View source for OpenStax University Physics/V1/Formulas (master)

[[File:OpenStax University Physics Volume 1 Formulas.pdf|right|thumb|These formulas print to 4 pages]]

<section begin=Introduction/>'''Introduction''':&nbsp;
{| class="wikitable" style="font-size:85%; text-align:center"
|+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
|-
| colspan="10" style="text-align: left; background: white; border=0px;" |  
|-
| 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.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Units_and_Measurement|Units_and_Measurement]]''':&nbsp;
The base SI units are mass: kg (kilogram); length: m (meter); time: s (second). <ref>[http://wiki.ubc.ca/index.php?title=Uncertainty_and_Error&oldid=81540 </ref>

Percent error is <math>(\delta A/A)\times 100%</math>
<section end=Units_and_Measurement/>

<section begin=Vectors/>[[File:Vector components with unit vectors.svg|160px|right]]'''2.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Vectors|Vectors]]''':&nbsp;[[w:special:permalink/811547880|Vector]] <math>\vec A = A_x \, \hat i+ A_y \, \hat j + A_z \, \hat k</math> involves [[w:Special:Permalink/816420484#Mathematics|components]] (A<sub>x</sub>,A<sub>y</sub>,A<sub>z</sub>) and <ref>three [[w:Special:Permalink/801387282|orthonormal]]</ref> [[w:Special:Permalink/805505822|unit vector]]s.<ref>[[File:Vectors A+B=C.png|140px|right]]</ref> &#9645;&nbsp;If <math>\vec A + \vec B =\vec C</math>, then A<sub>x</sub>+B<sub>x</sub>=C<sub>x</sub>, [[w:special:permalink/805356678|etc]], and [[w:special:permalink/816754783|vector subtraction]] is defined by <math>\vec B =\vec C - \vec A</math>.
&#9645;&nbsp;The&nbsp;two-dimensional [[w:Special:Permalink/816090976|displacement]]  from the origin is <math>\vec r = x\hat i + y\hat j</math>. The [[w:Special:Permalink/811018767#Vector_spaces|magnitude]] is <math>
A \equiv |\vec A|= \sqrt{A_x^2 + A_y^2}</math>. The [[w:Special:Permalink/817363346|angle]] ([[w:Special:Permalink/812844972|phase]]) is <math>\theta=\tan^{-1}{(y/x)} </math>.  
&#9645;&nbsp;[[w:Scalar multiplication|Scalar multiplication]] <math>\alpha\vec A = \alpha A_x \hat i 
     +  \alpha A_y \hat j+...\quad</math>
&#9645;&nbsp;Any&nbsp;vector divided by its [[w:special:permalink/812942205|magnitude]] is a [[w:special:permalink/805505822|unit&nbsp;vector]] and has [[w:special:permalink/817679405|unit]] magnitude: <math>|\hat V| = 1</math> where <math>\hat V \equiv \vec V/V</math>
&#9645;&nbsp;[[w:Dot product|Dot product]] <math>\vec A\cdot\vec B = AB\cos\theta =
     A_xB_x+A_yB_y+...\quad</math> and <math>\vec A\cdot\vec A = A^2</math>
&#9645;&nbsp;[[Cross product|Cross product]]  <math>\vec A = \vec B \times \vec C\Rightarrow</math> <math>A_\alpha = B_\beta C_\gamma -C_\gamma A_\beta</math> where <math>(\alpha,\beta,\gamma)</math> is any [[w:Special:Permalink/814107526#Vector_cross_product|cyclic permutation]] of <math>(x,y,z)</math>, i.e., (&alpha;,&beta;,&gamma;) represents either (x,y,z) or (y,z,x) or (z,x,y).
&#9645;&nbsp;{{nowrap|Cross-product}}  magnitudes obey <math>A=BC\sin\theta</math>  where <math>\theta</math> is the angle between <math>\vec B</math> and <math>\vec C</math>,  and <math>\vec A\perp\{\vec B,\vec C\}</math> by the [[w:Special:Permalink/813065999|right hand rule]]. 
&#9645;&nbsp;[[w:Special:Permalink/803981161|Vector identities]] <math> \;c (\mathbf{A}+\mathbf{B})=c\mathbf{A}+c\mathbf{B}\quad</math>
&#9645;&nbsp;<math> \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}\quad</math>
&#9645;&nbsp;<math> \mathbf{A}+(\mathbf{B}+\mathbf{C})=
    (\mathbf{A}+\mathbf{B})+\mathbf{C}\quad</math>
&#9645;&nbsp;<math> \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}\quad</math>
&#9645;&nbsp;<math> \mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A}\quad</math>
&#9645;&nbsp;<math> \left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C}
    +\mathbf{B}\cdot\mathbf{C} \quad</math>
<ref>
&#9645;&nbsp;<math> \left(\mathbf{A}+\mathbf{B}\right)\times\mathbf{C}=\mathbf{A}\times\mathbf{C}
   +\mathbf{B}\times\mathbf{C} \quad</math>  
&#9645;&nbsp;<math>\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=  \mathbf{B}\cdot\left(\mathbf{C}\times\mathbf{A}\right)=\left(\mathbf{A}\times\mathbf{B}\right)\cdot\mathbf{C}\quad</math>&#9645;&nbsp;<math> \mathbf{A\times} \left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)   \mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}\quad</math>  &#9645;&nbsp;<math> \mathbf{\left(A\times B\right)\cdot}\left(\mathbf{C} \times\mathbf{D}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\left(\mathbf{B}\cdot\mathbf{D}\right)-      \left(\mathbf{B}\cdot\mathbf{C}\right)\left(\mathbf{A}\cdot\mathbf{D}\right)</math> </ref>
<section end=Vectors/>

