x/v), \rightarrow +x; y=f(t+x/v), \leftarrow -x$
• Progressive sine wave: $y=A~sin(kx-\omega t)=A~sin(2\pi(x/\lambda-t/T))$

2.2: Waves on a String

• Speed of waves on a string: $v=\sqrt{T/\mu}$
• Transmitted power: $P_{av}=2\pi^{2}\mu vA^{2}\nu^{2}$
• Interference: $y_{1}=A_{1}sin(kx-\omega t)$, $y_{2}=A_{2}sin(kx-\omega t+\delta) \Rightarrow y=y_{1}+y_{2}=A~sin(kx-\omega t+\epsilon)$
• $A=\sqrt{{A_{1}}^{2}+{A_{2}}^{2}+2A_{1}A_{2}cos~\delta}$, $tan~\epsilon=\frac{A_{2}sin~\delta}{A_{1}+A_{2}cos~\delta}$
• $\delta = 2n\pi$ constructive; $(2n+1)\pi$ destructive.
• Standing Waves: $y_{1}=A_{1}sin(kx-\omega t), y_{2}=A_{2}sin(kx+\omega t) \Rightarrow y=(2A~cos~kx)sin~\omega t$
• String fixed at both ends:
• 1. Boundary conditions: y=0 at x=0 and x=L
• 2. Allowed Freq.: $L=n\frac{\lambda}{2} \Rightarrow \nu=\frac{n}{2L}\sqrt{\frac{T}{\mu}}, n=1,2,3...$
• 3. Fundamental/1st harmonics: $\nu_{0}=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$
• 4. 1st overtone/2nd harmonics: $\nu_{1}=\frac{2}{2L}\sqrt{\frac{T}{\mu}}$
• 5. 2nd overtone/3rd harmonics: $\nu_{2}=\frac{3}{2L}\sqrt{\frac{T}{\mu}}$
• 6. All harmonics are present.
• String fixed at one end:
• 1. Boundary conditions: y=0 at x=0
• 2. Allowed Freq.: $L=(2n+1)\frac{\lambda}{4} \Rightarrow \nu=\frac{2n+1}{4L}\sqrt{\frac{T}{\mu}}, n=0,1,2...$
• 3. Fundamental/1st harmonics: $\nu_{0}=\frac{1}{4L}\sqrt{\frac{T}{\mu}}$
• 4. 1st overtone/3rd harmonics: $\nu_{1}=\frac{3}{4L}\sqrt{\frac{T}{\mu}}$
• 5. 2nd overtone/5th harmonics: $\nu_{2}=\frac{5}{4L}\sqrt{\frac{T}{\mu}}$
• 6. Only odd harmonics are present.
• Sonometer: $\nu\propto\frac{1}{L}, \nu\propto\sqrt{T}, \nu\propto\frac{1}{\sqrt{\mu}}. \nu=\frac{n}{2L}\sqrt{\frac{T}{\mu}}$

2.3: Sound Waves

• Displacement wave: $s=s_{0}sin~\omega(t-x/v)$
• Pressure wave: $p=p_{0}cos~\omega(t-x/v)$, $p_{0}=(B\omega/v)s_{0}$
• Speed of sound waves: $v_{liquid}=\sqrt{\frac{B}{\rho}}, v_{gas}=\sqrt{\frac{\gamma P}{\rho}}$
• Intensity: $I=\frac{2\pi^{2}B}{v}s_{0}{}^{2}\nu^{2}=\frac{{p_{0}}^{2}v}{2B}=\frac{{p_{0}}^{2}}{2\rho v}$
• Standing longitudinal waves: $p_{1}=p_{0}sin~\omega(t-x/v), p_{2}=p_{0}sin~\omega(t+x/v) \Rightarrow p=2p_{0}cos~kx~sin~\omega t$
• Nodes at $x=(n+\frac{1}{2})\frac{\lambda}{2}$, Antinodes at $x=n\frac{\lambda}{2}$
• Closed organ pipe:
• 1. Boundary condition: y=0 at x=0
• 2. Allowed freq.: $L=(2n+1)\frac{\lambda}{4} \Rightarrow \nu=(2n+1)\frac{v}{4L}, n=0,1,2...$
• 3. Fundamental/1st harmonics: $\nu_{0}=\frac{v}{4L}$
• 4. 1st overtone/3rd harmonics: $\nu_{1}=3\nu_{0}=\frac{3v}{4L}$
• 5. 2nd overtone/5th harmonics: $\nu_{2}=5\nu_{0}=\frac{5v}{4L}$
• 6. Only odd harmonics are present.
• Open organ pipe:
• 1. Boundary condition: y=0 at x=0 (Note: PDF seems to have a typo for open pipe, but typically it's open at both ends)
• Allowed freq.: $L=n\frac{\lambda}{2} \Rightarrow \nu=n\frac{v}{2L}, n=1,2,...$
• 2. Fundamental/1st harmonics: $\nu_{0}=\frac{v}{2L}$
• 3. 1st overtone/2nd harmonics: $\nu_{1}=2\nu_{0}=\frac{2v}{2L}$
• 4. 2nd overtone/3rd harmonics: $\nu_{2}=3\nu_{0}=\frac{3v}{2L}$
• 5. All harmonics are present.
• Resonance column: $l_{1}+d=\frac{\lambda}{4}, l_{2}+d=\frac{3\lambda}{4} \Rightarrow v=2(l_{2}-l_{1})\nu$
• Beats: two waves of almost equal frequencies $\omega_{1}\approx\omega_{2}$
• $p=p_{1}+p_{2}=2p_{0}cos~\Delta\omega(t-x/v)sin~\omega(t-x/v)$, $\omega=(\omega_{1}+\omega_{2})/2, \Delta\omega=\omega_{1}-\omega_{2}$ (beats freq.)
• Doppler Effect: $\nu=\frac{v+u_{o}}{v-u_{s}}\nu$. v is speed of sound, $u_{o}$ speed of observer (positive towards source), $u_{s}$ speed of source (positive towards observer).

2.4: Light Waves

• Plane Wave: $E=E_{0}sin~\omega(t-\frac{x}{v}), I=I_{0}$
• Spherical Wave: $E=\frac{aE_{0}}{r}sin~\omega(t-\frac{r}{v}), I=\frac{I_{0}}{r^{2}}$
• Young's double slit experiment (YDSE):
• Path difference: $\Delta x=\frac{dy}{D}$
• Phase difference: $\delta=\frac{2\pi}{\lambda}\Delta x$
• Interference Conditions: $\delta = 2n\pi, \Delta x = n\lambda$ (constructive); $\delta = (2n+1)\pi, \Delta x = (n+\frac{1}{2})\lambda$ (destructive)
• Intensity: $I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}cos~\delta$
• $I_{max}=(\sqrt{I_{1}}+\sqrt{I_{2}})^{2}, I_{min}=(\sqrt{I_{1}}-\sqrt{I_{2}})^{2}$
• $I_{1}=I_{2}: I=4I_{0}cos^{2}\frac{\delta}{2}, I_{max}=4I_{0}, I_{min}=0$
• Fringe width: $w=\frac{\lambda D}{d}$
• Optical path: $\Delta x^{\prime}=\mu\Delta x$
• Interference of waves transmitted through thin film: $\Delta x=2\mu d = n\lambda$ (constructive), $(n+\frac{1}{2})\lambda$ (destructive)
• Diffraction from a single slit (Minima): $n\lambda=b~sin~\theta\approx b(y/D)$
• Resolution: $sin~\theta=\frac{1.22\lambda}{b}$
• Law of Malus: $I=I_{0}cos^{2}\theta$