Formula reference

\[ \text{prediction}=\text{model}(\text{measured inputs}) \]

\[ Q=Cx^a y^b,\qquad \frac{Q_2}{Q_1}=\left(\frac{x_2}{x_1}\right)^a\left(\frac{y_2}{y_1}\right)^b \]

\[ \frac{\Delta Q}{Q}\times100\approx a\left(\frac{\Delta x}{x}\times100\right)+b\left(\frac{\Delta y}{y}\times100\right) \]

\[ K=\frac{p^2}{2m},\qquad p=\sqrt{2mK} \]

\[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \theta_{\rm rad}=\theta_{\rm deg}\frac{\pi}{180} \]

\[ \sin2\theta=2\sin\theta\cos\theta,\quad \sin(A\pm B)=\sin A\cos B\pm\cos A\sin B \]

\[ s=r\theta,\qquad \sin\theta\approx\tan\theta\approx\theta,\quad \cos\theta\approx1-\frac{\theta^2}{2} \]

\[ (1+x)^n\approx1+nx\qquad(|x|\ll1) \]

\[ (1+x)^m(1+y)^n\approx1+mx+ny \]

\[ g_h=\frac{g}{(1+h/R)^2}\approx g\left(1-\frac{2h}{R}\right) \]

\[ y=mx+b,\qquad m=\frac{\Delta y}{\Delta x} \]

\[ m=\frac{y_2-y_1}{x_2-x_1},\qquad \frac{x}{a}+\frac{y}{b}=1 \]

\[ T^2=\frac{4\pi^2}{g}\,\ell\quad\Rightarrow\quad g=\frac{4\pi^2}{\text{slope of }T^2\text{ vs }\ell} \]

\[ \frac{d}{dt}(Ct^n)=nCt^{n-1},\qquad v=\frac{dx}{dt},\qquad a=\frac{dv}{dt} \]

\[ \frac{d}{dt}\sin(\omega t)=\omega\cos(\omega t),\quad \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx} \]

\[ F=-\frac{dU}{dx} \]

\[ \frac{dy}{dx}=0;\qquad \frac{d^2y}{dx^2}\lt0\Rightarrow\text{maximum},\quad \frac{d^2y}{dx^2}\gt0\Rightarrow\text{minimum} \]

\[ \left(a\sin\theta+b\cos\theta\right)_{\max}=\sqrt{a^2+b^2}\quad\text{at }\tan\theta=\frac ab \]

\[ F(\theta)=\frac{\mu mg}{\cos\theta+\mu\sin\theta},\qquad F_{\min}=\frac{\mu mg}{\sqrt{1+\mu^2}}\ \text{at}\ \tan\theta=\mu \]

\[ \int_{t_1}^{t_2}Ct^n\,dt=\frac{C}{n+1}(t_2^{n+1}-t_1^{n+1}),\qquad \Delta x=\int v\,dt \]

\[ \int\frac{dx}{x}=\ln x+C,\quad \int\sin x\,dx=-\cos x+C,\quad \int\cos x\,dx=\sin x+C \]

\[ \langle f\rangle=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}f(t)\,dt \]

\[ \ln(ab)=\ln a+\ln b,\quad \ln(a/b)=\ln a-\ln b,\quad \ln(e^x)=x \]

\[ y=y_0e^{-\lambda t},\quad t_{1/2}=\frac{\ln2}{\lambda}\approx\frac{0.693}{\lambda},\quad \frac{y}{y_0}=\left(\frac12\right)^{t/t_{1/2}} \]

\[ \ln x=2.303\log_{10}x \]

\[ 1\,\mathrm{N}=1\,\mathrm{kg\,m\,s^{-2}},\quad 1\,\mathrm{J}=1\,\mathrm{kg\,m^2\,s^{-2}},\quad 1\,\mathrm{km\,h^{-1}}=\frac5{18}\,\mathrm{m\,s^{-1}} \]

\[ 1\ \text{ly}=9.46\times10^{15}\ \text{m},\quad 1\ \text{AU}=1.496\times10^{11}\ \text{m},\quad 1\ \text{pc}=3.08\times10^{16}\ \text{m}=3.26\ \text{ly} \]

\[ 1\ \text{kWh}=3.6\times10^{6}\ \text{J},\quad 1\ \text{eV}=1.6\times10^{-19}\ \text{J},\quad 1\ \text{erg}=10^{-7}\ \text{J} \]

\[ D=\frac{b}{\theta},\qquad d=\alpha D \]

\[ D=\frac{vt}{2},\qquad t_{\text{film}}=\frac{V}{A} \]

\[ 1^\circ=\frac{\pi}{180}\ \text{rad},\quad 1^{\prime\prime}=\frac{\pi}{180\times3600}\ \text{rad}\approx4.85\times10^{-6}\ \text{rad} \]

\[ [F]=MLT^{-2},\quad [E]=ML^2T^{-2},\quad [G]=M^{-1}L^3T^{-2} \]

\[ \left(P+\frac{a}{V^2}\right)(V-b)=RT:\quad [a]=ML^5T^{-2},\ [b]=L^3 \]

\[ n_2=n_1\left(\frac{M_1}{M_2}\right)^a\left(\frac{L_1}{L_2}\right)^b\left(\frac{T_1}{T_2}\right)^c \]

\[ u_{\text{new}}=\alpha^a\beta^b\gamma^c\,u_{\text{SI}},\qquad n_{\text{new}}=\frac{n_{\text{SI}}}{\alpha^a\beta^b\gamma^c} \]

\[ 1\ \text{N}=10^5\ \text{dyne},\quad 1\ \text{J}=10^7\ \text{erg},\quad 1\ \text{Pa}=10\ \text{dyne cm}^{-2},\quad 1\ \text{kg m}^{-3}=10^{-3}\ \text{g cm}^{-3} \]

\[ T_p=C\ell^a g^b\ \Longrightarrow\ a=\tfrac12,\ b=-\tfrac12,\quad T_p=C\sqrt{\ell/g} \]

\[ F=6\pi\eta rv,\qquad T=2\pi\sqrt{\frac{m}{k}} \]

\[ \ell_P=\sqrt{\frac{hG}{c^3}} \]

\[ 0.00450=4.50\times10^{-3}\quad(3\ \text{significant figures}) \]

\[ \bar x=\frac1N\sum x_i,\quad \Delta x=\frac1N\sum|x_i-\bar x|,\quad \text{percentage uncertainty}=100\frac{\Delta x}{|\bar x|} \]

\[ a=\bar a\pm\overline{\Delta a},\qquad \text{percentage error}=\frac{\overline{\Delta a}}{\bar a}\times100\% \]

\[ z=x\pm y:\ \Delta z\leq\Delta x+\Delta y;\qquad z=Cx^a y^b:\ \frac{\Delta z}{|z|}\lesssim |a|\frac{\Delta x}{|x|}+|b|\frac{\Delta y}{|y|} \]

\[ Z=\frac{A^pB^q}{C^r}\ \Rightarrow\ \frac{\Delta Z}{Z}=p\frac{\Delta A}{A}+q\frac{\Delta B}{B}+r\frac{\Delta C}{C} \]

\[ \frac{\Delta R}{R^2}=\frac{\Delta R_1}{R_1^2}+\frac{\Delta R_2}{R_2^2} \]

\[ LC_v=1\,MSD-1\,VSD,\quad LC_s=\frac{p}{N},\quad x_{\rm true}=x_{\rm observed}-e_0 \]

\[ \text{Vernier: } x=\text{MSR}+n\times LC,\qquad \text{Screw gauge: } x=\text{LSR}+n\times\frac{p}{N} \]

\[ N\ \text{VSD}=(N-1)\ \text{MSD}\ \Rightarrow\ LC=\frac{1\ \text{MSD}}{N} \]

\[ e_{+}=+n\,LC,\qquad e_{-}=-(N-n)\,LC,\qquad x_{\text{true}}=x_{\text{obs}}-e \]

\[ x_{\text{true}}=(\text{MSR}+n\times LC)-e_0 \]

\[ \Delta x=x_f-x_i,\qquad d=\sum_j|\Delta x_j| \]

\[ d=\sum_k\left|x(t_k)-x(t_{k-1})\right|,\quad t_k=\text{times where }v=0 \]

\[ \bar v=\frac{\Delta x}{\Delta t},\quad \bar s=\frac{d}{\Delta t},\quad \bar s_{\rm equal\ distances}=\frac{2uv}{u+v} \]