<section begin=Motion_Along_a_Straight_Line/>'''3.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Motion_Along_a_Straight_Line|Motion_Along_a_Straight_Line]]''':&nbsp;
<ref>Delta as difference  <math>\Delta x =x_f-x_i \rightarrow dx\rightarrow 0</math> in limit of differential calculus.</ref>
&#9645;&nbsp;Average velocity <math>\bar v = \Delta x /\Delta t 
  \rightarrow v=dx/dt</math> (instantaneous velocity)  
&#9645;&nbsp;Acceleration <math>\bar a = \Delta v/\Delta t \rightarrow a=dv/dt</math>.
&#9645;&nbsp;[[w:special:permalink/804419966|WLOG]]&nbsp;set <math>\Delta t=t\;</math> and <math>\Delta x = x-x_0\;</math> if <math>t_i=0</math>. Then <math>\Delta v = v-v_0</math>, and   
<math>\; v(t) = \int_0^t a(t')dt' + v_0 </math>, <math>\;x(t) = \int_0^t v(t')dt' + x_0 = x_0 + \bar v t </math><ref>, where <math>\bar v = \frac 1 t \int_0^t v(t')dt'</math> is the [[w:special:permalink/803365903#Average_velocity|average velocity]]. </ref>
&#9645;&nbsp;At [[w:special:permalink/817212138#Uniform_acceleration|constant acceleration:]]
<math>\bar v = \tfrac{v_0+v}{2},\quad v=v_0+at,\quad x=x_0+ v_0t+\tfrac 1 2 at^2,\,</math> <math> v^2=v_0^2+2a\Delta x</math>.
&#9645;&nbsp;For [[w:special:permalink/816661718|free fall]], replace <math>x\rightarrow y</math> (positive up) and <math>a\rightarrow -g</math>, where <math>g</math> = 9.81 m/s<sup>2</sup> at [[w:special:permalink/813543013|Earth's surface]]).
<section end=Motion_Along_a_Straight_Line/>

<section begin=Motion_in_Two_and_Three_Dimensions/>'''4.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Motion_in_Two_and_Three_Dimensions|Motion_in_Two_and_Three_Dimensions]]''':&nbsp;
[[w:special:permalink/803365903#Instantaneous_velocity|Instantaneous&nbsp;velocity]]:  <math>\vec 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</math>
&#9645;&nbsp;<math>\vec v(t)= \lim_{\Delta t\rightarrow 0}{\tfrac{\Delta \vec r}{\Delta t}}
  =\lim_{\Delta t\rightarrow 0}{\tfrac{\vec r(t+\Delta t)-\vec r(t)}{\Delta t}}</math>, where <math>\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k</math>
&#9645;&nbsp;[[w:special:permalink/817212138|Acceleration]]  <math>\vec a = a_x\hat i + a_y\hat j + a_z\hat k</math>, where <math>a_x(t)=dv_x/dt=d^2x/dt^2</math>.
<ref>
&#9645;&nbsp;[[wikt:special:permalink/41993662|Average values]]:
<math>\vec v_{ave} = \tfrac{\Delta\vec r}{\Delta t}=\tfrac{\vec r(t_2)-\vec r(t_2)}{t_2-t_1}
   </math>, and <math>\;\vec a_{ave} = \tfrac{\Delta\vec v}{\Delta t}=
      \tfrac{\vec v(t_2)-\vec v(t_2)}{t_2-t_1}</math>
&#9645;&nbsp;[[w:special:permalink/816661718|Free&nbsp;fall]] [[w:special:permalink/811884285|time of flight]] <math>\,T_{of}=
  \tfrac{2(v_0\sin\theta_0)}{g}\,,\,</math> 
&#9645;&nbsp;[[w:special:permalink/809039681|Trajectory]] <math>   y=(\tan\theta_0)x - \left[\tfrac{g}{2(v_0\cos\theta_0)^2} \right]x^2\,,\,</math> &#9645;&nbsp;[[w:special:permalink/812780336|Range]] <math> R=\tfrac{v_0^2\sin2\theta_0}{g}</math></ref>
&#9645;&nbsp;[[w:special:permalink/815368658#Uniform_circular_motion|Uniform circular motion]]: position <math>\,\vec r(t)</math>, 
velocity <math>\,\vec v(t)=d\vec r(t)/dt</math>, 
and acceleration <math>\,\vec a(t)=d\vec v(t)/dt</math>: 
<math>\vec r = A\cos\omega t\hat i + A\sin\omega t\hat j \,,\,</math> <math>
\vec v = -A\omega\sin\omega t\hat i + A\omega\cos\omega t\hat j \,,\,</math> <math>
\vec a = -A\omega^2\cos\omega t\hat i - A\omega^2\sin\omega t\hat j\,. </math> Note that if <math>A=r</math> then <math>|\vec a|=a_C=\omega^2r = v^2/r</math> where <math>v\equiv|\vec v|=\omega r</math>.
<ref>&#9645;&nbsp;[[w:Special:Permalink/816722116#Tangential_and_centripetal_acceleration|Tangential and centripetal acceleration]] <math>\vec a = \vec a_c + \vec a_T</math> where <math>a_T=d|\vec v|/dt</math>.</ref>
&#9645;&nbsp;[[w:Special:Permalink/813700835|Relative motion]]:  
<ref><math>\,\vec r_{PS}=\vec r_{PS'}+\vec r_{S'S}</math>, </ref>
<math>\,\vec v_{PS}=\vec v_{PS'}+\vec v_{S'S}</math>, 
<ref><math>\,\vec v_{PC}=\vec v_{PA}+\vec v_{AB}+\vec v_{BC}</math>, <math>\,\vec a_{PS}=\vec a_{PS'}+\vec a_{S'S}</math></ref>
<section end=Motion_in_Two_and_Three_Dimensions/>

