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<big><big>Please do not use this template.  Instead go to [[Physics equations/Equations]] or a subpage and transclude from there. </big></big>


====SampleName====
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<onlyinclude>{{#ifeq:{{{transcludesection|SampleName}}}|SampleName|
* Foo
<section end=SampleName/>
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==---1 Introduction==

==---2 One dimensional kinematics==

====DefineDeltaVelocityAcceleration====
<onlyinclude>{{#ifeq:{{{transcludesection|DefineDeltaVelocityAcceleration}}}|DefineDeltaVelocityAcceleration|
*<math>\Delta x = x_2 - x_1</math> is the difference between two values of x (sometimes written as <math>x_f - x_i</math> or <math>x - x_0</math>)

Velocity is the rate at which position changes.  Acceleration is the rate at which velocity changes.   If the time interval is not infinitesimally small, we refer to these as "average" rates.  The '''average''' velocity or acceleration is often denoted by a bar above:  
*<math>\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x_f-x_i}{t_f-t_i}</math>, &nbsp;  &nbsp;  &nbsp; <math>\bar a  = \frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t_f-t_i}</math>.

Alternatives to <math>\bar{v}</math> to are the brakcet <math>\langle v\rangle</math> and the subscript <math>v_\text{ave}</math>. Instantaneous velocity and acceleration are derivatives, <math> v(t) = dx/dt</math>, &nbsp; <math> a(t)=dv/dt=d^2x/dt^2</math>, and occur in the limit that <math>\Delta x</math> and <math>\Delta x</math> are small.
<section end=DefineDeltaVelocityAcceleration/>
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====1DimUniformAccel====
<onlyinclude>{{#ifeq:{{{transcludesection|1DimUniformAccel}}}|1DimUniformAccel|
*<math>x(t) = x_0 + v_0t+\frac 1 2 at^2</math>

*<math>v = v_0+at</math>  
 
*<math>v^2  = v_0^2 + 2a\left( x - x_0 \right)</math>

*<math>x - x_0   = \frac{v_0+v}{2} = \bar vt</math>   &nbsp; (Note that <math>v_\mathrm{ave}  =\bar v = \frac{v_0+v}{2}</math> only if the acceleration is uniform)
<section end=1DimUniformAccel/>
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====1dMotionCALCULUS====
<onlyinclude>{{#ifeq:{{{transcludesection|1dMotionCALCULUS}}}|1dMotionCALCULUS|
The distinction between average and instantaneous velocity lies in whether the time interval, <math>(\Delta t \equiv t_2-t_1)</math> approaches zero:

:<math>\bar v = \frac{\Delta x}{\Delta t}\qquad v\equiv \lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}</math>; &nbsp; &nbsp; <math>\bar a = \frac{\Delta v}{\Delta t}\qquad a\equiv \lim_{\Delta t\rightarrow 0}\frac{\Delta v}{\Delta t}=\frac{dv}{dt}</math>

In the ''model'' of uniform acceleration, we take velocity to be a function of time, <math>v=v(t)</math>, and take the [[w:derivative|derivative]]:

:<math>x(t) = x_0 + v_0t+\frac 1 2 at^2</math>,

where <math>{x_0, v_0, a}</math> are three constants:

:<math>x_0 = x(t=0)</math> is the '''initial position'''(at time, <math>t=0</math>).
:<math>v_0 = v(t)</math> is the '''initial velocity''' (at time, <math>t=0</math>).
:<math> a </math> is the '''acceleration''', which remains '''uniform''' throughout all time (in this model).
<section end=1dMotionCALCULUS/>
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==---3 Two-Dimensional Kinematics==

====2DimKinematic====
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<onlyinclude>{{#ifeq:{{{transcludesection|2DimKinematic}}}|2DimKinematic|
Two dimensional motion is where an object undergoes motion along the <math>x</math> and <math>y</math> axes "at the same time." The position of an object in two-dimensional space is defined by its <math>(x,y)</math> coordinate.<ref>https://en.wikibooks.org/w/index.php?title=Fundamentals_of_Physics/Motion_in_Two_Dimensions&oldid=2602083</ref> By analogy with one-dimensional motion:

*<math>x=x_0+v_{0x}\Delta t+\frac{1}{2}a_x\Delta t^2</math>

*<math>y=y_0+v_{0y}\Delta t+\frac{1}{2}a_y\Delta t^2</math>.  

*<math>v_x=v_{0x}+a_x \Delta t</math>  

*<math>v_y=v_{0y}+a_y \Delta t</math>.  It is not uncommon to replace {{nowrap begin}}<math>\Delta t = t-{t_0}</math> {{nowrap end}} by t (i.e. to set the initial time, t<sub>0</sub> equal to zero.) In '''[[w:free fall|free fall]]''' it is customary to orient the coordinate system so that gravity points in the negative y-direction, so that
*a<sub>x</sub>=0 &nbsp; and &nbsp; a<sub>y</sub>= -g , where g ≈ 9.8 m/s<sup>2</sup> (at Earth's surface.)
<section end=2DimKinematic/>
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====DirectionOfMotion====
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<onlyinclude>{{#ifeq:{{{transcludesection|DirectionOfMotion}}}|DirectionOfMotion|
[[File:Free fall velocity vectors.gif|right]]
The direction of motion is measured with respect to the x axis.  At time equals t and 0, we have, respectively:
*<math> v_x =  v\cos\theta</math> &nbsp; &nbsp; <math> v_y =  v\sin\theta</math>
*<math> v_{x0} =  v_0\cos\theta_0</math> &nbsp; &nbsp; <math> v_{y0} =  v_0\sin\theta_0</math>
<section end=DirectionOfMotion/>
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====1dRelativeMotion====
<onlyinclude>{{#ifeq:{{{transcludesection|1dRelativeMotion}}}|1dRelativeMotion|
[[File:Relative motion man on train.gif|360px|right]]
*<math>\underbrace{\vec v_{M|E}}_{50\;km/hr}=\underbrace{\vec v_{M|T}}_{10\;km/hr}+\underbrace{\vec v_{T|E}}_{40\;km/hr}\,</math> is the velocity of the Man relative to Earth,<math>\vec v_{M|T}</math> is the velocity of the Man relative to the Train, and <math>\vec v_{T|E}</math> is the velocity of the Train relative to Earth.  If the speeds are relativistic, define u=v/c where c = 2.998x10<sup>8</sup>m/s is the speed of light, and
*<math>u_{A|O}=  \frac{u_{A|O'}+ u_{O'|O}}{1+ (u_{A|O\,'})(u_{O\,'|O})}</math>
<section end=1dRelativeMotion/>
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==---4 Dynamics: Force and Newton's Laws==

