OpenStax University Physics/V1/Formulas (master)

File:OpenStax University Physics Volume 1 Formulas.pdf

These formulas print to 4 pages

<section begin=Introduction/>Introduction: 

metric prefixes

da	h	k	M	G	T	P	E	Z	Y
deca	hecto	kilo	mega	giga	tera	peta	exa	zetta	yotta
1E+01	1E+02	1E+03	1E+06	1E+09	1E+12	1E+15	1E+18	1E+21	1E+24
d	c	m	µ	n	p	f	a	z	y
deci	centi	milli	micro	nano	pico	femto	atto	zepto	yocto
1E-01	1E-02	1E-03	1E-06	1E-09	1E-12	1E-15	1E-18	1E-21	1E-24

<section end=Introduction/>

<section begin=Units_and_Measurement/>1. Units_and_Measurement:  The base SI units are mass: kg (kilogram); length: m (meter); time: s (second).

Percent error is $(\delta A/A)\times 100\mathrm{\%}$ <section end=Units_and_Measurement/>

<section begin=Vectors/>

2. Vectors: Vector

$\overrightarrow{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}$

involves components (A_x,A_y,A_z) and unit vectors. ▭ If

$\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}$

, then A_x+B_x=C_x, etc, and vector subtraction is defined by

$\overrightarrow{B}=\overrightarrow{C}-\overrightarrow{A}$

.

▭ The two-dimensional displacement from the origin is $\overrightarrow{r}=x\hat{i}+y\hat{j}$. The magnitude is $A\equiv |\overrightarrow{A}|=\sqrt{A_x^2+A_y^2}$. The angle (phase) is $\theta ={\tan }^{-1}(y/x)$. ▭ Scalar multiplication $\alpha \overrightarrow{A}=\alpha A_x\hat{i}+\alpha A_y\hat{j}+...$ ▭ Any vector divided by its magnitude is a unit vector and has unit magnitude: $|\hat{V}|=1$ where $\hat{V}\equiv \overrightarrow{V}/V$ ▭ Dot product $\overrightarrow{A}\cdot \overrightarrow{B}=AB\cos \theta =A_x{B}_x+A_y{B}_y+...$ and $\overrightarrow{A}\cdot \overrightarrow{A}=A^2$ ▭ Cross product $\overrightarrow{A}=\overrightarrow{B}\times \overrightarrow{C}\Rightarrow$ $A_{\alpha }=B_{\beta }{C}_{\gamma }-C_{\gamma }{A}_{\beta }$ where $(\alpha ,\beta ,\gamma )$ is any cyclic permutation of $(x,y,z)$, i.e., (α,β,γ) represents either (x,y,z) or (y,z,x) or (z,x,y). ▭ Template:Nowrap magnitudes obey $A=BC\sin \theta$ where $\theta$ is the angle between $\overrightarrow{B}$ and $\overrightarrow{C}$, and $\overrightarrow{A}\perp \{\overrightarrow{B},\overrightarrow{C}\}$ by the right hand rule. ▭ Vector identities $c(𝐀+𝐁)=c𝐀+c𝐁$ ▭ $𝐀+𝐁=𝐁+𝐀$ ▭ $𝐀+(𝐁+𝐂)=(𝐀+𝐁)+𝐂$ ▭ $𝐀\cdot 𝐁=𝐁\cdot 𝐀$ ▭ $𝐀\times 𝐁=\mathbf{-}𝐁\times 𝐀$ ▭ $(𝐀+𝐁)\cdot 𝐂=𝐀\cdot 𝐂+𝐁\cdot 𝐂$ <section end=Vectors/>