\[ \bar v=\frac{3v_1v_2v_3}{v_1v_2+v_2v_3+v_3v_1} \]

\[ \bar v=\frac{\sum v_it_i}{\sum t_i}=\frac{\sum d_i}{\sum d_i/v_i} \]

\[ a=\frac{dv}{dt},\qquad \frac{d(v^2)}{dt}=2va \]

\[ \bar a=\frac{v_2-v_1}{t_2-t_1},\qquad a=\frac{d^2x}{dt^2} \]

\[ v=u+at,\quad s=ut+\tfrac12at^2,\quad v^2=u^2+2as,\quad s=\tfrac12(u+v)t \]

\[ d=ut_r+\frac{u^2}{2a} \]

\[ s=vt-\tfrac12at^2 \]

\[ s_n=u+\frac{a}{2}(2n-1) \]

\[ s_{n+1}-s_n=a,\qquad s_m-s_n=(m-n)\,a \]

\[ s_n=u-\frac{g}{2}(2n-1) \]

\[ v=\text{slope of }x(t),\quad a=\text{slope of }v(t),\quad \Delta x=\int v\,dt,\quad \Delta v=\int a\,dt \]

\[ \bar v_{\text{speed}}=\frac{\sum|\text{areas}|}{\text{total time}},\qquad \bar v=\frac{\sum\text{signed areas}}{\text{total time}} \]

\[ a=\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}\left(\frac{v^2}{2}\right) \]

\[ a=-kv\ \Rightarrow\ v=ue^{-kt},\qquad x_{\max}=\frac{u}{k} \]

\[ v=k\sqrt x\ \Rightarrow\ a=\frac{k^2}{2}\ (\text{constant}) \]

\[ v=u-gt,\quad y-y_0=ut-\tfrac12gt^2,\quad H=\frac{u^2}{2g},\quad T=\frac{2u}{g} \]

\[ t_1+t_2=\frac{2u}{g},\qquad t_1t_2=\frac{2h}{g},\qquad h=\tfrac12gt_1t_2 \]

\[ v=\sqrt{u^2+2gH} \]

\[ x_{A/B}=x_A-x_B,\quad v_{A/B}=v_A-v_B,\quad a_{A/B}=a_A-a_B \]

\[ t=\frac{L_1+L_2}{v_1\pm v_2} \]

\[ a_{\min}=\frac{(v_1-v_2)^2}{2d} \]

\[ \vec A=A_x\hat i+A_y\hat j,\quad A_x=A\cos\theta,\quad A_y=A\sin\theta,\quad A=\sqrt{A_x^2+A_y^2} \]

\[ \tan\theta=\frac{A_y}{A_x} \]

\[ A_{\parallel}=A\cos\varphi,\qquad A_{\perp}=A\sin\varphi \]

\[ \hat A=\frac{\vec A}{|\vec A|},\qquad \vec r=x\hat i+y\hat j+z\hat k,\qquad \Delta\vec r=(x_2-x_1)\hat i+(y_2-y_1)\hat j+(z_2-z_1)\hat k \]

\[ A=\sqrt{A_x^2+A_y^2+A_z^2} \]

\[ \cos\alpha=\frac{A_x}{A},\ \cos\beta=\frac{A_y}{A},\ \cos\gamma=\frac{A_z}{A},\qquad \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 \]

\[ \vec R=\vec A+\vec B,\qquad R^2=A^2+B^2+2AB\cos\theta \]

\[ \tan\alpha=\frac{B\sin\theta}{A+B\cos\theta} \]

\[ |\vec A+\vec B|=2A\cos\frac{\theta}{2},\qquad |\vec A-\vec B|=2A\sin\frac{\theta}{2} \]

\[ \vec A\cdot\vec B=AB\cos\theta=A_xB_x+A_yB_y+A_zB_z \]

\[ \cos\theta=\frac{A_xB_x+A_yB_y+A_zB_z}{AB} \]

\[ A_{B}=A\cos\theta=\frac{\vec A\cdot\vec B}{B} \]

\[ |\vec A\pm\vec B|^2=A^2+B^2\pm2\,\vec A\cdot\vec B \]

\[ \vec A\times\vec B=AB\sin\theta\,\hat n,\qquad \hat i\times\hat j=\hat k,\quad \vec B\times\vec A=-\vec A\times\vec B \]

\[ \text{parallelogram}=|\vec A\times\vec B|,\qquad \text{triangle}=\tfrac12|\vec A\times\vec B| \]

\[ (\vec A\cdot\vec B)^2+|\vec A\times\vec B|^2=A^2B^2,\qquad \tan\theta=\frac{|\vec A\times\vec B|}{\vec A\cdot\vec B} \]

\[ \sum\vec F=0\ \Longleftrightarrow\ \sum F_x=0,\ \sum F_y=0,\ \sum F_z=0 \]

\[ \vec F_3=-(\vec F_1+\vec F_2),\qquad |F_1-F_2|\le F_3\le F_1+F_2 \]

\[ \frac{F_1}{\sin\alpha}=\frac{F_2}{\sin\beta}=\frac{F_3}{\sin\gamma} \]

\[ T=\frac{W}{2\cos\theta} \]

\[ \vec r=\vec r_0+\vec u t+\tfrac12\vec a t^2,\qquad \vec v=\vec u+\vec a t \]

\[ \vec v=\frac{dx}{dt}\hat i+\frac{dy}{dt}\hat j,\qquad \vec a=\frac{d^2x}{dt^2}\hat i+\frac{d^2y}{dt^2}\hat j \]

\[ a_t=\frac{dv}{dt}=\frac{\vec v\cdot\vec a}{v} \]

\[ x=u\cos\theta\,t,\quad y=u\sin\theta\,t-\tfrac12gt^2,\quad T=\frac{2u\sin\theta}{g},\quad H=\frac{u^2\sin^2\theta}{2g},\quad R=\frac{u^2\sin2\theta}{g} \]

\[ R=4H\cot\theta,\qquad H=\frac{gT^2}{8} \]

\[ R_{\max}=\frac{u^2}{g}\ \ (\theta=45^\circ),\qquad H_{45^\circ}=\frac{u^2}{4g}=\frac{R_{\max}}{4} \]

\[ K_{\rm top}=K\cos^2\theta \]

\[ y=x\tan\theta-\frac{gx^2}{2u^2\cos^2\theta}=x\tan\theta\left(1-\frac{x}{R}\right) \]

\[ H_1+H_2=\frac{u^2}{2g},\qquad R=4\sqrt{H_1H_2},\qquad T_1T_2=\frac{2R}{g} \]

\[ \tan\theta=b,\qquad R=\frac{b}{c},\qquad H=\frac{b^2}{4c} \]

\[ t=\sqrt{\frac{2h}{g}},\qquad R=u_x\sqrt{\frac{2h}{g}},\qquad v_y=-gt \]

\[ v=\sqrt{u^2+2gh},\qquad \tan\varphi=\frac{\sqrt{2gh}}{u} \]

\[ y=\frac{gx^2}{2u^2},\qquad \tan\varphi=2\tan\alpha \]

\[ T=\frac{2u\sin\theta}{g\cos\alpha},\qquad R_{\rm up}=\frac{2u^2\sin\theta\cos(\theta+\alpha)}{g\cos^2\alpha} \]

\[ R_{\max,\rm up}=\frac{u^2}{g(1+\sin\alpha)},\qquad R_{\max,\rm down}=\frac{u^2}{g(1-\sin\alpha)} \]

\[ t=\frac{2u\tan\alpha}{g},\qquad L=\frac{2u^2\tan\alpha}{g\cos\alpha} \]

\[ \cot\theta=2\tan\alpha \]

\[ \vec v_{A/G}=\vec v_{A/B}+\vec v_{B/G},\quad t_{\min}=\frac{d}{u},\quad t_{\rm opposite}=\frac{d}{\sqrt{u^2-w^2}} \]

\[ \vec v_{R/M}=\vec v_R-\vec v_M,\qquad \tan\theta=\frac{v_M}{v_R}\ \ (\text{vertical rain}) \]

\[ x=\frac{(w-u\sin\theta)\,d}{u\cos\theta},\qquad x_{\min}=\frac{d\sqrt{w^2-u^2}}{u}\ \ (u

Symbols and assumptions: d river width (m); u swimmer speed relative to water (m/s); w current speed (m/s); θ heading measured upstream from the perpendicular.