====NewtonsThreeLaws====
<onlyinclude>{{#ifeq:{{{transcludesection|NewtonsThreeLaws}}}|NewtonsThreeLaws|*<math> m\vec a = \sum\vec F_j\;\quad\;\vec F_{ij}=-\vec F_{ji}</math>
&nbsp;&nbsp;&nbsp;<math>F_x = F \cos\theta\;\quad\;F_y = F\sin\theta\quad\;</math>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>F_x^2+F_y^2 = F^2\;\quad\;\tan\theta=\frac{\sin\theta}{\cos\theta}</math>
<section end=NewtonsThreeLaws/>
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====WeightSimple====
<onlyinclude>{{#ifeq:{{{transcludesection|WeightSimple}}}|WeightSimple|
* <math>m\vec g</math> is the force of gravity on an object of mass, m.  It is called weight, and at Earth/s surface , g&nbsp;≈&nbsp;9.8&nbsp;m/s<sup>2</sup>.
<section end=WeightSimple/>
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====NormalFrictionRamp====
<onlyinclude>{{#ifeq:{{{transcludesection|NormalFrictionRamp}}}|NormalFrictionRamp|
[[File:Free body.svg|280px|right]]
* The '''[[w:Normal (geometry)|normal]]''' to a surface is the direction perpendicular to that surface.
* <math>\vec N</math> is the ''normal force'', which is the component of the ''contact force'' that is perpendicular to the surface.
* <math>\vec f</math> is force of ''friction'', which is the component of the ''contact force'' parallel to the surface.

If &theta; is the angle of an inclined plane's inclination with respect to the horizontal, then (depending on how the rotated coordinate system is defined):
* <math>\pm mg\cos\theta</math> is the component of weight in the normal direction.
* <math>\pm mg\sin\theta</math> is the component of weight perpendicular to the normal direction.
<section end=NormalFrictionRamp/>
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====ThreeTensions====
<onlyinclude>{{#ifeq:{{{transcludesection|ThreeTensions}}}|ThreeTensions|
[[File:Threetensions.jpg|240px|right]]
*The x and y components of the three forces on the small grey circle at the center are:
:<math>T_{1x}=-T_1\cos\theta_1 </math> ,&nbsp;  &nbsp;  &nbsp;  &nbsp; <math> T_{1y}=T_1\sin\theta_1</math>
:<math>T_{2x}=0 </math> ,&nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp;  &nbsp;<math>T_{2y}=-mg</math>
:<math>T_{3x}=T_3\cos\theta_3 </math> ,&nbsp;  &nbsp;  &nbsp;  &nbsp;  &nbsp; <math>T_{3y}=T_3\sin\theta_3</math>
<section end=ThreeTensions/>
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==---5 Friction, Drag, and Elasticity==

====FrictionKineticStatic====
<onlyinclude>{{#ifeq:{{{transcludesection|FrictionKineticStatic}}}|FrictionKineticStatic|
*<math>f_k=\mu_k N</math> is the force friction when an object is sliding on a surface, where <math>\mu_k</math> ("mew-sub-k") is  the kinetic coefficient of friction, and ''N'' is the normal force.
*<math>f_s\le \mu_sN</math> establishes the maximum possible friction (called static friction) that can occur before the object begins to slide. Usually <math>\mu_s > \mu_k</math>.  

<small>Also, [[w: Drag (physics)|air drag]] often depends on speed, an effect this model fails to capture.</small>
<section end=FrictionKineticStatic/>
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==---6 Uniform Circular Motion and Gravitation==

====UniformCircularMotion====
<onlyinclude>{{#ifeq:{{{transcludesection|UniformCircularMotion}}}|UniformCircularMotion|
* <math>a=\frac{v^2}{r}=\omega v =\omega^2 r</math> is the acceleration of '''[[w:Circular_motion#Uniform|uniform circular motion]]''', where v is speed, r is radius, and &omega; is the '''[[w:angular frequency|angular frequency]]'''.
* <math>v=\omega r =2\pi r/T</math>, where T is period.
<section end=UniformCircularMotion/>
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====UniformCircularMotionDerive====
<onlyinclude>{{#ifeq:{{{transcludesection|UniformCircularMotionDerive}}}|UniformCircularMotionDerive|
[[File:UniCircMot gv52.png|thumb|200px|uniform circular motion (here the Latin ''d'' was used instead of the Greek ''Δ'']]
Using the figure we define the distance traveled by a particle during a brief time interval, <math>\Delta t</math>, and the (vector) change in velocity: 


1 &nbsp; &nbsp; <math>\Delta\ell= |\vec r_2 - \vec r_1|</math>, and <math>\Delta v= |\vec v_2 - \vec v_1|</math>


2 &nbsp; &nbsp; <math>\Delta\ell= v\Delta t</math> (rate times time equals distance).