<section begin=Newton's_Laws_of_Motion/>'''5.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Newton's_Laws_of_Motion|Newton's_Laws_of_Motion]]''':&nbsp;
<ref>Newton's  2<sup>nd</sup> [[w:Newton's laws of motion|Law]] </ref><math>\;m\vec a = d\vec p/dt =\sum \vec F_j</math>, where <math>\vec p =m\vec v</math> is [[w:momentum|momentum]], <ref><math>m</math> is [[w:mass|mass]], and </ref> <math>\sum \vec F_j</math> is the sum of all [[w:force|force]]s  This sum  needs only include external forces <ref>because all internal forces cancel by the 3<sup>rd</sup> law  </ref><math>\vec F_{AB}=-\vec F_{BA}</math>.<ref>  The 1<sup>st</sup> law is that velocity is constant if the net force is zero.</ref>
[[File:Free body1.3.svg|right|150px]]
&#9645;&nbsp;[[w:Weight|Weight]]<math>=\vec w=m\vec g</math>. 
&#9645;&nbsp;[[w:normal force|normal force]]<ref> is a [[w:Vector projection|component]] of the [[w:contact force|contact force]] by the surface. If the only forces are contact and weight,</ref> <math>|\vec N|=N=mg\cos\theta</math><ref> where <math>\theta</math> is the angle of [[w:inclined plane|incline]].  </ref>
&#9645;&nbsp;<ref>[[w:Hooke's law|Hooke's law]] </ref><math>F=-kx</math> where <math>k</math> is the spring constant.
<section end=Newton's_Laws_of_Motion/>

<section begin=Applications_of_Newton's_Laws/>'''6.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Applications_of_Newton's_Laws|Applications_of_Newton's_Laws]]''':&nbsp;
<math>f_s\le\mu_sN \text{ and } f_k=\mu_kN</math>: <math>\,f=</math> [[w:special:permalink/809506463|friction]], <math>\mu_{s,k}=</math> [[w:Special:Permalink/809506463#Coefficient_of_friction|coefficient]] of  ([[w:Special:Permalink/809506463#Static_friction|static]],[[w:Special:Permalink/809506463#Kinetic_friction|kinetic]]) friction, <math>N=</math> [[w:special:permalink/807513920|normal force]].
&#9645;&nbsp;[[w:special:permalink/816185814|Centripetal force]]<math>F_c=m v^2/r=mr\omega^2</math> for [[w:special:permalink/815368658#Uniform_circular_motion|uniform circular motion]]. [[w:special:permalink/816138492|Angular velocity]] <math>\omega</math> is measured in [[w:special:permalink/813819117|radian]]s per second.
<ref>&#9645;&nbsp;Ideal angle of [[w:special:permalink/816470254|banked curve]]:  <math>\tan\theta=v^2/(rg)</math> for curve of radius <math>r</math> banked at angle <math>\theta</math>.</ref>&#9645;&nbsp;[[w:special:permalink/816431002|Drag equation]] <math>F_D=\tfrac 1 2 C \rho A v^2</math> where <math>C=</math>  [[w:special:permalink/815514267|Drag coefficient]], <math>\rho=</math> [[w:special:permalink/815142720|mass density]], <math>A=</math> area, <math>v=</math> speed. [[w:special:permalink/815636949#Sphere_in_a_fluid|Holds approximately]] for large [[w:special:permalink/815636949|Reynold's number]]<ref> <math>=\mathrm{Re} = \rho v L/ \eta</math>, where <math>\eta =</math>[[w:special:permalink/817872541#Dynamic_viscosity,_%CE%BC|dynamic viscosity]]; <math>L=</math> characteristic length.  &#9645;&nbsp;[[w:special:permalink/816664617|Stokes's law]] models a sphere of radius <math>r</math> at small Reynold's number: <math>F_s=6\pi r\eta v</math>.</ref>
<section end=Applications_of_Newton's_Laws/>