<section begin=Motion_Along_a_Straight_Line/>3. Motion_Along_a_Straight_Line:  ▭ Average velocity $\overline{v}=\mathrm{\Delta }x/\mathrm{\Delta }t\to v=dx/dt$ (instantaneous velocity) ▭ Acceleration $\overline{a}=\mathrm{\Delta }v/\mathrm{\Delta }t\to a=dv/dt$. ▭ WLOG set $\mathrm{\Delta }t=t$ and $\mathrm{\Delta }x=x-x_0$ if $t_i=0$. Then $\mathrm{\Delta }v=v-v_0$, and $v(t)=\underset{0}{\overset{t}{\int }}a(t^{\prime })d{t}^{\prime }+v_0$, $x(t)=\underset{0}{\overset{t}{\int }}v(t^{\prime })d{t}^{\prime }+x_0=x_0+\overline{v}t$ ▭ At constant acceleration: $\overline{v}=\frac{v_0+v}{2},v=v_0+at,x=x_0+v_{0}t+\frac{1}{2}a{t}^2,$ $v^2=v_0^2+2a\mathrm{\Delta }x$. ▭ For free fall, replace $x\to y$ (positive up) and $a\to -g$, where $g$ = 9.81 m/s² at Earth's surface). <section end=Motion_Along_a_Straight_Line/>

<section begin=Motion_in_Two_and_Three_Dimensions/>4. Motion_in_Two_and_Three_Dimensions:  Instantaneous velocity: $\overrightarrow{v}(t)=v_x(t)\hat{i}+v_y(t)\hat{j}+v_z(t)\hat{k}=\frac{dx}{dt}\hat{i}+\frac{dy}{dt}\hat{j}+\frac{dz}{dt}\hat{k}$ ▭ $\overrightarrow{v}(t)=\underset{\mathrm{\Delta }t\to 0}{\lim }\frac{\mathrm{\Delta }\overrightarrow{r}}{\mathrm{\Delta }t}=\underset{\mathrm{\Delta }t\to 0}{\lim }\frac{\overrightarrow{r}(t+\mathrm{\Delta }t)-\overrightarrow{r}(t)}{\mathrm{\Delta }t}$, where $\overrightarrow{r}(t)=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$ ▭ Acceleration $\overrightarrow{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$, where $a_x(t)=d{v}_x/dt=d^{2}x/d{t}^2$. ▭ Uniform circular motion: position $\overrightarrow{r}(t)$, velocity $\overrightarrow{v}(t)=d\overrightarrow{r}(t)/dt$, and acceleration $\overrightarrow{a}(t)=d\overrightarrow{v}(t)/dt$: $\overrightarrow{r}=A\cos \omega t\hat{i}+A\sin \omega t\hat{j},$ $\overrightarrow{v}=-A\omega \sin \omega t\hat{i}+A\omega \cos \omega t\hat{j},$ $\overrightarrow{a}=-A{\omega }^2\cos \omega t\hat{i}-A{\omega }^2\sin \omega t\hat{j}.$ Note that if $A=r$ then $|\overrightarrow{a}|=a_C={\omega }^{2}r=v^2/r$ where $v\equiv |\overrightarrow{v}|=\omega r$. ▭ Relative motion: ${\overrightarrow{v}}_{PS}={\overrightarrow{v}}_{P{S}^{\prime }}+{\overrightarrow{v}}_{S^{\prime }S}$, <section end=Motion_in_Two_and_Three_Dimensions/>

<section begin=Newton's_Laws_of_Motion/>5. Newton's_Laws_of_Motion:  $m\overrightarrow{a}=d\overrightarrow{p}/dt=\sum {\overrightarrow{F}}_j$, where $\overrightarrow{p}=m\overrightarrow{v}$ is momentum, $\sum {\overrightarrow{F}}_j$ is the sum of all forces This sum needs only include external forces ${\overrightarrow{F}}_{AB}=-{\overrightarrow{F}}_{BA}$.