3 &nbsp; &nbsp; <math>\Delta\vec v =\vec a\Delta t</math> (definition of acceleration).


4 &nbsp; &nbsp; <math>\Delta v = a\Delta t</math> (taking the absolute value of both sides).


5 &nbsp; &nbsp; <math>\frac{\Delta v}{v} = \frac{\Delta\ell}{r}</math> (by similar triangles). Substituting (2) and (4) yields:


6 &nbsp; &nbsp; <math>\frac{a\Delta t}{ v} =\frac{v\Delta t}{ r}</math>, which leads to <math>\frac{a}{v} = \frac{v}{r}</math>, and therefore:


7 &nbsp; &nbsp; <math>a=\frac{v^2}{r}</math>
<section end=UniformCircularMotionDerive/>
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====FundamentalConstantsGravity====
<onlyinclude>{{#ifeq:{{{transcludesection|FundamentalConstantsGravity}}}|FundamentalConstantsGravity| 
*<math>G \,</math> ≈ 6.674×10<sup>-11</sup> m<sup>3</sup>·kg<sup>−1</sup>·s<sup>−2</sup> is Newton's universal constant of gravity.
*<math>g=\frac{GM_\oplus }{R_\oplus^2}</math>≈ 9.8 m·s<sup>-2</sup> where ''M<sub>⊕</sub>'' and ''R<sub>⊕</sub>'' are Earth's mass and radius, respectively. (g is called the acceleration of gravity).
<section end=FundamentalConstantsGravity/>
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<small>taken from [https://en.wikipedia.org/w/index.php?title=Physical_constant&oldid=586893645 Physical constants]</small>


====NewtonUniversalLawScalar====
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<onlyinclude>{{#ifeq:{{{transcludesection|NewtonUniversalLawScalar}}}|NewtonUniversalLawScalar|
* <math>F = G \frac{mM}{r^2}= mg^*</math> is the force of gravity between two objects, where the universal constant of gravity is {{nowrap begin}}G ≈ 6.674 × 10<sup>-11</sup> m<sup>3</sup>·kg<sup>−1</sup>·s<sup>−2</sup>{{nowrap end}}.  If, {{nowrap begin}}M ={{nowrap end}}{{nowrap begin}}M<sub>⊕</sub> ≈ 5.97 × 10<sup>24</sup> kg{{nowrap end}}, and {{nowrap begin}}R ={{nowrap end}}{{nowrap begin}}R<sub>⊕</sub> ≈ 6.37 × 10<sup>6</sup> kg{{nowrap end}}, then <math>g^*</math> = {{nowrap begin}}g ≈ 9.8 m/s<sup>2</sup>{{nowrap end}} is the acceleration of gravity at Earth's surface.
<section end=NewtonUniversalLawScalar/>
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====NewtonKeplerThirdDerive====
<onlyinclude>{{#ifeq:{{{transcludesection|NewtonKeplerThirdDerive}}}|NewtonKeplerThirdDerive|
# &nbsp; &nbsp; <math>ma=m\frac{v^2}{r}=\frac{mMG}{r^2}</math>, where m is the mass of the orbiting object, and M>>m is the mass of the central body, and r is the radius (assuming a circular orbit).
# &nbsp; &nbsp; <math>vT=2\pi r</math>, where m is the mass of the orbiting object, and M>>m is the mass of the central body, and r is the radius (assuming a circular orbit).  After some algebra:
# &nbsp; &nbsp; <math> r^3=\frac{MG}{4\pi^2}T^2</math>
<section end=NewtonKeplerThirdDerive/>
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====NewtonKeplerThirdGeneralized====
<onlyinclude>{{#ifeq:{{{transcludesection|NewtonKeplerThirdGeneralized}}}|NewtonKeplerThirdGeneralized|
* <math>a^3=\frac{(M+m)G}{4\pi^2}T^2</math>, is valid for objects of comparable mass, where ''T'' is the period, ''(m+M)'' is the sum of the masses, and ''a'' is the semimajor axis: {{nowrap begin}}''a'' = ½(r<sub>min</sub>+r<sub>max</sub>){{nowrap end}}  where ''r<sub>min</sub>'' and ''r<sub>max</sub>'' are the minimum and maximum separations between the moving bodies, respectively.
<section end=NewtonKeplerThirdGeneralized/>
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==---7 Work and Energy==

=====EnergyConservation=====
<!--Copy and paste before modifying.-->
<onlyinclude>{{#ifeq:{{{transcludesection|EnergyConservation}}}|EnergyConservation|
<!--*<math>KE = \frac 1 2 m v^2\qquad\left[\frac{kg\cdot m^2}{s^2}\right]</math> is kinetic energy, where ''m'' is mass and ''v'' is speed.-->
*<math>KE = \frac 1 2 m v^2\,</math> is kinetic energy, where ''m'' is mass and ''v'' is speed..
*<math>U_g = mgy</math> is gravitational potential energy,where y is height, and <math>g = 9.80\;\frac{m}{s^2}</math> is the gravitational acceleration at Earth's surface.
*<math>U_s = \frac 1 2 k_s x^2</math> is the potential energy stored in a spring with [[w:Hooke's_law|spring constant]] <math>k_s</math>.
*<math>\sum KE_f + \sum PE_f = \sum KE_i + \sum PE_i -Q</math> relates the final energy to the initial energy.  If energy is lost to heat or other nonconservative force, then Q>0.
<section end=EnergyConservation/>
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=====WorkBasic=====
<onlyinclude>{{#ifeq:{{{transcludesection|WorkBasic}}}|WorkBasic|
*<math>W = F \ell\cos\theta=\vec F\cdot\vec\ell</math> (measured in Joules) is the work done by a force <math>F</math> as it moves an object a distance <math>\ell</math>.  The angle between the force and the displacement is &theta;.
*<math>\sum \vec F\cdot\Delta\ell</math> describes the work if the force is not uniform.  The steps, <math>\Delta\vec\ell</math>, taken by the particle are assumed small enough that the force is approximately uniform over the small step.  If force and displacement are parallel, then the work becomes the area under a curve of F(x) versus x.
*<math>P=\frac{\vec F \cdot \vec\Delta \ell}{\Delta t}=\vec F\cdot\vec v</math> is the power (measured in Watts) is the rate at which work is done.  ('''''v''''' is velocity.)
<section end=WorkBasic/>
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==---8 Linear Momentum and Collisions==