<section begin=Work_and_Kinetic_Energy/>'''7.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Work_and_Kinetic_Energy|Work_and_Kinetic_Energy]]''':&nbsp;
[[w:special:permalink/813870961|Infinitesimal]] [[w:special:permalink/813216443|work]]<ref> done by force: </ref> <math>
dW=\vec F\cdot d\vec r = |\vec F|\,|d\vec r|\cos\theta</math> leads to the
[[w:special:permalink/816232070|path integral]] <math>W_{AB}=\int_A^B\vec F\cdot d\vec r</math>
&#9645;&nbsp;Work done from A&rarr;B by friction <math>-f_k|\ell_{AB}|,\;</math>gravity <math>-mg(y_B-y_A),\;</math> and [[w:special:permalink/812496433|spring]] <math>-\tfrac 1 2 k\left(x_B^2-x_A^2\right)</math>
&#9645;&nbsp;[[w:simple:special:permalink/5779584|Work-energy theorem]]:  <ref>The work done on a particle is</ref> <math>W_{net}=K_B-K_A</math> where [[w:special:permalink/815290642|kinetic energy]] <math>=K=\tfrac 1 2 mv^2 = \frac{p^2}{2m}</math>.
&#9645;&nbsp;[[w:special:permalink/813752583|Power]]<math>=P=dW/dt=\vec F\cdot \vec v</math>.
<section end=Work_and_Kinetic_Energy/>

<section begin=Potential_Energy_and_Conservation_of_Energy/>'''8.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Potential_Energy_and_Conservation_of_Energy|Potential_Energy_and_Conservation_of_Energy]]''':&nbsp;
[[w:Potential Energy|Potential Energy]]:  <math>\Delta U_{AB} = U_B-U_A=-W_{AB}</math>; [[w:Potential Energy|PE]] at <math>\vec r</math> [[wikt:special:permalink/47006216|WRT]]  <math>\vec r_0</math> is <math>\Delta U = U(\vec r)-U(\vec r_0)</math>
<math>U=mgy+\mathcal{C}</math> ([[w:special:permalink/813543013|gravitational PE Earth's surface]].  <math>U=\tfrac 1 2 k x^2+\mathcal{C}</math> (ideal [[w:special:permalink/812496433|spring]])
&#9645;&nbsp;[[w:special:permalink/817399341|Conservative force]]:  <math>\oint\vec F_{\text{cons}}\cdot d\vec r = 0</math>. In [[w:Two-dimensional space|2D]], <math>\vec F(x,y)</math> is conservative [[w:special:permalink/810680090|if and only if]] <math>
\vec F = -(\partial U/\partial x) \,\hat i  -(\partial U/\partial y) \,\hat j \iff
\partial F_x/\partial y = \partial F_y/\partial x</math>
&#9645;&nbsp;[[w:special:permalink/815049073|Mechanical energy]] is conserved if no non-conservative forces are present: 
<math>0=W_{nc,AB}=\Delta (K+U)_{AB} =\Delta E_{AB}</math>
<section end=Potential_Energy_and_Conservation_of_Energy/>

<section begin=Linear_Momentum_and_Collisions/>'''9.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Linear_Momentum_and_Collisions|Linear_Momentum_and_Collisions]]''':&nbsp;
<math>\vec F(t) = d\vec p/dt \text{, where } \vec p=m\vec v</math> is [[w:special:permalink/815223684|momentum]].
&#9645;&nbsp;[[w:special:permalink/805705989|Impulse-momentum]] theorem <math>\vec J  = F_{ave}\Delta t = \int_{t_i}^{t_f} \vec F dt = \Delta\vec p</math>.
&#9645;&nbsp;For&nbsp;2&nbsp;particles [[w:special:permalink/817819796|in 2D]]
<math>\text{If } \vec F_{ext}=0 \text{ then } \sum_{j=1}^N\vec p_j =0 \Rightarrow p_{f,\alpha}=p_{1,i,\alpha}+p_{2,i,\alpha}</math>  where (&alpha;,&beta;)=(x,y)
&#9645;&nbsp;[[w:special:permalink/817301310|Center&nbsp;of&nbsp;mass]]: <math>\vec r_{CM}=
  \tfrac 1 M \sum_{j=1}^N m_j\vec r_j \rightarrow \tfrac 1 M \int\vec r dm, </math> <math>\vec v_{CM}= \tfrac{\,d}{dt}\vec r_{CM}</math>, and <math>\vec p_{CM}=\sum_{j=1}^N m_j\vec v_j =M\vec v_{CM}.</math>
&#9645;&nbsp;<math>\vec F = \tfrac{\,d}{dt}\vec p_{CM}=m\vec a_{CM}=\sum_{j=1}^Nm_j\vec a_j</math>
<ref>&#9645;&nbsp;[[w:special:permalink/809804270|Rocket&nbsp;equation]] <math>mdv=-udm\Rightarrow \Delta v = u\ln(m_f/m_i)</math> where u is the gas speed [[wikt:special:permalink/47006216|WRT]] the rocket.</ref>
<section end=Linear_Momentum_and_Collisions/>