▭ Weight$=\overrightarrow{w}=m\overrightarrow{g}$. ▭ normal force $|\overrightarrow{N}|=N=mg\cos \theta$ ▭ $F=-kx$ where $k$ is the spring constant. <section end=Newton's_Laws_of_Motion/>

<section begin=Applications_of_Newton's_Laws/>6. Applications_of_Newton's_Laws:  $f_s\le {\mu }_{s}N\text{ and }{f}_k={\mu }_{k}N$: $f=$ friction, ${\mu }_{s,k}=$ coefficient of (static,kinetic) friction, $N=$ normal force. ▭ Centripetal force$F_c=m{v}^2/r=mr{\omega }^2$ for uniform circular motion. Angular velocity $\omega$ is measured in radians per second. ▭ Drag equation $F_D=\frac{1}{2}C\rho A{v}^2$ where $C=$ Drag coefficient, $\rho =$ mass density, $A=$ area, $v=$ speed. Holds approximately for large Reynold's number <section end=Applications_of_Newton's_Laws/>

<section begin=Work_and_Kinetic_Energy/>7. Work_and_Kinetic_Energy:  Infinitesimal work $dW=\overrightarrow{F}\cdot d\overrightarrow{r}=|\overrightarrow{F}||d\overrightarrow{r}|\cos \theta$ leads to the path integral $W_{AB}=\underset{A}{\overset{B}{\int }}\overrightarrow{F}\cdot d\overrightarrow{r}$ ▭ Work done from A→B by friction $-f_k|{\ell }_{AB}|,$gravity $-mg(y_B-y_A),$ and spring $-\frac{1}{2}k(x_B^2-x_A^2)$ ▭ Work-energy theorem: $W_{net}=K_B-K_A$ where kinetic energy $=K=\frac{1}{2}m{v}^2=\frac{p^2}{2m}$. ▭ Power$=P=dW/dt=\overrightarrow{F}\cdot \overrightarrow{v}$. <section end=Work_and_Kinetic_Energy/>

<section begin=Potential_Energy_and_Conservation_of_Energy/>8. Potential_Energy_and_Conservation_of_Energy:  Potential Energy: $\mathrm{\Delta }{U}_{AB}=U_B-U_A=-W_{AB}$; PE at $\overrightarrow{r}$ WRT ${\overrightarrow{r}}_0$ is $\mathrm{\Delta }U=U(\overrightarrow{r})-U({\overrightarrow{r}}_0)$ $U=mgy+𝒞$ (gravitational PE Earth's surface. $U=\frac{1}{2}k{x}^2+𝒞$ (ideal spring) ▭ Conservative force: $\oint {\overrightarrow{F}}_{\text{cons}}\cdot d\overrightarrow{r}=0$. In 2D, $\overrightarrow{F}(x,y)$ is conservative if and only if $\overrightarrow{F}=-(\partial U/\partial x)\hat{i}-(\partial U/\partial y)\hat{j}⟺\partial F_x/\partial y=\partial F_y/\partial x$ ▭ Mechanical energy is conserved if no non-conservative forces are present: $0=W_{nc,AB}=\mathrm{\Delta }(K+U{)}_{AB}=\mathrm{\Delta }{E}_{AB}$ <section end=Potential_Energy_and_Conservation_of_Energy/>

<section begin=Linear_Momentum_and_Collisions/>9. Linear_Momentum_and_Collisions:  $\overrightarrow{F}(t)=d\overrightarrow{p}/dt\text{, where }\overrightarrow{p}=m\overrightarrow{v}$ is momentum. ▭ Impulse-momentum theorem $\overrightarrow{J}=F_{ave}\mathrm{\Delta }t=\underset{t_i}{\overset{t_f}{\int }}\overrightarrow{F}dt=\mathrm{\Delta }\overrightarrow{p}$. ▭ For 2 particles in 2D $\text{If }{\overrightarrow{F}}_{ext}=0\text{ then }\sum _{j=1}^N{\overrightarrow{p}}_j=0\Rightarrow p_{f,\alpha }=p_{1,i,\alpha }+p_{2,i,\alpha }$ where (α,β)=(x,y) ▭ Center of mass: ${\overrightarrow{r}}_{CM}=\frac{1}{M}\sum _{j=1}^N{m}_j{\overrightarrow{r}}_j\to \frac{1}{M}\int \overrightarrow{r}dm,$ ${\overrightarrow{v}}_{CM}=\frac{d}{dt}{\overrightarrow{r}}_{CM}$, and ${\overrightarrow{p}}_{CM}=\sum _{j=1}^N{m}_j{\overrightarrow{v}}_j=M{\overrightarrow{v}}_{CM}.$ ▭ $\overrightarrow{F}=\frac{d}{dt}{\overrightarrow{p}}_{CM}=m{\overrightarrow{a}}_{CM}=\sum _{j=1}^N{m}_j{\overrightarrow{a}}_j$ <section end=Linear_Momentum_and_Collisions/>