====MomentumConservation====
<onlyinclude>{{#ifeq:{{{transcludesection|MomentumConservation}}}|MomentumConservation|
*<math> \vec p = m\vec v</math> is momentum, where m is mass and <math> \vec v</math> is velocity.  Momentum is conserved if the net '''external force''' is zero.  The net momemtum is conserved '''if the net external force equal zero''':
*<math>\sum \vec p_f = \sum \vec p_i</math>.  In a simple, one dimensional case with only two particles:
*<math> m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'</math> , where the prime denotes 'final'.  
To avoid subscripts and superscripts, seek ways to simplify the formula.  For example if the collision is [[w:Inelastic_collision#Perfectly_inelastic_collision|perfectly inelastic]] (i.e. they stick), then it is more convenient to write:
*<math> m_1v_1 + m_2v_2 = (m_1 + m_2)v'</math>.
<section end=MomentumConservation/>
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==---9 Statics and Torque==
====TorqueSimple====
<onlyinclude>{{#ifeq:{{{transcludesection|TorqueSimple}}}|TorqueSimple|
[[File:Torque, position, and force.svg|right|300px]] 
* <math>\tau = rF\sin\theta,\!</math> is the [[w:torque|torque]] caused by a force, F, exerted at a distance ,r, from the axis.  The angle between '''''r''''' and '''''F''''' is &theta;.
[[File:Torque formulas with lever arm OR perp- F and r.jpg|left|240px]]
<nowiki>&nbsp;* </nowiki><math>\tau = rF_\perp\,</math> where <math>F_\perp=F\sin\theta</math> where is the component of '''''F''''' that is perpendicular to '''''r'''''.

<nowiki>&nbsp;* </nowiki><math>\tau = r_\perp F\,</math> where <math>r_\perp=r\sin\theta</math> 

The {{lw|SI units|SI unit}} for torque is the {{lw|newton metre|}} (N·m). It would be inadvisable to call this a Joule, even though a Joule is also a (N·m). The symbol for torque is typically ''τ'', the [[Greek alphabet|Greek letter]] ''[[w:tau|tau]]''. When it is called moment, it is commonly denoted ''M''.<ref>https://en.wikipedia.org/w/index.php?title=Torque&oldid=582917749</ref>    The lever arm is defined as either ''r'', or ''r''<sub>⊥</sub> .  Labeling ''r'' as the ''[[wikt:lever arm|lever arm]]'' allows ''[[wikt:moment arm|moment arm]]'' to be reserved for ''r''<sub>⊥</sub>. 
<section end=TorqueSimple/>
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====TorqueCrossProduct====
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<onlyinclude>{{#ifeq:{{{transcludesection|TorqueCrossProduct}}}|TorqueCrossProduct|
*<math>\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F},</math> uses the [[w:cross product|cross product]] to define torque as a vector.
<section end=TorqueCrossProduct/>
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==---10 Rotational Motion and Angular Momentum==
====[[Template:Physeq1/RotationalLinearEqnsMotionTable]]====
{{Physeq1/RotationalLinearEqnsMotionTable}}
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====[[Template:Physeq1/RotationalLinearAnalogyTable]]====
{{Physeq1/RotationalLinearAnalogyTable}}
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====[[Template:Physeq1/MomentOfInertia]]====
{{hidden|This table is large.  Click to view it.|{{Physeq1/MomentOfInertia}}}}
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====[[Template:Physeq1/MomentOfInertiaShort]]====
{{Physeq1/MomentOfInertiaShort}}
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====Arclength====
<onlyinclude>{{#ifeq:{{{transcludesection|Arclength}}}|Arclength|
[[File:Radians Angle Definition.svg|right|100px]]
*<math>s=r\theta_\mathrm{rad}\approx r\frac{\theta_\mathrm{deg}}{57.3}\,</math> &nbsp; is the [[w:arclength|arclength]] of a portion of a circle of radius ''r'' described the angle ''θ''. The two forms allow  ''θ'' to be measured in either degrees or radians (2π rad = 360 deg). The lengths ''r'' and ''s'' must be measured in the same units. 
<section end=Arclength/>
}}</onlyinclude>
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====RadianDegreeRevolutionFreqOmegaPeriod====
<onlyinclude>{{#ifeq:{{{transcludesection|RadianDegreeRevolutionFreqOmegaPeriod}}}|RadianDegreeRevolutionFreqOmegaPeriod|
* <math>2\pi\;rad=360\;deg=1\;rev</math> relates the radian, degree, and revolution.

*<math>f = \frac{\#\,\text{revs}}{\#\,\text{secs}}</math> is the number of revolutions per second, called '''[[w:frequency|frequency]]'''.