\[ \vec F_{\rm net}=\frac{d\vec p}{dt}=m\vec a\quad(\text{constant }m),\qquad \vec p=m\vec v \]

\[ F=v\frac{dm}{dt} \]

\[ \vec F_{\rm av}=\frac{\Delta\vec p}{\Delta t} \]

\[ W=mg,\qquad W_{\parallel}=mg\sin\theta,\qquad W_{\perp}=mg\cos\theta \]

\[ N_{12}=\frac{m_2F}{m_1+m_2} \]

\[ N=mg-F\sin\theta\ \ (\text{pull up}),\qquad N=mg+F\sin\theta\ \ (\text{push down}) \]

\[ T(x)=\frac{M}{L}xg\ \ (\text{hanging rope}),\qquad T(x)=\frac{x}{L}F\ \ (\text{rope pulled on a smooth floor}),\qquad F_s=kx \]

\[ T(x)=\left(m+\frac{M}{L}x\right)g \]

\[ \frac{1}{k_s}=\frac{1}{k_1}+\frac{1}{k_2},\qquad k_p=k_1+k_2,\qquad k_{\rm piece}=nk \]

\[ a=\frac{(m_2-m_1)g}{m_1+m_2},\qquad T=\frac{2m_1m_2g}{m_1+m_2} \]

\[ a=\frac{m_2g}{m_1+m_2},\qquad T=\frac{m_1m_2g}{m_1+m_2} \]

\[ F_{\rm clamp}=2T\cos\frac{\phi}{2} \]

\[ a_{\rm rel}=\frac{(m_2-m_1)(g+a_0)}{m_1+m_2},\qquad T=\frac{2m_1m_2(g+a_0)}{m_1+m_2} \]

\[ N=m(g+a)\ \text{(upward acceleration)},\quad N=m(g-a)\ \text{(downward acceleration)},\quad \vec F_{\rm pseudo}=-m\vec a_{\rm frame} \]

\[ g_{\rm eff}=g+a_0,\qquad T_{\rm pend}=2\pi\sqrt{\frac{L}{g_{\rm eff}}} \]

\[ T_{\rm cable}=M(g+a_0) \]

\[ \vec F_{\rm pseudo}=-m\vec a_0,\qquad \tan\theta=\frac{a_0}{g}\ \ (\text{pendulum in an accelerating car}) \]

\[ g_{\rm eff}=\sqrt{g^2+a_0^2},\qquad T=m\sqrt{g^2+a_0^2} \]

\[ a_{\rm rel}=g\sin\theta-a_0\cos\theta,\qquad N=m(g\cos\theta+a_0\sin\theta) \]

\[ \vec J=\int\vec F\,dt=\Delta\vec p,\qquad \vec F_{\rm av}=\frac{\Delta\vec p}{\Delta t},\qquad \vec P_f=\vec P_i\quad(J_{\rm ext}=0) \]

\[ V=\frac{mv}{M} \]

\[ |\Delta\vec p|=2mv\cos\theta \]

\[ \sum F_r=\frac{mv^2}{r},\qquad \tan\theta=\frac{v^2}{rg}\quad\text{(frictionless bank)} \]

\[ v_{\max}=\sqrt{rg\,\frac{\mu+\tan\theta}{1-\mu\tan\theta}} \]

\[ v_{\max}=\sqrt{\mu rg}\ \ (\text{level road}),\qquad v_{\max}=\sqrt{gr}\ \ (\text{top of a hump}) \]

\[ v_{\rm top,min}=\sqrt{gr},\qquad v_{\rm bottom,min}=\sqrt{5gr} \]

\[ 0\leq |f_s|\leq\mu_sN,\qquad f_{s,\max}=\mu_sN \]

\[ mg\;\le\;\sqrt{N^2+f^2}\;\le\;mg\sqrt{1+\mu_s^2} \]

\[ \tan\lambda=\frac{f_{s,\max}}{N}=\mu_s \]

\[ f_k=\mu_kN,\qquad W_f=-f_ks\quad\text{(fixed surface, opposite sliding)} \]

\[ s=\frac{u^2}{2\mu_kg},\qquad t=\frac{u}{\mu_kg} \]

\[ Q=\mu_kN\,s_{\rm rel} \]

\[ \tan\theta_r=\mu_s,\qquad a=g(\sin\theta-\mu_k\cos\theta)\quad\text{(sliding downhill)} \]

\[ mg(\sin\theta-\mu_s\cos\theta)\;\le\;F\;\le\;mg(\sin\theta+\mu_s\cos\theta) \]

\[ \mu_k=\tan\theta\left(1-\frac{1}{n^2}\right) \]

\[ a_{\rm up}=g(\sin\theta+\mu_k\cos\theta),\qquad a_{\rm down}=g(\sin\theta-\mu_k\cos\theta) \]

\[ N=mg-P\sin\theta,\qquad P\cos\theta=\mu_s(mg-P\sin\theta) \]

\[ P=\frac{\mu mg}{\cos\theta+\mu\sin\theta},\qquad P_{\min}=\frac{\mu mg}{\sqrt{1+\mu^2}}\ \text{at}\ \tan\theta=\mu \]

\[ P=\frac{\mu mg}{\cos\theta-\mu\sin\theta}\qquad(\text{no motion possible if }\tan\theta\ge 1/\mu) \]

\[ a_{\max}=\mu_sg\quad\text{(upper block)},\qquad v_{\max}=\sqrt{\mu_srg}\quad\text{(level turn)} \]

\[ a=\frac{(m_2-\mu_km_1)g}{m_1+m_2},\qquad T=m_2(g-a),\qquad \text{moves only if } m_2>\mu_sm_1 \]

\[ v_{\max}=\sqrt{rg\,\frac{\tan\theta+\mu_s}{1-\mu_s\tan\theta}},\qquad v_{\min}=\sqrt{rg\,\frac{\tan\theta-\mu_s}{1+\mu_s\tan\theta}} \]

\[ F_{\max}=\mu_s(m+M)g\ \ (\text{force on lower block}),\qquad F_{\max}=\mu_smg\,\frac{m+M}{M}\ \ (\text{force on upper block}) \]

\[ a_m=\mu_kg,\qquad a_M=\frac{F-\mu_kmg}{M},\qquad t_{\rm off}=\sqrt{\frac{2L}{a_M-a_m}} \]

\[ F_{\max}=(\mu_1+\mu_2)(m+M)g \]

\[ f_{\rm roll}=\mu_rN,\qquad \mu_r\ll\mu_k<\mu_s \]

\[ W=\vec F\cdot\vec s=Fs\cos\theta,\qquad W=\int_{x_i}^{x_f}F_x\,dx \]

\[ W=\vec F\cdot\Delta\vec r=F_x\Delta x+F_y\Delta y+F_z\Delta z \]

\[ 1\ \text{J}=10^7\ \text{erg},\quad 1\ \text{kWh}=3.6\times10^6\ \text{J},\quad 1\ \text{eV}=1.6\times10^{-19}\ \text{J} \]

\[ K=\tfrac12mv^2,\qquad W_{\rm net}=K_f-K_i \]

\[ K=\frac{p^2}{2m},\qquad p=\sqrt{2mK} \]

\[ \int_{x_i}^{x_f}F_{\rm net}\,dx=\tfrac12mv^2-\tfrac12mu^2 \]

\[ W_c=-\Delta U,\quad F_x=-\frac{dU}{dx},\quad U_g=mgh,\quad U_s=\tfrac12kx^2 \]

\[ \frac{dU}{dx}=0;\quad \frac{d^2U}{dx^2}>0\ \text{stable},\quad <0\ \text{unstable},\quad =0\ \text{neutral} \]

\[ K(x)=E-U(x)\ge 0 \]

\[ F=-kx,\qquad W_{\rm spring}=-\tfrac12k\left(x_2^2-x_1^2\right),\qquad U=\tfrac12kx^2=\frac{F^2}{2k} \]

\[ \frac1{k_s}=\frac1{k_1}+\frac1{k_2},\qquad k_p=k_1+k_2,\qquad kL=\text{constant} \]

\[ x_{\max}=v\sqrt{\frac mk}\ \ (\text{smooth}),\qquad \tfrac12mv^2=\tfrac12kx^2+\mu mgx\ \ (\text{rough}) \]

\[ K_i+U_i+W_{nc}=K_f+U_f \]

\[ v_{\rm bottom}=\sqrt{2gL(1-\cos\theta_0)},\qquad T_{\rm bottom}=mg(3-2\cos\theta_0) \]

\[ W=\frac{MgL}{2n^2} \]