<section begin=Fixed-Axis_Rotation/>'''10.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Fixed-Axis_Rotation|Fixed-Axis_Rotation]]''':&nbsp;
[[File:Theta in radians minimalistic.svg|95px|right]][[File:Torque lever arm w point of application and line of action.svg|right|175px]][[File:Parallel axis theorem.svg|right|150px]]
<math>\theta=s/r\,</math> is angle in [[w:special:permalink/813819117|radian]]s,<math>\,\omega=d\theta/dt\,</math> is [[w:special:permalink/816138492|angular velocity]];
&#9645;&nbsp;<math>\,v_t=\omega r=ds/dt\,</math> is [[w:special:permalink/817347057#Tangential_speed|tangential speed]]. [[w:special:permalink/812691617|Angular acceleration]] is <math>\alpha=d\omega/dt=d^2\theta/dt^2\,</math>. <math>a_t=\alpha r= d^2s/dt^2\,</math> is the tangential acceleration.
&#9645;&nbsp;[[special:permalink/1794295|Constant angular acceleration]] <math>\bar\omega=\tfrac 1 2 (\omega_0+\omega_f)\,</math> is average angular velocity. 
&#9645;&nbsp;<math>\theta_f=\theta_0+\bar\omega t = \theta_0+\omega_0t+\tfrac 1 2 \alpha t^2\,.</math> 
&#9645;&nbsp;<math>\omega_f=\omega_0+\alpha t.\,</math> <math>\omega_f^2=\omega_0^2 + 2\alpha\Delta\theta\,.</math>
&#9645;&nbsp;Total [[w:special:permalink/817212138|acceleration]] is [[w:special:permalink/817212138#Tangential_and_centripetal_acceleration|centripetal plus tangential]]: <math>\vec a = \vec a_c + \vec a_t.\,</math> 
&#9645;&nbsp;[[w:special:permalink/805979890|Rotational kinetic energy]] is <math>K=\tfrac 1 2 I\omega^2,\,</math> where <math>I =\sum_j m_j r_j^2\rightarrow\int r^2 dm</math> is the [[w:special:permalink/815953609|Moment of inertia]].
&#9645;&nbsp;[[b:special:permalink/3355995#Parallel_axis_theorem|parallel axis theorem]] <math>I_{parallel-axis}=I_{center\,of\,mass}+md^2</math>
&#9645;&nbsp;Restricting&nbsp;ourselves to [[w:special:permalink/813639271|fixed axis rotation]],  <math>r</math> is the distance from a fixed axis; the sum of [[w:special:permalink/813286461|torque]]s, <math>\vec\tau=\vec r\times\vec F</math> requires only one component, summed as <math>\tau_{net}=\sum\tau_j=\sum r_{\perp_j}F_j=I\alpha</math>. 
&#9645;&nbsp;Work done by a torque is <math>dW=\left(\sum\tau_j\right)d\theta</math>.  The [[w:simple:special:permalink/5779584|Work-energy theorem]] is 
<math>K_B-K_A=W_{AB}=\int_{\theta_A}^{\theta_B}\left(\sum_j\tau_j\right)d\theta</math>.  
&#9645;&nbsp;[[w:special:permalink/813752583#Mechanical_power|Rotational power]] <math>=P=\tau\omega</math>.
<section end=Fixed-Axis_Rotation/>

<section begin=Angular_Momentum/>'''11.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Angular_Momentum|Angular_Momentum]]''':&nbsp;
[[w:special:permalink/817301310|Center of mass]]
([[w:special:permalink/813228997#Physics_of_simple_rolling|rolling without slip]])  <math>d_{CM}= r\theta,\;</math> <math>v_{CM}= r\omega,\;</math><math>a_{MC}=R\alpha=
\tfrac{mg\sin\theta/}{m+\left(I_{cm}/r^2\right)}
</math>
&#9645;&nbsp;[[w:special:permalink/811230219|Total angular momentum]] and net [[w:special:permalink/813286461|torque]]: <math>d\vec L/dt = \sum \vec\tau </math> <math>= \vec l_1+\vec l_2 + ...;</math> <math>\vec l = \vec r\times\vec p\,</math> for a single particle.  <math>L_{total}=I\omega.</math>
&#9645;&nbsp;[[w:special:permalink/817667800|Precession]] of a [[w:special:permalink/817751726|top]] <math>\omega_P=mrg/(I\omega).
</math><section end=Angular_Momentum/>