*<math>T = \frac{\#\,\text{secs}}{\#\,\text{revs}}</math> is the number of seconds per revolution, called '''[[w:period|period]]'''.  Obviously <math>fT = 1</math>.

*<math>\omega = \frac{\Delta\theta}{\Delta t}</math> is called '''[[w:angular frequency|angular frequency]]''' (ω is called omega).  Obviously <math>\omega T = 2\pi</math>
<section end=RadianDegreeRevolutionFreqOmegaPeriod/>
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====RotationalUniformAccel====
<onlyinclude>{{#ifeq:{{{transcludesection|RotationalUniformAccel}}}|RotationalUniformAccel|
[[File:Radians_Angle_Definition.svg|right|160px]]
*<math>\theta=\frac s r</math> is the angle (in radians) where s is arclength and r is radius.

*<math> \omega = \frac{d\theta}{dt} </math> (or &Delta;&theta;/&Delta;t), called ''angular velocity'' is the rate at which &theta; changes.

*<math> \alpha= \frac{d\omega}{dt} </math> (or &Delta;&omega;/&Delta;t), called ''angular acceleration'' is the rate at which &omega; changes.

The equations of uniform angular acceleration are:
*<math>\theta(t) = \theta_0 + \omega_0t+\frac 1 2 \alpha t^2</math>

*<math>\omega = \omega_0+\alpha t</math>  

*<math>\omega ^2  = \omega_0^2 + 2\alpha\left( \theta - \theta_0 \right)</math>

*<math>\theta - \theta_0   = \frac{\omega_0+\omega}{2} = \bar \omega t</math>   &nbsp; (Note that <math>\omega_\mathrm{ave}  =\bar \omega = \frac{\omega_0+\omega}{2}</math> only if the angular acceleration is uniform)
<section end=RotationalUniformAccel/>
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====AngularMotionEnergyMomentum====
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<onlyinclude>{{#ifeq:{{{transcludesection|AngularMotionEnergyMomentum}}}|AngularMotionEnergyMomentum|
*<math>KE_{rot}= \frac 1 2 \sum m_n v_n^2 = \frac 1 2 \sum m_n (\omega r_n)^2 =\frac 1 2 I \omega^2</math> is the kinetic of a rigidly rotating object, where
*<math>I=\sum m_n r_n^2</math> is the [[w:moment of inertia|moment of inertia]], equal to <math>MR^2</math> for a hoop of radius R and mass M (assuming the axis is through the center).  For a solid disk, the moment of inertia equals <math>\frac 1 2 MR^2</math>.  
*The generalization of {{nowrap begin}}''F=ma''{{nowrap end}} for rotational motion through a fixed axis is {{nowrap begin}}<big>''&tau; = I&alpha;''</big> {{nowrap end}}  , where ''&tau;'' (called tau) is '''[[w:torque|torque]]'''.  If the force is perpendicular to r, then {{nowrap begin}}<big>''&tau; = r F''</big> {{nowrap end}}  
*The total '''[[w:angular momentum|angular momentum]]''', {{nowrap begin}} <big>''L<sub>net</sub> = &Sigma; I&omega;''</big>{{nowrap end}} is '''[[w:conservation law|conserved]]''' if no net external torque is acting on a system.
<section end=AngularMotionEnergyMomentum/>
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==---11 Fluid Statics==
====PressureVersusDepth====
<!--Copy and paste before modifying.-->
<onlyinclude>{{#ifeq:{{{transcludesection|PressureVersusDepth}}}|PressureVersusDepth|
[[File:Archimedes gv52.png|thumb|120px|Pressure is the weight per unit area of the fluid above a point.]]
A fluid's '''[[w:pressure|pressure]]''' is ''F/A'' where ''F'' is force and A is a (flat) area.  The pressure at depth, <math>h</math> below the surface is the weight (per area) of the fluid above that point.  As shown in the figure, this implies:

:<math> P = P_0 + \rho g h  </math>  

where <math>P_0</math> is the pressure at the top surface, <math>h</math> is the depth, and <math>\rho</math>  is the mass density of the fluid.  In many cases, only the difference between two pressures appears in the final answer to a question, and in such cases it is permissible to set the pressure at the top surface of the fluid equal to zero.  In many applications, it is possible to artificially set <math>P_0</math> equal to zero, for example at atmospheric pressure.  The resulting pressure is called the gauge pressure, for <math> P_{gauge} = \rho gh  </math> below the surface of a body of water.   
<section end=PressureVersusDepth/>
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====Archimedes====
<!--Copy and paste before modifying.-->
<onlyinclude>{{#ifeq:{{{transcludesection|Archimedes}}}|Archimedes|
Pascal's principle does not hold if two fluids are separated by a seal that prohibits fluid flow (as in the case of the piston of an internal combustion engine).  Suppose the upper and lower fluids shown in the figure are not sealed, so that a fluid of mass density <math>\rho_{flu}</math> comes to equilibrium above and below an object.  Let the object have a mass density of  <math>\rho_{obj}</math> and a volume of <math>A\Delta h</math>, as shown in the figure.  The net (bottom minus top) force on the object due to the fluid is called the buoyant force:

:<math>\rm{buoyant}\;\rm{force}=(A\Delta h )(\rho_{flu})g\,</math>, 

and is directed upward.  The volume in this formula, AΔh, is called the volume of the displaced fluid, since placing the volume into a fluid at that location requires the removal of that amount of fluid.  Archimedes principle states:

:'''''A body wholly or partially submerged in a fluid is buoyed up by a force equal to the weight of the displaced fluid.'''''