<section begin=Fixed-Axis_Rotation/>10. Fixed-Axis_Rotation: 

$\theta =s/r$ is angle in radians,$\omega =d\theta /dt$ is angular velocity; ▭ $v_t=\omega r=ds/dt$ is tangential speed. Angular acceleration is $\alpha =d\omega /dt=d^2\theta /d{t}^2$. $a_t=\alpha r=d^{2}s/d{t}^2$ is the tangential acceleration. ▭ Constant angular acceleration $\overline{\omega }=\frac{1}{2}({\omega }_0+{\omega }_f)$ is average angular velocity. ▭ ${\theta }_f={\theta }_0+\overline{\omega }t={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2.$ ▭ ${\omega }_f={\omega }_0+\alpha t.$ ${\omega }_f^2={\omega }_0^2+2\alpha \mathrm{\Delta }\theta .$ ▭ Total acceleration is centripetal plus tangential: $\overrightarrow{a}={\overrightarrow{a}}_c+{\overrightarrow{a}}_t.$ ▭ Rotational kinetic energy is $K=\frac{1}{2}I{\omega }^2,$ where $I=\sum _j{m}_j{r}_j^2\to \int r^{2}dm$ is the Moment of inertia. ▭ parallel axis theorem $I_{parallel-axis}=I_{centerofmass}+m{d}^2$ ▭ Restricting ourselves to fixed axis rotation, $r$ is the distance from a fixed axis; the sum of torques, $\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$ requires only one component, summed as ${\tau }_{net}=\sum {\tau }_j=\sum r_{{\perp }_j}{F}_j=I\alpha$. ▭ Work done by a torque is $dW=(\sum {\tau }_j)d\theta$. The Work-energy theorem is $K_B-K_A=W_{AB}=\underset{{\theta }_A}{\overset{{\theta }_B}{\int }}\left(\sum _j{\tau }_j\right)d\theta$. ▭ Rotational power $=P=\tau \omega$. <section end=Fixed-Axis_Rotation/>

<section begin=Angular_Momentum/>11. Angular_Momentum:  Center of mass (rolling without slip) $d_{CM}=r\theta ,$ $v_{CM}=r\omega ,$$a_{MC}=R\alpha =\frac{mg\sin \theta /}{m+(I_{cm}/r^2)}$ ▭ Total angular momentum and net torque: $d\overrightarrow{L}/dt=\sum \overrightarrow{\tau }$ $={\overrightarrow{l}}_1+{\overrightarrow{l}}_2+...;$ $\overrightarrow{l}=\overrightarrow{r}\times \overrightarrow{p}$ for a single particle. $L_{total}=I\omega .$ ▭ Precession of a top ${\omega }_P=mrg/(I\omega ).$<section end=Angular_Momentum/>

<section begin=Static_Equilibrium_and_Elasticity/>12. Static_Equilibrium_and_Elasticity: Equilibrium $\sum {\overrightarrow{F}}_j=0=\sum {\overrightarrow{\tau }}_j.$ Stress = elastic modulus · strain (analogous to Force = k · Δ x ) ▭ (Young's , Bulk , Shear) modulus: $(\frac{F_{\perp }}{A}=Y\cdot \frac{\mathrm{\Delta }L}{L_0},\mathrm{\Delta }p=B\cdot \frac{-\mathrm{\Delta }V}{V_0},\frac{F_{\parallel }}{A}=S\cdot \frac{\mathrm{\Delta }x}{L_0})$ <section end=Static_Equilibrium_and_Elasticity/>