<section begin=Static_Equilibrium_and_Elasticity/>'''12.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Static_Equilibrium_and_Elasticity|Static_Equilibrium_and_Elasticity]]''':&nbsp;[[w:special:permalink/816725496|Equilibrium]] <math>\sum \vec F_j=0=\sum\vec \tau_j.\,</math>
[[w:special:permalink/816481752|''Stress = elastic modulus '''&middot;''' strain'']] (analogous to [[w:special:permalink/812496433|''Force = k''' &middot;''' &Delta; x'']] )
&#9645; ([[w:special:permalink/817765684|Young's]] , [[w:special:permalink/817765634|Bulk]] , [[w:special:permalink/817765611|Shear]]) modulus:  <math>\left(
\tfrac{F_\perp}{A}=Y\cdot\tfrac{\Delta L}{L_0}\,,\;
\Delta p          =B\cdot\tfrac{-\Delta V}{V_0}\,,\;
\tfrac{F_\parallel}{A}=S\cdot\tfrac{\Delta x}{L_0}\right)
</math> 
<section end=Static_Equilibrium_and_Elasticity/>

<section begin=Gravitation/>'''13.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Gravitation|Gravitation]]''':&nbsp;
[[w:special:permalink/Newton's law of universal gravitation|Newton's law of gravity]] <math>\vec F_{12}=G\tfrac{m_1m_2}{r^2}\hat r_{12}</math>
&#9645;&nbsp;[[w:special:permalink/Gravity of Earth|Earth's gravity]] <math>g=G\tfrac{M_E}{r^2}</math>
&#9645;&nbsp;[[w:special:permalink/812588328|Gravitational PE beyond Earth]] <math>U
   =-G\tfrac{M_Em}{r}</math>
&#9645;&nbsp;[[w:special:permalink/815049073|Energy conservation]] <math>\tfrac 1 2 mv_1^2-G\tfrac{Mm}
                 {r_1}=\tfrac 1 2 mv_2^2-G\tfrac{Mm}{r_2}</math>
&#9645;&nbsp;[[w:special:permalink/816136252|Escape velocity]] <math>v_{esc}
     =\sqrt{\tfrac{2GM_E}{r}}</math>
&#9645;&nbsp;[[w:special:permalink/784641407|Orbital speed]] <math>v_{orbit}=\sqrt{\tfrac{GM_E}{r}}</math>
&#9645;&nbsp;[[w:special:permalink/812107697|Orbital period]] <math>T=2\pi\sqrt{\tfrac{r^3}{GM_E}}</math>
&#9645;&nbsp;Energy in circular orbit <math>E=K+U=-\tfrac{GmM_E}{2r}</math>
&#9645;&nbsp;[[w:special:permalink/815570035|Conic section]] <math>\tfrac{\alpha}{r}=1+e\,\!\cos\theta</math>
&#9645;&nbsp;[[w:special:permalink/816915223#Third_law_Of_Kepler's|Kepler's third law]]<math>T^2=\tfrac{4\pi^2}{GM}a^3</math>
&#9645;&nbsp;[[w:special:permalink/813599962|Schwarzschild radius]] <math>R_S=\tfrac{2GM}{c^2}</math>
<section end=Gravitation/>

<section begin=Fluid_Mechanics/>'''14.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Fluid_Mechanics|Fluid_Mechanics]]''':&nbsp;
[[w:Mass density|Mass density]] <math>\rho=m/V\;</math>
&#9645; [[w:Pressure|Pressure]] <math>P=F/A\;</math>
&#9645; [[w:Vertical pressure variation|Pressure vs depth/height]] (constant density)<math>\,p=p_o+\rho gh \Leftarrow dp/dy = -\rho g</math> 
&#9645; [[w:Absolute scale|Absolute]] vs [[w:Pressure_measurement#Gauge|gauge pressure]] <math>\,p_{abs}=p_g+p_{atm} \;</math> 
&#9645; [[w:Pascal's law|Pascal's principle]]: <math>\,F/A\,</math> depends only on depth, not on orientation of [[w:Surface_area|A]]. 
&#9645; [[w:Volumetric flow rate|Volume flow rate]] <math>Q=dV/dt\;</math> 
&#9645; [[w:Continuity equation|Continuity equation]] <math>\rho_1A_1v_1=\rho_2A_2v_2</math><math>\Rightarrow A_1v_1=A_2v_2\text{ if }\rho=const.\;</math> 
<section end=Fluid_Mechanics/>