Note that if <math>\rho_{obj}=\rho_{flu}</math>, the buoyant force exactly cancels the force of gravity.  A fluid element within a stationary fluid will remain stationary.  But if the two densities are not equal, a third force (in addition to weight and the buoyant force) is required to hold the object at that depth.  If an object is floating or partially submerged, the volume of the displaced fluid equals the volume of that portion of the object which is below the waterline.
<section end=Archimedes/>
}}</onlyinclude>
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==---12 Fluid Dynamics==
====ContinuityPipe====
<onlyinclude>{{#ifeq:{{{transcludesection|ContinuityPipe}}}|ContinuityPipe|
[[File:FlowRate gv52.png|thumb|180px|A fluid element speeds up if the area is constricted.]]
*<math>\frac{\Delta V}{ \Delta t } =\dot V=Av = Q</math> the volume flow for incompressible fluid flow if viscosity and turbulence are both neglected.  The average velocity is <math>v</math> and <math>A</math> is the cross sectional area of the pipe.  As shown in the figure, <math>v_1A_1=v_2A_2</math> because <math>Av</math> is constant along the developed flow.  To see this, note that the volume of pipe is <math>\Delta V=A\Delta x</math> along a distance <math>\Delta x</math>. And, <math>v = \Delta x / \Delta t</math> is the volume of fluid that passes a given point in the pipe during a time <math>\Delta t</math>.
<section end=ContinuityPipe/>
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====ContinuityCALCULUS====
<onlyinclude>{{#ifeq:{{{transcludesection|ContinuityCALCULUS}}}|ContinuityCALCULUS|
*<math>A_1v_1 =  A_2v_2=0 \rightarrow\oint \vec v\cdot\hat n dA = 0 \rightarrow \nabla\cdot\vec v = 0 \;</math> is the generalization of the continuity equation for incompressible fluid flow in three dimensions, where <math>\hat n </math> is the outward unit vector and the integral is over the entire surface. 
<section end=ContinuityCALCULUS/>
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====Bernoulli====
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<onlyinclude>{{#ifeq:{{{transcludesection|Bernoulli}}}|Bernoulli|
 *<math>P_1 + \rho g y_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \frac{1}{2}\rho v_2^2</math> is [[w:Bernoulli's principle|Bernoulli's equation]], where <math>P</math> is pressure, <math>\rho</math> is density, and <math>y</math> is height.  This holds for [[w:inviscid flow|inviscid flow]].
<section end=Bernoulli/>
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==---13 Temperature, Kinetic Theory, and Gas Laws==

====TemperatureConversion====
<onlyinclude>{{#ifeq:{{{transcludesection|TemperatureConversion}}}|TemperatureConversion|
* <math>T_C=T_K-273.15</math> converts from Celsius to Kelvins.
* <math>T_F=\frac 9 5 T_C+32</math> converts from Celsius to Fahrenheit.
<section end=TemperatureConversion/>
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====ThermodynamicConstants====
<onlyinclude>{{#ifeq:{{{transcludesection|ThermodynamicConstants}}}|ThermodynamicConstants|
* Boltzmann's constant is k<sub>B</sub>&nbsp;≈ {{nowrap|1.38 × 10<sup>-23</sup> [[w:Joule|J]] [[w:Kelvin|K]]<sup>−1</sup> }} and the gas constant is R&nbsp;≈ {{val|8.314}}[[w:Joule|J]] [[w:Kelvin|K]]<sup>−1</sup> [[w:Mole (unit)|mol]]<sup>−1</sup>. 
* The atomic mass unit ≈ {{nowrap|1.66 × 10<sup>-27</sup> kg}} is the approximate mass of protons and neutrons in the atom.  (The proton/electron mass ratio is approximately 1836)
<section end=ThermodynamicConstants/>
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====IdealGasLaw====
<onlyinclude>{{#ifeq:{{{transcludesection|IdealGasLaw}}}|IdealGasLaw|
* <math>PV=nRT=Nk_BT</math> is the '''[[w:ideal gas law|ideal gas law]]''', where P is pressure, V is volume, n is the number of moles and N is the number of atoms or molecules.  Temperature must be measured on an absolute scale (e.g. Kelvins).  
<section end=IdealGasLaw/>
}}</onlyinclude>
Call with  <nowiki>{{Physeq1|transcludesection=IdealGasLaw}}N<sub>A</sub>k<sub>B</sub>=R where N<sub>A</sub>= {{nowrap|6.02 × 10<sup>23</sup>}} is the Avogadro number. Boltzmann's constant can also be written in eV and Kelvins: k<sub>B</sub> ≈{{nowrap|8.6 × 10<sup>-5</sup> eV/deg}}.</nowiki>