\[ P_{\rm av}=\frac{W}{\Delta t},\quad P=\vec F\cdot\vec v,\quad \eta=\frac{P_{\rm useful}}{P_{\rm input}} \]

\[ v=\sqrt{\frac{2Pt}{m}}\propto t^{1/2},\qquad x\propto t^{3/2},\qquad F=\frac{P}{v}\propto t^{-1/2} \]

\[ P=(mg\sin\theta+f)\,v,\qquad v_{\max}=\frac{P}{f}\ \ (\text{level road}) \]

\[ m_1u_1+m_2u_2=m_1v_1+m_2v_2,\quad e=\frac{v_2-v_1}{u_1-u_2},\quad V_{\rm stick}=\frac{m_1u_1+m_2u_2}{m_1+m_2} \]

\[ \Delta K=\frac{m_1m_2\,(u_1-u_2)^2}{2(m_1+m_2)},\qquad \frac{\Delta K}{K_i}=\frac{m_2}{m_1+m_2}\ (u_2=0) \]

\[ v=\frac{m+M}{m}\sqrt{2gh} \]

\[ m_1v_1=m_2v_2,\qquad \frac{K_1}{K_2}=\frac{m_2}{m_1} \]

\[ e=\frac{v_2-v_1}{u_1-u_2},\qquad h_n=e^{2n}h,\qquad v_n=e^n\sqrt{2gh} \]

\[ H_{\rm total}=h\,\frac{1+e^2}{1-e^2},\qquad T_{\rm total}=\sqrt{\frac{2h}{g}}\;\frac{1+e}{1-e} \]

\[ \Delta K=\frac{m_1m_2}{2(m_1+m_2)}\,(1-e^2)\,(u_1-u_2)^2 \]

\[ v_t'=v_t,\quad v_n'=e\,v_n,\quad \tan\theta'=\frac{\tan\theta}{e} \]

\[ v_{\rm top,min}=\sqrt{gr},\qquad v_{\rm bottom,min}=\sqrt{5gr} \]

\[ T=\frac{mu^2}{r}+mg(3\cos\theta-2),\qquad T_{\rm bottom}-T_{\rm top}=6mg \]

\[ u_{\rm rod}=\sqrt{4gr},\qquad h_{\rm loop}=\tfrac52R,\qquad \cos\theta_{\rm leave}=\tfrac23\ (\text{sphere}) \]

\[ v_1=\frac{(m_1-m_2)u_1+2m_2u_2}{m_1+m_2},\quad v_2=\frac{2m_1u_1+(m_2-m_1)u_2}{m_1+m_2} \]

\[ \frac{K_2}{K_1}=\frac{4m_1m_2}{(m_1+m_2)^2},\qquad \frac{K_1^{\prime}}{K_1}=\left(\frac{m_1-m_2}{m_1+m_2}\right)^2 \]

\[ v_2-v_1=-(u_2-u_1) \]

\[ \vec R=\frac{\sum m_i\vec r_i}{M},\quad M=\sum m_i,\quad \vec F_{\rm ext}=M\vec A_{CM} \]

\[ \begin{array}{ll}\text{Semicircular ring (wire)} & 2R/\pi\ \text{from centre}\\ \text{Semicircular disc} & 4R/3\pi\ \text{from centre}\\ \text{Hemispherical shell} & R/2\ \text{from centre}\\ \text{Solid hemisphere} & 3R/8\ \text{from centre}\\ \text{Solid cone} & h/4\ \text{from base}\\ \text{Hollow cone (surface)} & h/3\ \text{from base}\end{array} \]

\[ X=\frac{M_1x_1-M_2x_2}{M_1-M_2} \]

\[ \omega=\omega_0+\alpha t,\quad \theta=\omega_0t+\tfrac12\alpha t^2,\quad v=r\omega,\quad a_t=r\alpha \]

\[ a=\sqrt{(r\alpha)^2+(\omega^2r)^2}=r\sqrt{\alpha^2+\omega^4} \]

\[ \omega=\frac{2\pi N}{60},\qquad \theta_n=\omega_0+\frac{\alpha}{2}(2n-1) \]

\[ \vec\tau=\vec r\times\vec F,\quad |\tau|=Fd_\perp,\quad \sum\vec F=0,\quad \sum\vec\tau=0 \]

\[ \vec\tau=\begin{vmatrix}\hat i&\hat j&\hat k\\ x&y&z\\ F_x&F_y&F_z\end{vmatrix},\qquad \tau_z=xF_y-yF_x \]

\[ F_1d_1=F_2d_2 \]

\[ \sum\vec F=0,\qquad \sum\vec\tau=0,\qquad \mu_{\min}=\frac{1}{2\tan\theta}\ \ (\text{ladder, smooth wall}) \]

\[ \mu>\frac{b}{2h}\ (\text{push at height }h\text{ tips first}),\qquad \tan\theta_{\rm topple}=\frac bh\ (\text{incline}) \]

\[ v_{\rm overturn}=\sqrt{\frac{grd}{2h}} \]

\[ I=\sum m_ir_i^2,\quad I_{\rm ring}=MR^2,\quad I_{\rm disc}=\tfrac12MR^2,\quad I_{\rm sphere}=\tfrac25MR^2,\quad I_{\rm rod,CM}=\tfrac1{12}M\ell^2 \]

\[ \begin{array}{lll}\text{Body} & \text{Axis} & I\\ \text{Thin ring} & \text{central, }\perp\text{ plane} & MR^2\\ \text{Thin ring} & \text{diameter} & MR^2/2\\ \text{Disc / solid cylinder} & \text{central, }\perp\text{ plane (own axis)} & MR^2/2\\ \text{Disc} & \text{diameter} & MR^2/4\\ \text{Hollow cylinder} & \text{own axis} & MR^2\\ \text{Solid sphere} & \text{diameter} & 2MR^2/5\\ \text{Hollow sphere} & \text{diameter} & 2MR^2/3\\ \text{Rod} & \text{centre, }\perp\text{ rod} & ML^2/12\\ \text{Rod} & \text{end, }\perp\text{ rod} & ML^2/3\\ \text{Rectangular plate } a\times b & \text{centre, }\perp\text{ plate} & M(a^2+b^2)/12\end{array} \]

\[ I=Mk^2,\qquad k_{\rm ring}=R,\ k_{\rm disc}=\frac{R}{\sqrt2},\ k_{\rm sphere}=\sqrt{\tfrac25}\,R,\ k_{\rm rod,c}=\frac{L}{2\sqrt3} \]

\[ I=I_{CM}+Md^2,\qquad I_z=I_x+I_y\quad\text{(plane lamina in xy)} \]

\[ \begin{array}{lll}\text{Body} & \text{Axis} & I\\ \text{Ring} & \text{tangent, in plane} & 3MR^2/2\\ \text{Ring} & \text{tangent, }\perp\text{ plane} & 2MR^2\\ \text{Disc} & \text{tangent, in plane} & 5MR^2/4\\ \text{Disc} & \text{tangent, }\perp\text{ plane} & 3MR^2/2\\ \text{Solid sphere} & \text{tangent} & 7MR^2/5\\ \text{Hollow sphere} & \text{tangent} & 5MR^2/3\\ \text{Square plate} & \text{diagonal} & Ma^2/12\end{array} \]

\[ \tau_{\rm net}=I\alpha,\quad K_{\rm rot}=\tfrac12I\omega^2,\quad W=\int\tau\,d\theta,\quad P=\tau\omega \]

\[ a=\frac{mg}{m+I/R^2},\qquad T=\frac{I}{R^2}\,a \]

\[ \alpha_0=\frac{3g}{2L},\qquad \omega_{\rm vertical}=\sqrt{\frac{3g}{L}},\qquad v_{\rm end}=\sqrt{3gL} \]

\[ \vec L=\vec r\times\vec p,\quad L=I\omega,\quad \tau_{\rm ext}=\frac{dL}{dt},\quad I_i\omega_i=I_f\omega_f\quad(\tau_{\rm ext}=0) \]

\[ \omega=\frac{I_1\omega_1+I_2\omega_2}{I_1+I_2},\qquad \Delta K=\frac{I_1I_2(\omega_1-\omega_2)^2}{2(I_1+I_2)} \]

\[ \int\tau\,dt=\Delta L,\qquad \frac{dA}{dt}=\frac{L}{2m} \]

\[ v_{CM}=\omega R,\quad K=\tfrac12Mv_{CM}^2+\tfrac12I_{CM}\omega^2,\quad a=\frac{g\sin\theta}{1+I_{CM}/(MR^2)} \]