<section begin=Gravitation/>13. Gravitation:  Newton's law of gravity ${\overrightarrow{F}}_{12}=G\frac{m_1{m}_2}{r^2}{\hat{r}}_{12}$ ▭ Earth's gravity $g=G\frac{M_E}{r^2}$ ▭ Gravitational PE beyond Earth $U=-G\frac{M_{E}m}{r}$ ▭ Energy conservation $\frac{1}{2}m{v}_1^2-G\frac{Mm}{r_1}=\frac{1}{2}m{v}_2^2-G\frac{Mm}{r_2}$ ▭ Escape velocity $v_{esc}=\sqrt{\frac{2G{M}_E}{r}}$ ▭ Orbital speed $v_{orbit}=\sqrt{\frac{G{M}_E}{r}}$ ▭ Orbital period $T=2\pi \sqrt{\frac{r^3}{G{M}_E}}$ ▭ Energy in circular orbit $E=K+U=-\frac{Gm{M}_E}{2r}$ ▭ Conic section $\frac{\alpha }{r}=1+e\cos \theta$ ▭ Kepler's third law$T^2=\frac{4{\pi }^2}{GM}{a}^3$ ▭ Schwarzschild radius $R_S=\frac{2GM}{c^2}$ <section end=Gravitation/>

<section begin=Fluid_Mechanics/>14. Fluid_Mechanics:  Mass density $\rho =m/V$ ▭ Pressure $P=F/A$ ▭ Pressure vs depth/height (constant density)$p=p_o+\rho gh\Leftarrow dp/dy=-\rho g$ ▭ Absolute vs gauge pressure $p_{abs}=p_g+p_{atm}$ ▭ Pascal's principle: $F/A$ depends only on depth, not on orientation of A. ▭ Volume flow rate $Q=dV/dt$ ▭ Continuity equation ${\rho }_1{A}_1{v}_1={\rho }_2{A}_2{v}_2$$\Rightarrow A_1{v}_1=A_2{v}_2\text{ if }\rho =const.$ <section end=Fluid_Mechanics/>

<section begin=Oscillations/>15. Oscillations:  Frequency $f$, period $T$ and angular frequency $\omega :$ $fT=1,\omega T=2\pi$ ▭ Simple harmonic motion $x(t)=A\cos (\omega t+\varphi ),$ $v(t)=-A\omega \sin (\omega t+\varphi ),$ $a(t)=-A{\omega }^2\cos (\omega t+\varphi )$ also models the x-component of uniform circular motion. ▭ For $(A,\omega )$ positive: $x_{max}=A,v_{max}=A\omega ,a_{max}=A{\omega }^2$ ▭ Mass-spring $\omega =2\pi /T=2\pi f=\sqrt{k/m};$ ▭ Energy $E_{Tot}=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2=\frac{1}{2}k{A}^2\Rightarrow$$v=\pm \sqrt{\frac{k}{m}(A^2-x^2)}$ ▭ Simple pendulum $\omega \approx \sqrt{g/L}$ ▭ Physical pendulum $\tau =-MgL\sin \theta \approx -MgL\theta \Rightarrow$$\omega =\sqrt{mgL/I}$ and $L$ measures from pivot to CM. ▭ Torsional pendulum $\tau =-\kappa \theta$$\Rightarrow \omega =\sqrt{I/\kappa }$ ▭ Damped harmonic oscillator $m\frac{d^{2}x}{d{t}^2}=-kx-b\frac{dx}{dt}$$\Rightarrow x=A_0{e}^{\frac{b}{2m}t}\cos (\omega t+\varphi )$ where $\omega =\sqrt{{\omega }_0^2-{\left(\frac{b}{2m}\right)}^2}$ and ${\omega }_0=\sqrt{\frac{k}{m}}.$ ▭ Forced harmonic oscillator (MIT wiki!)] $m\frac{d^{2}x}{d{t}^2}=-kx-b\frac{dx}{dt}+F_0\sin \omega t$$\Rightarrow x=A{e}^{\frac{b}{2m}t}\cos (\omega t+\varphi )$ where $A=\frac{F_0}{\sqrt{m^2(\omega -{\omega }_0{)}^2+b^2{\omega }^2}}$. <section end=Oscillations/>