=====AverageTranslationalKineticEnergyGas=====
<onlyinclude>{{#ifeq:{{{transcludesection|AverageTranslationalKineticEnergyGas}}}|AverageTranslationalKineticEnergyGas|
* <math>\frac 3 2 k_B T =\frac 1 2 mv_{rms}^2</math> is the average translational kinetic energy per "atom" of a 3-dimensional ideal gas. 
*<math>v_{rms}=\sqrt{\frac{3k_BT}{m}}=\sqrt{\overline{v^2}}</math> is the root-mean-square speed of atoms in an ideal gas.
<section end=AverageTranslationalKineticEnergyGas/>
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=====EnergyIdealGas=====
<onlyinclude>{{#ifeq:{{{transcludesection|EnergyIdealGas}}}|EnergyIdealGas|
* <math>E = \frac{\varpi}{2}Nk_BT</math> is the total energy of an ideal gas, where <math>\varpi=3\;</math> [[w:Degrees_of_freedom_(mechanics)|degrees of freedom]] a three-dimensional monatomic gas.
<section end=EnergyIdealGas/>
}}</onlyinclude>
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==---14 Heat and Heat Transfer==
====SpecificLatentHeat====
<onlyinclude>{{#ifeq:{{{transcludesection|SpecificLatentHeat}}}|SpecificLatentHeat|
* <math>Q = mc_S\Delta T</math> is the heat required to change the temperature of a substance of mass, m.  The change in temperature is &Delta;T.  The '''[[w:Heat capacity|specific heat]]''', c<sub>S</sub>, depends on the substance (and to some extent, its temperature and other factors such as pressure).  Heat is the transfer of energy, usually from a hotter object to a colder one.  The units of specfic heat are ''energy/mass/degree'', or {{nowrap begin}}J/(kg-degree){{nowrap end}}.
* <math>Q = mL</math> is the heat required to change the phase of a a mass, m, of a substance (with no change in temperature).  The '''[[w:latent heat|latent heat]]''', L, depends not only on the substance, but on the nature of the phase change for any given substance.  L<sub>F</sub> is called the latent heat of fusion, and refers to the melting or freezing of the substance.  L<sub>V</sub> is called the latent heat of vaporization, and refers to evaporation or condensation of a substance.
<section end=SpecificLatentHeat/>
}}</onlyinclude>
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====PowerRateHeatTransfer====
<onlyinclude>{{#ifeq:{{{transcludesection|PowerRateHeatTransfer}}}|PowerRateHeatTransfer|
The '''rate of heat transfer''' is Q/t (or dQ/dt) and has units of "power": {{nowrap begin}}1 Watt = 1 W = 1J/s{{nowrap end}} 
*<math>\frac{kA}{d}\Delta T</math> is rate of heat transfer for a material of thermal conductivity, k, of area, A, and thickness, d.  (In this model, the thickness is assumed uniform over the area, and no heat flows through the sides.) The '''[[w:thermal conductivity|thermal conductivity]]''' is a property of the substance used to insulate, or subdue, the flow of heat.
<section end=PowerRateHeatTransfer/>
}}</onlyinclude>
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====StefanBoltzmannLaw====
<onlyinclude>{{#ifeq:{{{transcludesection|StefanBoltzmannLaw}}}|StefanBoltzmannLaw|
* <math> \sigma AeT^{4}</math> is the power radiated by a surface of area, A, at a temperature, T, measured on an absolute scale such as Kelvins.  The emissivity, e, varies from 1 for a black body to 0 for a perfectly reflecting surface.  The Stefan-Boltzmann constant is <math>\sigma\approx 5.67 \times 10^{-8}\, \mathrm{J\, s^{-1}m^{-2}K^{-4}}</math>.
<section end=StefanBoltzmannLaw/>
}}</onlyinclude>
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==---15 Thermodynamics==