\[ K=\tfrac12Mv^2\left(1+\frac{k^2}{R^2}\right),\qquad \frac{K_{\rm rot}}{K}=\frac{k^2/R^2}{1+k^2/R^2} \]

\[ v=\frac{u}{1+k^2/R^2},\qquad t=\frac{u-v}{\mu_kg} \]

\[ a=\frac{g\sin\theta}{1+k^2/R^2},\qquad v=\sqrt{\frac{2gh}{1+k^2/R^2}},\qquad f=\frac{Mg\sin\theta\,(k^2/R^2)}{1+k^2/R^2} \]

\[ t=\sqrt{\frac{2L\,(1+k^2/R^2)}{g\sin\theta}},\qquad \mu_{\min}=\tan\theta\,\frac{k^2/R^2}{1+k^2/R^2} \]

\[ h=\frac{v^2}{2g}\left(1+\frac{k^2}{R^2}\right) \]

\[ F=\frac{GMm}{r^2},\qquad g_r=\frac{GM}{r^2},\qquad \vec g_{\rm net}=\sum\vec g_i \]

\[ x=\frac{d\sqrt{m_1}}{\sqrt{m_1}+\sqrt{m_2}} \]

\[ g=\frac{GM}{R^2}=\frac43\pi G\rho R \]

\[ F_{\rm net}=\sqrt3\,\frac{Gm^2}{a^2} \]

\[ g_h=g_0\left(\frac{R}{R+h}\right)^2,\qquad g_d=g_0\left(1-\frac dR\right) \]

\[ g_h\approx g\left(1-\frac{2h}{R}\right),\qquad \frac{\Delta g}{g}\approx\frac{2h}{R} \]

\[ d = 2h\quad(h,\,d\ll R) \]

\[ g'=g-\omega^2R\cos^2\lambda \]

\[ \Delta g=\omega^2R\cos^2\lambda \]

\[ \omega=\sqrt{\frac gR},\qquad T=2\pi\sqrt{\frac Rg}\approx 84\ \text{min} \]

\[ V=-\frac{GM}{r},\quad U=mV=-\frac{GMm}{r},\quad g_r=-\frac{dV}{dr}\ \text{(signed outward component)} \]

\[ \Delta U=\frac{mgh}{1+h/R} \]

\[ U=-\sum_{\text{pairs}}\frac{Gm_im_j}{r_{ij}} \]

\[ V=-\frac{GM}{\sqrt{a^2+x^2}} \]

\[ v_o=\sqrt{\frac{GM}{r}},\quad T=2\pi\sqrt{\frac{r^3}{GM}},\quad E=-\frac{GMm}{2r},\quad K=\frac{GMm}{2r} \]

\[ v_o\approx\sqrt{gR}\approx 8\ \text{km/s},\qquad T_0=2\pi\sqrt{R/g}\approx 84\ \text{min} \]

\[ \Delta E=\frac{GMm}{2}\left(\frac1{r_1}-\frac1{r_2}\right) \]

\[ L=mv_or=m\sqrt{GMr} \]

\[ N=m(g_{\rm local}-a),\qquad a=g_{\rm local}\ \Rightarrow\ N=0 \]

\[ N=m(g+a)\ \text{(accelerating up)},\qquad N=m(g-a)\ \text{(accelerating down)} \]

\[ g_{\rm art}=\omega^2r \]

\[ v_e=\sqrt{\frac{2GM}{r}}=\sqrt{2g_0R}\quad\text{(surface }r=R) \]

\[ v_e=R\sqrt{\frac{8\pi G\rho}{3}} \]

\[ h=\frac{Rv^2}{v_e^2-v^2}\qquad(v

Symbols and assumptions: h greatest height above the surface (m); R planet radius (m); v launch speed (m/s); v_e surface escape speed (m/s). Exact for a spherical planet without air; reduces to v²/2g when v ≪ v_e.

\[ \sigma = \frac{F}{A} \]

\[ \text{strain} = \frac{\text{change in size}}{\text{original size}} \]

\[ \text{stress} \propto \text{strain} \quad\Rightarrow\quad \text{stress} = k \times \text{strain} \]

\[ Y = \frac{\text{tensile (or compressive) stress}}{\text{longitudinal strain}} = \frac{F/A}{\Delta L/L} = \frac{F\,L}{A\,\Delta L} = \frac{MgL}{\pi r^{2}\,\Delta L} \]

\[\Delta L=\int_0^L\frac{Mgx}{LAY}\,dx=\frac{MgL}{2AY}=\frac{\rho gL^2}{2Y}\]

\[ G = \frac{\text{shearing stress}}{\text{shearing strain}} = \frac{F/A}{\Delta x/L} = \frac{F\,L}{A\,\Delta x} \approx \frac{F}{A\,\theta} \]

\[ B = -\frac{p}{\Delta V/V} = -\frac{V\,\Delta p}{\Delta V} \]

\[ k = \frac{1}{B} = -\frac{1}{\Delta p}\,\frac{\Delta V}{V} \]

\[ \frac{\Delta V}{V} = 3\,\frac{\Delta r}{r} \;\Rightarrow\; \frac{\Delta r}{r} = -\frac{p}{3B} \]

\[ \sigma = -\frac{\text{lateral strain}}{\text{longitudinal strain}} = -\frac{\Delta d/d}{\Delta L/L} \]

\[ \frac{\Delta V}{V} = (1-2\sigma)\,\frac{\Delta L}{L} \]

\[ U = \tfrac12 F\,\Delta l = \tfrac12\,\frac{YA}{L}\,(\Delta l)^{2} \]

\[ u = \tfrac12\,\text{stress}\times\text{strain} = \tfrac12 Y(\text{strain})^{2} = \frac{(\text{stress})^{2}}{2Y} \]

\[ \text{strain} = \frac{L\alpha\Delta T}{L} = \alpha\Delta T,\qquad \text{stress} = Y\alpha\Delta T,\qquad F = YA\alpha\Delta T \]

\[ \frac{Mg}{\pi r^{2}} \le \sigma_y \;\Rightarrow\; r \ge \sqrt{\frac{Mg}{\pi\sigma_y}} = \sqrt{\frac{10^{4}\times9.8}{3.14\times3\times10^{8}}} \approx 1\ \text{cm} \]

\[ \delta = \frac{W\,l^{3}}{4\,b\,d^{3}\,Y} \]

\[ h\rho g \le \sigma_\text{limit} \;\Rightarrow\; h_\text{max} = \frac{30\times10^{7}}{3\times10^{3}\times10} = 10^{4}\ \text{m} = 10\ \text{km} \]

\[ \rho = \frac{m}{V}, \qquad \text{RD} = \frac{\rho}{1000\ \text{kg m}^{-3}} \]

\[ \rho_{\text{equal }V} = \frac{\rho_1 + \rho_2}{2}, \qquad \rho_{\text{equal }m} = \frac{2\rho_1\rho_2}{\rho_1 + \rho_2} \]

\[P_1=P_2\quad\text{at equal height in a connected fluid at rest}\]

\[ P = P_0 + \rho g h \]

\[ P_2 - P_1 = \rho g (h_2 - h_1) \]

\[ F = \underbrace{\frac{\rho g H}{2}}_{\text{average pressure}} \times (H w) = \frac{\rho g H^2 w}{2} \]

\[A_1h_1+A_2h_2=(A_1+A_2)h_f,\qquad h_f=\frac{A_1h_1+A_2h_2}{A_1+A_2}\]

\[W_g=\frac{\rho g}{2}\left[A_1h_1^2+A_2h_2^2-(A_1+A_2)h_f^2\right]=\frac{\rho gA_1A_2}{2(A_1+A_2)}(h_1-h_2)^2\]

\[W_g=\frac{\rho gA}{4}(h_1-h_2)^2\]

\[ P_0 = \rho_{Hg}\, g\, h \]

\[ P_{\text{gas}} = P_0 + \rho g h \qquad (h = \text{difference in liquid levels}) \]