<section begin=Oscillations/>'''15.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Oscillations|Oscillations]]''':&nbsp;
[[w:special:permalink/816470873|Frequency]] <math>f</math>,
[[w:special:permalink/816596611|period]] <math>T</math> and
[[w:special:permalink/812078558|angular frequency]] <math>
\omega\,:\;</math> <math>fT=1\,,\quad\omega T=2\pi</math>
&#9645;&nbsp;[[w:Simple harmonic motion|Simple harmonic motion]] <math>x(t)=A\cos(\omega t+\phi),\,
 </math> <math>v(t)=-A\omega\sin(\omega t+\phi),\,</math> <math>
  a(t)=-A\omega^2\cos(\omega t+\phi)</math> also [[:File:Circle cos sin.gif|models]] the [[w:special:permalink/775579368#Example:_circular_motion|x-component of uniform circular motion.]]
&#9645;&nbsp;For <math>(A,\omega)</math> positive: <math>\,x_{max}=A,\;v_{max}=A\omega,\;a_{max}=A\omega^2 </math>
&#9645;&nbsp;[[w:special:permalink/817152019#Spring/mass_system|Mass-spring]] <math>
  \omega=2\pi/T= 2\pi f=\sqrt{k/m};\,</math>
&#9645;&nbsp;Energy <math>E_{Tot}=\tfrac 1 2 kx^2 + \tfrac 1 2 mv^2 = \tfrac 1 2 kA^2 \Rightarrow </math><math>v = \pm\sqrt{\tfrac k m \left(A^2-x^2\right)}</math>
&#9645;&nbsp;[[w:special:permalink/814155125#Small-angle_approximation|Simple pendulum]] <math>
   \omega\approx\sqrt{g/L}</math>
&#9645;&nbsp;[[w:special:permalink/816585815#Compound_pendulum|Physical pendulum]] <math>\tau=-MgL\sin\theta\approx -MgL\theta
   \Rightarrow \;</math><math>\omega=\sqrt{mgL/I}</math> and <math>L</math> measures from pivot to [[w:special:permalink/817301310|CM]].
&#9645;&nbsp;[[w:special:permalink/813724377#Torsional_harmonic_oscillators|Torsional pendulum]] <math>\tau=-\kappa\theta</math><math>
   \Rightarrow \omega =\sqrt{I/\kappa}</math>
&#9645;&nbsp;[[w:special:permalink/817152019#Damped_harmonic_oscillator|Damped&nbsp;harmonic&nbsp;oscillator]] <math>
     m\tfrac{d^2x}{dt^2}=-kx-b\tfrac{dx}{dt}
 </math><math>
   \Rightarrow x=A_0e^{\frac{b}{2m}t}\cos{(\omega t + \phi)}</math> where <math>
   \omega=\sqrt{\omega_0^2-\left(\tfrac{b}{2m}\right)^2}</math> and <math>
\omega_0=\sqrt\tfrac k m.</math>
&#9645;&nbsp;<ref>[https://scripts.mit.edu/~srayyan/PERwiki/index.php?title=Module_3_--_Damped_and_Driven_Harmonic_Oscillations&oldid=7055 </ref>Forced harmonic oscillator ('''MIT wiki!''')] <math>
     m\tfrac{d^2x}{dt^2}=-kx-b\tfrac{dx}{dt}+F_0\sin\omega t
 </math><math>
   \Rightarrow x=Ae^{\frac{b}{2m}t}\cos{(\omega t + \phi)}</math> where <math>
   A= \tfrac{F_0}{\sqrt{  m^2(\omega-\omega_0)^2   +   b^2\omega^2}}</math>.
<section end=Oscillations/>

<section begin=Waves/>'''16.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Waves|Waves]]''':&nbsp;
<ref>[http://wiki.ubc.ca/index.php?title=Waves_and_the_Doppler_Effect&oldid=218637</ref> Wave speed] ([[w:special:permalink/803796426|phase velocity]]) <math>v=\lambda/T=\lambda f = \omega/k</math> where <math>k=2\pi/\lambda</math> is [[w:special:permalink/817826685|wavenumber]].
&#9645;&nbsp;[[w:special:permalink/803796426|Wave]] and [[w:special:permalink/780823430|pulse]] speed of a [[w:special:permalink/814359291|stretched string]] <math>=\sqrt{F_T/\mu}</math> where <math>F_T</math> is [[w:special:permalink/816052440|tension]] and <math>\mu</math> is [[w:special:permalink/807879225|linear mass density]].
&#9645;&nbsp;Speed of a [[w:special:permalink/817484953#Speed_of_sound_in_liquids|compression wave]]  in a fluid <math>v=\sqrt{B/\rho}.</math>
&#9645;&nbsp;[[w:special:permalink/806706267#General_form|Periodic travelling wave]] <math>y(x,t)=A\sin(kx\mp\omega t)</math>  travels in the positive/negative direction.  The phase is <math>kx\mp\omega t</math> and the amplitude is <math>A</math>.
&#9645;&nbsp;The [[w:special:permalink/815556041|resultant]] of two waves with identical [[w:special:permalink/812844972#Formula_for_phase_of_an_oscillation_or_a_wave|amplitude and frequency]] <math>
y_R(x,t) = \left[2A  \cos\left( \tfrac\phi 2 \right) \right]\sin\left(kx-\omega t + \tfrac\phi 2\right)
</math> where <math>\phi</math> is the [[w:special:permalink/812844972#Phase_difference|phase shift]].
&#9645;&nbsp;This [[w:special:permalink/818029462#Scalar_wave_equation_in_one_space_dimension|wave equation]] <math>\partial^2 y/\partial t^2 = v_w^2\,\partial^2 y/\partial x^2</math>  is [[w:special:permalink/802142961|linear]] in <math>y=y(x,t)</math>
&#9645;&nbsp;Power in a tranverse stretched string wave <math>P_{ave}=\tfrac 1 2 \mu A^2\omega^2v</math>.
&#9645;&nbsp;[[w:special:permalink/783373011|Intensity]] of a [[w:special:permalink/814598681|plane wave]] <math>I= P/A \Rightarrow \tfrac{P}{4\pi r^2}</math> in a [[w:special:permalink/818029462#Monochromatic_spherical_wave|spherical wave]].
&#9645;&nbsp;[[w:simple:special:permalink/5120212|Standing wave]] <math>y(x,t)=A\sin(kx)\cos(\omega t+\phi)</math> For symmetric [[w:special:permalink/818029462#Problems_with_boundaries|boundary conditions]] <math>\lambda_n=2\pi/k_n=\tfrac 2\pi L</math> <math>n=1,2,3,...</math>, or equivalently <math>f=nf_1</math> where <math>f_1=\tfrac{v}{2L} </math> is the [[w:special:permalink/817756341|fundamental frequency]].
<section end=Waves/>