<section begin=Waves/>16. Waves:  Wave speed] (phase velocity) $v=\lambda /T=\lambda f=\omega /k$ where $k=2\pi /\lambda$ is wavenumber. ▭ Wave and pulse speed of a stretched string $=\sqrt{F_T/\mu }$ where $F_T$ is tension and $\mu$ is linear mass density. ▭ Speed of a compression wave in a fluid $v=\sqrt{B/\rho }.$ ▭ Periodic travelling wave $y(x,t)=A\sin (kx\mp \omega t)$ travels in the positive/negative direction. The phase is $kx\mp \omega t$ and the amplitude is $A$. ▭ The resultant of two waves with identical amplitude and frequency $y_R(x,t)=[2A\cos \left(\frac{\varphi }{2}\right)]\sin (kx-\omega t+\frac{\varphi }{2})$ where $\varphi$ is the phase shift. ▭ This wave equation ${\partial }^{2}y/\partial t^2=v_w^2{\partial }^{2}y/\partial x^2$ is linear in $y=y(x,t)$ ▭ Power in a tranverse stretched string wave $P_{ave}=\frac{1}{2}\mu A^2{\omega }^{2}v$. ▭ Intensity of a plane wave $I=P/A\Rightarrow \frac{P}{4\pi r^2}$ in a spherical wave. ▭ Standing wave $y(x,t)=A\sin (kx)\cos (\omega t+\varphi )$ For symmetric boundary conditions ${\lambda }_n=2\pi /k_n=\frac{2}{\pi }L$ $n=1,2,3,...$, or equivalently $f=n{f}_1$ where $f_1=\frac{v}{2L}$ is the fundamental frequency. <section end=Waves/>

<section begin=Sound/>17. Sound:  Pressure and displacement fluctuations in a sound wave $P=\mathrm{\Delta }{P}_{max}\sin (kx\mp \omega t+\varphi )$ and $s=s_{max}\cos (kx\mp \omega t+\varphi )$ ▭ Speed of sound in a fluid $v=f\lambda =\sqrt{\beta /\rho }$, ▭ in a solid $\sqrt{Y/\rho }$, ▭ in an idal gas $\sqrt{\gamma RT/M}$, ▭ in air $331\frac{m}{s}\sqrt{\frac{T_K}{273K}}=331\frac{m}{s}\sqrt{1+\frac{T_C}{273^{o}C}}$ ▭ Decreasing intensity spherical wave $I_2=I_1{\left(\frac{r_1}{r_2}\right)}^2$ ▭ Sound intensity $I=\frac{⟨P⟩}{A}=\frac{{\left(\mathrm{\Delta }{P}_{max}\right)}^2}{2\rho v}$ ▭  ...level $10{\log }_{10}I/I_0$ ▭ Resonance tube One end closed: ${\lambda }_n=\frac{4}{n}L,$ $f_n=n\frac{v}{4L},$ $n=1,3,5,...$ ▭ Both ends open: ${\lambda }_n=\frac{2}{n}L,$ $f_n=n\frac{v}{2L},$ $n=1,2,3,...$ ▭ Beat frequency $f_{beat}=|f_2-f_1|$ ▭ (nonrelativistic) Doppler effect $f_O=f_s\frac{v\pm v_o}{v\mp v_s}$ where $v$ is the speed of sound, $v_s$ is the velocity of the source, and $v_o$ is the velocity of the observer. ▭ Angle of shock wave $\sin \theta =v/v_s=1/M$ where $v$ is the speed of sound, $v_s$ is the speed of the source, and $M$ is the Mach number.

I=∫r²dm for a hoop, disk, cylinder, box, plate, rod, and spherical shell or solid can be found from this figure.

<section end=Sound/>