\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \;\Rightarrow\; F_2 = F_1 \frac{A_2}{A_1} = F_1\left(\frac{d_2}{d_1}\right)^2 \]

\[ A_1 x_1 = A_2 x_2, \qquad F_1 x_1 = F_2 x_2 \]

\[ F_B = \rho_l\, V_{sub}\, g \]

\[ \rho_b V g = \rho_l V_{sub} g \;\Rightarrow\; \frac{V_{sub}}{V} = \frac{\rho_b}{\rho_l} \]

\[ W_{app} = W - F_B = W\left(1 - \frac{\rho_l}{\rho_b}\right) \]

\[ \rho_b V = \rho_1 V_1 + \rho_2 V_2 \]

\[ A_1 v_1 = A_2 v_2 = Q \]

\[ v \propto \frac{1}{A} \propto \frac{1}{d^2}, \qquad \frac{v_1}{v_2} = \left(\frac{d_2}{d_1}\right)^2 \]

\[ A v = \sum_i A_i v_i, \qquad A_1 v_1 = n\,a\,v_2 \]

\[ P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant} \]

\[ \frac{P}{\rho} + \frac{v^2}{2} + g h = \text{const} \]

\[ \frac{P}{\rho g} + \frac{v^2}{2g} + h = \text{const} \]

\[ v = \sqrt{2 g h} \]

\[ v = \sqrt{2gh + \frac{2(P - P_0)}{\rho}} \]

\[ R = v\sqrt{\frac{2y}{g}} = 2\sqrt{h\,y} = 2\sqrt{h(H-h)} \]

\[ t = \frac{A}{a}\sqrt{\frac{2H}{g}} \;\propto\; \sqrt{H} \]

\[ F = \rho a v^2 = 2\rho a g h \]

\[ A_1v_1 = A_2v_2, \qquad P_1 - P_2 = \tfrac12\rho\,(v_2^2 - v_1^2) \]

\[ v_1 = \sqrt{\frac{2\,\Delta P}{\rho\left[(A_1/A_2)^2 - 1\right]}}, \qquad Q = A_1 v_1 = A_1A_2\sqrt{\frac{2\,\Delta P}{\rho\,(A_1^2 - A_2^2)}} \]

\[P_\text{bottom}-P_\text{top}=\frac12\rho(v_\text{top}^{2}-v_\text{bottom}^{2}),\qquad F_\text{lift}=\Delta P\,A\]

\[\tan\theta=\frac{|a|}{g},\qquad |\Delta h|=\frac{|a|\ell}{g}\]

\[P-P_0=\rho(g+a)h,\qquad F_B=\rho_\ell V_\text{sub}(g+a)\]

\[ F = -\,\eta A \frac{dv}{dx} \]

\[ \frac{dv}{dx} = \frac{v}{d} \;\;(\text{uniform film}) \]

\[ \eta = \frac{F/A}{v/d} \;\Rightarrow\; [\mathrm{M\,L^{-1}\,T^{-1}}] \]

\[Q=\frac{\pi r^4\Delta P}{8\eta\ell}\]

\[v(s)=\frac{\Delta P}{4\eta\ell}(r^2-s^2),\quad v_\text{max}=2\bar v\]

\[ F_v = 6\pi\eta r v \]

\[ v_t = \frac{2 r^2 (\rho - \sigma) g}{9\eta} \]

\[ F_v = W - B = \tfrac43\pi r^3(\rho-\sigma)g = Mg\left(1-\tfrac{\sigma}{\rho}\right) \]

\[ R = n^{1/3}r,\quad v' = n^{2/3}\,v \]

\[ v = \frac{2r^2\sigma g}{9\eta} \;\Rightarrow\; \eta = \frac{2r^2\sigma g}{9v} \]

\[ R_e = \frac{\rho v d}{\eta} = \text{inertia/viscosity scale} \]

\[ T = \frac{F}{L} \]

\[ F = 2TL \]

\[ W = T \cdot 2\,\Delta A \]

\[ W = 8\pi T\,(r_2^2 - r_1^2) \]

\[ E = 4\pi r^2 T \]

\[ \Delta E = 4\pi r^2 T\, n\,(1 - n^{-1/3}) = 4\pi R^2 T\,(n^{1/3}-1) \]

\[ S_{la}\cos\theta + S_{sl} = S_{sa} \;\Rightarrow\; \cos\theta = \frac{S_{sa}-S_{sl}}{S_{la}} \]

\[ P_{in}-P_{out} = \frac{2T}{r} \]

\[ P_{in}-P_{out} = \frac{4T}{r} \]

\[ P_{in} = P_0 + \rho g h + \frac{2T}{r} \]

\[ r = \frac{r_1 r_2}{r_2 - r_1} \]

\[ n = \frac{PV}{RT_K} = \frac{\left(P_0 + \frac{4T}{r}\right)\frac43\pi r^3}{RT_K} \]

\[ h = \frac{2T\cos\theta}{r\rho g} \]

\[ \pi r^2 h\rho g = 2\pi r T\cos\theta \]

\[ T = \frac{r h \rho g}{2\cos\theta} \;\xrightarrow{\theta\,\approx\,0}\; \frac{rh\rho g}{2} \]

\[T=\frac{\rho grh}{2\cos\theta}\approx\frac{\rho grh}{2}\quad(\theta\approx0)\]

\[ a=-\omega^2x,\qquad F=-kx,\qquad \omega=\sqrt{k/m} \]

\[ k=\left.\frac{d^2U}{dx^2}\right|_{x_0},\qquad \omega=\sqrt{\frac km} \]

\[ \sin^2\omega t=\tfrac12-\tfrac12\cos2\omega t\quad\Rightarrow\quad T=\frac{\pi}{\omega} \]

\[ x=A\cos(\omega t+\phi),\qquad T=\frac{2\pi}{\omega},\qquad f=\frac1T \]

\[ A=\sqrt{x_0^2+\frac{v_0^2}{\omega^2}},\qquad \tan\phi=-\frac{v_0}{\omega x_0} \]

\[ t=\frac{T}{2\pi}\sin^{-1}\!\left(\frac xA\right) \]

\[ v=-A\omega\sin(\omega t+\phi),\quad v^2=\omega^2(A^2-x^2),\quad a=-\omega^2x \]

\[ v_{\max}=A\omega,\quad a_{\max}=A\omega^2,\quad \frac{a_{\max}}{v_{\max}}=\omega \]

\[ \frac{x^2}{A^2}+\frac{v^2}{A^2\omega^2}=1 \]

\[ \omega^2=\frac{v_1^2-v_2^2}{x_2^2-x_1^2} \]

\[ E=\tfrac12kA^2,\quad U=\tfrac12kx^2,\quad K=\tfrac12k(A^2-x^2) \]

\[ E=\tfrac12m\omega^2A^2=2\pi^2mf^2A^2 \]

\[ x=\frac{A}{\sqrt{n+1}} \]

\[ f_{K}=f_{U}=2f \]

\[ T=2\pi\sqrt{m/k_{\rm eff}},\quad k_{\parallel}=k_1+k_2,\quad \frac1{k_{\rm series}}=\frac1{k_1}+\frac1{k_2},\quad x_0=\frac{mg}{k} \]

\[ k_i=k\,\frac{L}{l_i} \]

\[ T=2\pi\sqrt{\frac{\mu}{k}},\qquad \mu=\frac{m_1m_2}{m_1+m_2} \]

\[ T=2\pi\sqrt{\frac{x_0}{g}} \]

\[ T=2\pi\sqrt{\ell/g},\qquad g_{\rm eff}=g+a\quad\text{(lift accelerating upward)} \]

\[ T=2\pi\sqrt{\frac{\ell}{\sqrt{g^2+a^2}}} \]

\[ T=2\pi\sqrt{\frac{\ell}{g(1-\sigma/\rho)}} \]

\[ T=2\pi\sqrt{\frac{1}{g\left(\frac1\ell+\frac1R\right)}}\ \xrightarrow{\ \ell\to\infty\ }\ 2\pi\sqrt{\frac Rg} \]

\[ T_{\rm float}=2\pi\sqrt{\frac{m}{\sigma Ag}}=2\pi\sqrt{\frac hg},\qquad T_{\rm tunnel}=2\pi\sqrt{\frac Rg} \]

\[ T=2\pi\sqrt{\frac{L\rho}{\sigma g}} \]

\[ v_{\max}=\sqrt{gR}\approx8\ \text{km/s} \]

\[ m\ddot x+b\dot x+kx=F_0\cos(\omega_dt),\qquad \omega_0=\sqrt{k/m} \]

\[ A(t)=A_0e^{-bt/2m},\qquad E(t)=E_0e^{-bt/m} \]

\[ A=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega_d^2)^2+(b\omega_d/m)^2}} \]

\[ x=a\sin\omega t+b\cos\omega t,\quad A=\sqrt{a^2+b^2},\qquad T_{U}=2\pi\sqrt{\frac{L}{2g}} \]