<section begin=Sound/>'''17.&nbsp;[[OpenStax_University_Physics/V1/Formulas_(master)#Sound|Sound]]''':&nbsp;
[[w:special:permalink/816047502#Mathematical_definition|Pressure]] and [[w:special:permalink/793294895|displacement]] fluctuations in a sound wave
<math>P=\Delta P_{max}\sin(kx\mp\omega t +\phi)</math> and <math>s=s_{max}\cos(kx\mp\omega t +\phi)</math>
&#9645;&nbsp;[[w:special:permalink/817484953|Speed of sound]] in a fluid <math>v=f\lambda=\sqrt{\beta/\rho}</math>,
&#9645;&nbsp;in a solid <math>\sqrt{Y/\rho}</math>,
&#9645;&nbsp;in an idal gas <math>\sqrt{\gamma RT/M}</math>,
&#9645;&nbsp;in air <math>331\tfrac m s\sqrt{\tfrac{T_K}{273\,K}}=331\tfrac m s\sqrt{1+\tfrac{T_C}{273^oC}}</math>
&#9645;&nbsp;[[b:special:permalink/3343070#Monopole|Decreasing intensity spherical wave]] <math>I_2=I_1\left(\tfrac{r_1}{r_2}\right)^2</math>
&#9645;&nbsp;[[w:special:permalink/813411440|Sound intensity]] <math>I=\tfrac{\langle P\rangle}{A}=\tfrac{\left(\Delta P_{max}\right)^2}{2\rho v}</math>
&#9645;&nbsp; ...level <math>10\log_{10}{I/I_0}</math>
&#9645;&nbsp;[[w:special:permalink/815621756#Cylinders|Resonance tube]] One end closed:
<math>\lambda_n=\tfrac 4 n L,</math>
<math>f_n=n\tfrac {v}{4L},</math>
<math>n=1,3,5,...</math>
&#9645;&nbsp;Both ends open:
<math>\lambda_n=\tfrac 2 n L,</math>
<math>f_n=n\tfrac {v}{2L},</math>
<math>n=1,2,3,...</math>
&#9645;&nbsp;[[w:special:permalink/812533875|Beat frequency]] <math>f_{beat}=|f_2-f_1|</math>
&#9645;&nbsp;([[w:special:permalink/81767778|nonrelativistic]])&nbsp;[[w:special:permalink/817362507|Doppler effect]] <math>f_O=f_s\tfrac{v\pm v_o}{v\mp v_s}</math> where <math>v</math> is the speed of sound, <math>v_s</math> is the velocity of the source, and <math>v_o</math> is the velocity of the observer.
&#9645;&nbsp;[[w:special:permalink/812305584#Mach_Angle|Angle]] of [[w:special:permalink/817997612|shock wave]]    <math>\sin\theta=v/v_s=1/M</math> where <math>v</math> is the [[w:special:permalink/817484953|speed of sound]], <math>v_s</math> is the speed of the source, and <math>M</math> is the [[w:special:permalink/815724415|Mach number]].
[[File:Five basic moments of inertia.svg|thumb|600px|right|I=&int;r<sup>2</sup>dm for a hoop, disk, cylinder, box, plate, rod, and spherical shell or solid [[c:File:Five basic moments of inertia.svg#Summary|''can be found'']] from this figure.]]
<section end=Sound/>

==Footnotes==
{{cot|These will not available as you take your test}}<references/>
{{cob}}
[[Category:User/Guy vandegrift/Guild]]