\[ \tan\phi=\frac{A_2\sin\delta}{A_1+A_2\cos\delta} \]

\[ A=2a\cos\frac{\delta}{2} \]

\[ y=A\sin(kx-\omega t+\phi),\quad k=\frac{2\pi}{\lambda},\quad \omega=2\pi f,\quad v_w=\frac{\omega}{k}=f\lambda \]

\[ \Delta\phi=\frac{2\pi}{\lambda}\Delta x,\qquad \Delta\phi=\frac{2\pi}{T}\Delta t \]

\[ v_p=-v\,\frac{\partial y}{\partial x},\qquad \frac{v_{p,\max}}{v}=Ak=\frac{2\pi A}{\lambda} \]

\[ v_{\rm string}=\sqrt{\frac{F_T}{\mu}},\qquad v_{\rm sound}=\sqrt{\frac{\gamma p}{\rho}}=\sqrt{\frac{\gamma RT}{M}} \]

\[ v=\sqrt{\frac{T}{\rho A}}=\sqrt{\frac{\text{stress}}{\rho}} \]

\[ v=\sqrt{gx},\qquad t_{\rm total}=2\sqrt{\frac Lg} \]

\[ v_{\rm rod}=\sqrt{\frac{Y}{\rho}},\qquad v_{\rm fluid}=\sqrt{\frac{B}{\rho}} \]

\[ v_{\rm Newton}=\sqrt{\frac p\rho},\qquad v_{\rm Laplace}=\sqrt{\frac{\gamma p}{\rho}}=\sqrt{\frac{\gamma RT}{M}} \]

\[ v_t\approx v_0+0.61\,t \]

\[ \frac{v_1}{v_2}=\sqrt{\frac{\gamma_1T_1M_2}{\gamma_2T_2M_1}} \]

\[ \frac{v_{\rm sound}}{v_{\rm rms}}=\sqrt{\frac\gamma3} \]

\[ I=\tfrac12\rho v\omega^2A^2=2\pi^2\rho vf^2A^2 \]

\[ I_{\rm point}=\frac{P}{4\pi r^2}\propto\frac1{r^2},\qquad I_{\rm line}\propto\frac1r \]

\[ \beta=10\log_{10}\frac{I}{I_0},\qquad \beta_2-\beta_1=10\log_{10}\frac{I_2}{I_1} \]

\[ y_{\rm net}=y_1+y_2,\quad A_R^2=A_1^2+A_2^2+2A_1A_2\cos\delta,\quad I\propto A^2 \]

\[ I_R=I_1+I_2+2\sqrt{I_1I_2}\cos\delta \]

\[ \frac{I_{\max}}{I_{\min}}=\left(\frac{\sqrt{I_1}+\sqrt{I_2}}{\sqrt{I_1}-\sqrt{I_2}}\right)^2=\left(\frac{A_1+A_2}{A_1-A_2}\right)^2 \]

\[ \Delta x=n\lambda\ \text{(max)},\qquad \Delta x=\left(n+\tfrac12\right)\lambda\ \text{(min)} \]

\[ y=2A\sin(kx)\cos(\omega t),\quad \Delta x_{NN}=\frac\lambda2,\quad \Delta x_{NA}=\frac\lambda4 \]

\[ x_{\rm N}=\frac{n\lambda}{2},\qquad x_{\rm A}=\frac{(2n+1)\lambda}{4}\qquad(n=0,1,2,\ldots) \]

\[ A(x)=2A|\sin kx| \]

\[ f_n=\frac{nv}{2L}\ \text{(fixed string or open pipe)},\qquad f_n=\frac{(2n-1)v}{4L}\ \text{(one closed end)} \]

\[ f_{\rm closed}=\frac{v}{4(L+e)},\qquad f_{\rm open}=\frac{v}{2(L+2e)},\qquad e\approx0.6r \]

\[ v=2f(l_2-l_1),\qquad e=\frac{l_2-3l_1}{2} \]

\[ f_1=f_{n+1}-f_n\ \text{(string or open pipe)},\qquad f_1=\frac{f_{\rm next}-f_{\rm prev}}{2}\ \text{(closed pipe)} \]

\[ f_b=|f_1-f_2| \]

\[ y=2A\cos\left(\pi\Delta f\,t\right)\sin\left(2\pi\bar f\,t\right),\quad \Delta f=f_1-f_2,\ \bar f=\frac{f_1+f_2}{2} \]

\[ T_{\rm beat}=\frac{1}{|f_1-f_2|} \]

\[ f'=f\frac{v+v_o}{v-v_s}\quad\text{(source and observer approaching)} \]

\[ f'=f\,\frac{v\pm v_o}{v\mp v_s} \]

\[ f''=f\,\frac{v+u}{v-u} \]

\[ \frac{f'_{\rm app}}{f'_{\rm rec}}=\frac{v+v_s}{v-v_s} \]

\[ T_K=t_C+273.15,\qquad \frac{t_F-32}{9}=\frac{t_C}{5} \]

\[ \frac{X-\text{LFP}_X}{\text{UFP}_X-\text{LFP}_X}=\frac{Y-\text{LFP}_Y}{\text{UFP}_Y-\text{LFP}_Y} \]

\[ T=273.16\ \text{K}\times\frac{P}{P_{\rm tr}} \]

\[ \Delta L=\alpha L_0\Delta T,\quad \Delta A=\beta A_0\Delta T,\quad \Delta V=\gamma V_0\Delta T,\quad \beta\approx2\alpha,\quad\gamma\approx3\alpha \]

\[ \gamma=\alpha_x+\alpha_y+\alpha_z,\qquad \beta_{xy}=\alpha_x+\alpha_y \]

\[ \frac{\Delta I}{I}=2\alpha\Delta T,\qquad \frac{\Delta\omega}{\omega}=-2\alpha\Delta T \]

\[ R\approx\frac{d}{(\alpha_1-\alpha_2)\Delta T} \]

\[ \sigma_{\rm th}=Y\alpha\Delta\theta,\quad F=YA\alpha\Delta\theta,\quad \frac{\Delta T}{T}=\tfrac12\alpha\Delta\theta \]

\[ \Delta t=\tfrac12\,\alpha\,|\Delta\theta|\times86400\ \text{s per day} \]

\[ L_{\rm true}=L_{\rm read}\,[1+\alpha_s(\theta-\theta_0)] \]

\[ L_1\alpha_1=L_2\alpha_2 \]

\[ \gamma_{\rm app}=\gamma_l-\gamma_v,\qquad \rho_T=\frac{\rho_0}{1+\gamma_l\Delta T} \]

\[ \Delta V_{\rm overflow}=V_0(\gamma_l-\gamma_v)\Delta T \]

\[ F_B\approx F_{B0}\,[1+(\gamma_s-\gamma_l)\Delta T] \]

\[ Q=mc\Delta T=C\Delta T,\qquad \sum Q_i=0\quad\text{(isolated mixture)} \]

\[ T_f=\frac{\sum_i m_ic_iT_i}{\sum_i m_ic_i} \]

\[ W=\frac{m\,c}{c_w} \]

\[ Q=\int_{T_1}^{T_2}m\,c(T)\,dT \]

\[ Q_{\rm phase}=mL,\qquad Q_{\rm total}=\sum mc\Delta T+\sum mL \]

\[ \frac{dT}{dt}=\frac{P}{mc},\qquad t_{\rm plateau}=\frac{mL}{P} \]

\[ L_f=3.36\times10^5\ \mathrm{J/kg}=80\ \mathrm{cal/g},\qquad L_v=2.26\times10^6\ \mathrm{J/kg}=540\ \mathrm{cal/g} \]

\[ Q_{\rm avail}=m_wc_w(T_w-0),\qquad Q_{\rm need}=m_ic_i(0-T_i)+m_iL_f \]

\[ m_s\,(L_v+c_w\times100)=m_iL_f \]

\[ m_{\rm melt}=\frac{Q_{\rm avail}-m_ic_i(0-T_i)}{L_f} \]

\[ \dot Q=\frac{\kappa A(T_h-T_c)}{L},\qquad R_{th}=\frac{L}{\kappa A},\qquad R_{\rm series}=\sum R_i \]

\[ \kappa_{\rm series}=\frac{L_1+L_2}{L_1/\kappa_1+L_2/\kappa_2},\qquad \kappa_{\rm parallel}=\frac{\kappa_1A_1+\kappa_2A_2}{A_1+A_2} \]

\[ T_J=\frac{\frac{\kappa_1}{L_1}T_1+\frac{\kappa_2}{L_2}T_2}{\frac{\kappa_1}{L_1}+\frac{\kappa_2}{L_2}} \]

\[ t=\frac{\rho L}{2\kappa\theta}\left(x_2^2-x_1^2\right) \]

\[ P_{\rm net}=\varepsilon\sigma A(T^4-T_s^4),\qquad \lambda_{\max}T=b \]

\[ H=hA\,(T-T_0) \]

\[ E=\varepsilon\sigma T^4,\quad P_{\rm net}=\varepsilon\sigma A\,(T^4-T_0^4),\quad \lambda_mT=b \]

\[ \frac{E_\lambda}{a_\lambda}=E_{\lambda,\rm black},\qquad \varepsilon=a \]

\[ -\frac{dT}{dt}=\frac{\varepsilon\sigma A}{mc}\left(T^4-T_0^4\right) \]

\[ L=4\pi R^2\sigma T^4 \]

\[ \frac{dT}{dt}=-\lambda(T-T_s),\qquad T-T_s=(T_0-T_s)e^{-\lambda t} \]

\[ \frac{T_1-T_2}{t}=K\left[\frac{T_1+T_2}{2}-T_0\right] \]

\[ \ln\frac{T_1-T_0}{T_2-T_0}=kt \]

\[ \frac{m_1c_1+W}{t_1}=\frac{m_2c_2+W}{t_2} \]

\[ pV=nRT=Nk_BT,\qquad N=nN_A,\qquad R=N_Ak_B \]

\[ P=\frac{\rho RT}{M}\quad\Rightarrow\quad \rho=\frac{PM}{RT} \]

\[ n_N=\frac{N}{V}=\frac{P}{k_BT} \]

\[ p=\frac13\rho\langle v^2\rangle=\frac13\rho v_{rms}^2 \]

\[ P=\frac13\,\frac{mN}{V}\,v_{rms}^2=\frac13\,m\,n_N\,v_{rms}^2 \]

\[ PV=\frac23E,\qquad P=\frac23\,\frac{E}{V},\qquad \frac12mv_{rms}^2=\frac32k_BT \]

\[ \langle K_{tr}\rangle=\tfrac32k_BT,\quad v_{rms}=\sqrt{\frac{3RT}{M}},\quad \bar v=\sqrt{\frac{8RT}{\pi M}},\quad v_{mp}=\sqrt{\frac{2RT}{M}} \]

\[ v_{rms}=\sqrt{\frac{3k_BT}{m}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3P}{\rho}} \]

\[ v_{rms}:\bar v:v_{mp}=\sqrt3:\sqrt{\tfrac{8}{\pi}}:\sqrt2\approx1.73:1.60:1.41 \]

\[ v_s=\sqrt{\frac{\gamma RT}{M}}=\sqrt{\frac{\gamma}{3}}\;v_{rms} \]

\[ U=\tfrac f2nRT,\quad C_V=\tfrac f2R,\quad C_P=C_V+R,\quad \gamma=\frac{C_P}{C_V}=\frac{f+2}{f} \]

\[ \bar\varepsilon=\frac f2k_BT\ \text{per molecule},\qquad U=\frac f2nRT,\qquad f=3N-k \]

\[ C_P-C_V=R,\qquad c_p-c_v=\frac{R}{M},\qquad \gamma=\frac{C_P}{C_V}=1+\frac{2}{f} \]

\[ C_V=\frac{R}{\gamma-1},\qquad C_P=\frac{\gamma R}{\gamma-1} \]

\[ \frac{\Delta U}{Q}=\frac1\gamma,\qquad \frac{W}{Q}=1-\frac1\gamma=\frac{2}{f+2} \]

\[ C\approx3R\approx25\ \text{J mol}^{-1}\text{K}^{-1} \]

\[ p_{\rm total}=\sum p_i,\qquad p_i=\frac{n_iRT}{V},\qquad \frac{p_i}{p_{\rm total}}=\frac{n_i}{\sum n_j} \]

\[ M_{mix}=\frac{n_1M_1+n_2M_2}{n_1+n_2}=\frac{m_1+m_2}{\dfrac{m_1}{M_1}+\dfrac{m_2}{M_2}} \]

\[ T=\frac{f_1n_1T_1+f_2n_2T_2}{f_1n_1+f_2n_2} \]

\[ \frac{n_1+n_2}{\gamma_{mix}-1}=\frac{n_1}{\gamma_1-1}+\frac{n_2}{\gamma_2-1} \]

\[ \ell_{mfp}=\frac1{\sqrt2\pi d^2 n_N}=\frac{k_BT}{\sqrt2\pi d^2p} \]

\[ \nu=\frac{\bar v}{\lambda}=\sqrt2\,\pi d^2n_N\bar v,\qquad \tau=\frac1\nu=\frac{\lambda}{\bar v} \]

\[ \Delta U=Q-W,\qquad \Delta U=nC_V\Delta T\quad\text{(ideal gas, constant }C_V) \]

\[ U=\frac{f}{2}nRT,\quad C_V=\frac{f}{2}R,\quad C_P=\left(\frac{f}{2}+1\right)R,\quad \gamma=1+\frac{2}{f} \]

\[ W=\int_{V_i}^{V_f}p_{\rm ext}\,dV,\quad W=p(V_f-V_i)\quad\text{(constant pressure)} \]

\[ W=\tfrac{1}{2}(P_1+P_2)(V_2-V_1) \]

\[ W_{\rm net}=\pm\,(\text{area enclosed on } P\text{–}V)=Q_{\rm net} \]

\[ Q_V=nC_V\Delta T,\quad Q_P=nC_P\Delta T,\quad W_P=nR\Delta T,\quad C_P-C_V=R \]

\[ \frac{\Delta U}{Q}=\frac{1}{\gamma},\qquad \frac{W}{Q}=1-\frac{1}{\gamma} \]

\[ c_P-c_V=\frac{R}{M} \]

\[ W_{\rm rev}=nRT\ln\frac{V_f}{V_i},\quad Q=W_{\rm rev},\quad \Delta U=0\quad\text{(ideal isothermal)} \]

\[ W=nRT\ln\frac{P_1}{P_2}=2.303\,nRT\log_{10}\frac{V_2}{V_1} \]

\[ \left(\frac{dP}{dV}\right)_{T}=-\frac{P}{V},\qquad B_T=P \]

\[ pV^\gamma=\text{constant},\quad TV^{\gamma-1}=\text{constant},\quad W=\frac{p_iV_i-p_fV_f}{\gamma-1},\quad U_{1i}+U_{2i}=U_{1f}+U_{2f}\ \text{(isolated rigid pair)} \]

\[ P^{1-\gamma}T^{\gamma}=\text{constant},\qquad \frac{T_2}{T_1}=\left(\frac{P_2}{P_1}\right)^{(\gamma-1)/\gamma} \]

\[ W=\frac{nR(T_1-T_2)}{\gamma-1}=-\Delta U \]

\[ \frac{(dP/dV)_{\rm adia}}{(dP/dV)_{\rm iso}}=\gamma,\qquad B_S=\gamma P,\quad B_T=P \]

\[ PV^{n}=\text{constant},\quad W=\frac{\mu R(T_1-T_2)}{n-1}=\frac{P_1V_1-P_2V_2}{n-1},\quad C=C_V+\frac{R}{1-n} \]

\[ C=\frac{R}{\gamma-1}-\frac{R}{n-1} \]

\[ C=C_V+\frac{P}{\mu}\frac{dV}{dT} \]

\[ TV^{\,n-1}=\text{constant},\qquad n=1-\frac{1}{k}\ \ \text{for } V\propto T^{k} \]

\[ \Delta U_{\rm cycle}=0,\quad W=Q_h-Q_c,\quad \eta=\frac{W}{Q_h}=1-\frac{Q_c}{Q_h} \]

\[ \frac{Q_2}{Q_1}=\frac{T_2}{T_1},\qquad Q_1=nRT_1\ln\frac{V_B}{V_A} \]

\[ \eta=\frac{W_{\rm net}}{\sum Q_{\rm absorbed}} \]

\[ \eta_C=1-\frac{T_c}{T_h},\qquad COP=\frac{Q_c}{W},\qquad COP_C=\frac{T_c}{T_h-T_c} \]

\[ \beta=\frac{1-\eta}{\eta},\qquad \text{COP}_{\rm hp}=\frac{Q_1}{W}=\beta+1=\frac{T_1}{T_1-T_2} \]

\[ Q_1=Q_2+W \]