====StateVariables====
<onlyinclude>{{#ifeq:{{{transcludesection|StateVariables}}}|StateVariables|
[[File:Piston gv52.png|thumb]]
* Pressure (P), Energy (E), Volume (V), and Temperature (T) are state variables ([[w:State_function|state functions]]called state functions).  The number of particles (N) can also be viewed as a state variable.
*  Work (W), Heat (Q) '''are not state variables.'''
<section end=StateVariables/>
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====EntropyMonatomic====
<onlyinclude>{{#ifeq:{{{transcludesection|EntropyMonatomic}}}|EntropyMonatomic|
* <math> S (V,T)= \frac{3 Nk_B}{2}\ln T+ Nk_B\ln V + constant</math>, is the [[w:entropy|entropy]] of an [[w:Ideal_gas|ideal ]], [[w:monatomic_gas|monatomic]] gas. The constant is arbitrary only in [[w:classical_physics|classical]] (non-quantum) thermodynamics. Since it is a function of state variables, entropy is also a [[w:state function|state function]]. 
<section end=EntropyMonatomic/>
}}</onlyinclude>
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====HeatEngine====
<!--Copy and paste before modifying.-->
<onlyinclude>{{#ifeq:{{{transcludesection|HeatEngine}}}|HeatEngine|
[[File:Simple heat engine.jpg|220px|right]]
[[File:PdV work cycle.gif|right|120px]]
A point on a [[w:Pressure volume diagram|PV diagram]] define's the system's pressure (P) and volume (V). Energy (E) and pressure (P) can be deduced from [[w:Category:Equations_of_state|equations of state]]: {{nowrap begin}}E=E(V,P) and T=T(V,P){{nowrap end}}. If the piston moves, or if heat is added or taken from the substance, energy (in the form of work and/or heat) is added or subtracted.  If the path returns to its original point on the PV-diagram (e.g., 12341 along the rectantular path shown), and if the process is [[w:Quasistatic_process|quasistatic]], all state variables {{nowrap|(P, V, E, T)}} return to their original values, and the final system is indistinguishable from its original state.   
* Net work done equals area enclosed by the loop. This are is often written as a closed line integral:
*<math> \oint P \ dV = </math> work done on or by the engine each cycle.
*<math>Work = Q_{in} - Q_{out} </math>: The net heat ''Q<sub>in</sub>'' that enters at each cycle equals the work done ''W<sub>out</sub>''.
**''Remember:'' Area ''"under"'' is the work to get from one point to the other; Area ''"inside"'' is the total work per cycle.
<section end=HeatEngine/>
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====IsothermalWork====
<onlyinclude>{{#ifeq:{{{transcludesection|IsothermalWork}}}|IsothermalWork|
In an isothermal expansion (contraction), temperature, T, is constant.  Hence P=nRT/V and substitution yields,
*<math>\int_{Vi}^{Vf}PdV=\int_{Vi}^{Vf}nRT\frac{dV}{V}=nRT\int_{Vi}^{Vf}\frac{dV}{V}=nRT\ln\frac{V_f}{V_i}</math>
<section end=IsothermalWork/>
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====SecondLawThermo====
<onlyinclude>{{#ifeq:{{{transcludesection|SecondLawThermo}}}|SecondLawThermo|
* <math>\Delta W=-F\Delta x= ( -P\cdot\mathrm{Area} ) \left( \frac{\Delta V}{\mathrm{Area}} \right) = -P\Delta V</math> is the work done on a system of pressure P by a piston of voulume V.  If ΔV>0 the substance is expanding as it exerts an outward force, so that ΔW<0 and the substance is doing work on the universe; ΔW>0 whenever the universe is doing work on the system. 
* <math>\Delta Q</math> is the amount of heat (energy) that flows into a system. It is positive if the system is placed in a ''heat bath'' of higher temperature.  If this process is reversible, then the heat bath is at an infinitesimally higher temperature and a finite &Delta;Q takes an infinite amount of time.
* <math>\Delta E=\Delta Q-P\Delta V</math> is the change in energy ([[w:First Law of Thermodynamics|First Law of Thermodynamics]]).
<section end=SecondLawThermo/>
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==---16 Oscillatory Motion and Waves==
====algebraSHO====
<onlyinclude>{{#ifeq:{{{transcludesection|algebraSHO}}}|algebraSHO|
*<math> x = X\cos{   \frac{2\pi t}{T}   }</math> describes oscillatory motion with period T.  The amplitude, or maximum displacement is <math>X</math>.  Alternative notation includes the use of  <math>x_0</math> instead of <math>X</math>).  Using by  <math> \omega_0T=2\pi</math> allows us to write this in terms of angular frequency, &omega;<sub>0</sub>:
* <math> x(t) = x_0\cos\left(\omega_0 t - \varphi\right)</math> , where we have introduced a phase shift to permit both sine and cosine waves.  For example, <math>\cos\left(\omega_0 t - \varphi\right)= \sin\omega_0 t</math>.
*<math>\omega_0=\sqrt{\frac{k_s}{m}}=\frac{2\pi}{T}</math> holds for a mass-spring system with mass, m, and spring constant, k<sub>s</sub>.
*<math>\omega_0=\sqrt{\frac{g}{L}}=\frac{2\pi}{T}</math> holds for a low amplitude pendulum of length, L, in a gravitational field, g.
*<math>PE=\frac 1 2 k_s x^2</math> is the potential energy of a mass spring system.  This equation can also be used for a pendulum if we replace the spring constant <math>k_s</math> by an ''effective'' spring constant <math>k_{eff} = mg/L</math>.
<section end=algebraSHO/>
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====SHO====
<onlyinclude>{{#ifeq:{{{transcludesection|SHO}}}|SHO|
* <math> x(t) = x_0\cos\left(\omega_0 t - \varphi\right) =X\cos\left(\omega_0 t - \varphi\right)</math> describes an oscillating variable.   The velocity and acceleration are:
*  <math>v(t) = dx/dt = -\omega_0 x_0\sin\left(\omega_0 t - \varphi\right)=v_0\cos\left(\omega_0 t - \varphi\right)</math>, where <math>v_0=\omega_0 x_0</math>. The acceleration is given by:
*<math> a(t) = dv/dt =-\omega_0 x_0\cos\left(\omega_0 t - \varphi\right)=a_0\cos\left(\omega_0 t - \varphi\right)</math>, where 
*<math>a_0=\omega_0 v_0=\omega_0^2 x_0</math>.  Note also that the maximum force obeys,  <math>F_0=m a_0</math>, and that
*<math>E=\frac 1 2 mv_0^2=\frac 1 2 k_s x_0^2</math> is the total energy (which also equals the maximum kinetic energy, as well as the maximum potential energy (with k<sub>s</sub> being the spring constant).
* x(t) obeys the [[w:Linear_homogeneous_differential_equation|linear homogeneous differential equation]] (ODE), <math>\frac{d^2x}{dt^2} = -\omega_0^2 x(t)</math>, with ω  being a (constant) parameter of the ODE.  
<section end=SHO/>
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====FrequencyWavelengthSpeed====
<onlyinclude>{{#ifeq:{{{transcludesection|FrequencyWavelengthSpeed}}}|FrequencyWavelengthSpeed|
[[File:StandingWaves.png|right|200px]]
*<math>f\lambda =  v_p</math> relates the frequency, f, wavelength, &lambda;,and the the phase speed, v<sub>p</sub> of the wave (also written as v<sub>w</sub>)   This phase speed is the speed of individual crests, which for sound and light waves also equals the speed at which a [[w:wave packet|wave packet]] travels. 
*<math>L = \frac{n\lambda_n}{2}</math> describes the n-th normal mode vibrating wave on a string that is fixed at both ends (i.e. has a node at both ends).  The mode number, n = 1, 2, 3,..., as shown in the figure.
<section end=FrequencyWavelengthSpeed/>
}}</onlyinclude>

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==---17 Physics of Hearing==
====SpeedSound====
<onlyinclude>{{#ifeq:{{{transcludesection|SpeedSound}}}|SpeedSound|
*<math>v_s =\sqrt{   \frac{T}{273}    }\cdot 331 \text{m/s}</math> is the the approximate speed near Earth's surface, where the temperature, T, is measured in Kelvins.  A theoretical calculation is <math>v_s=\sqrt{\frac{\gamma k_B T}{m}}</math> where <math>\gamma =\frac{\varpi+2}{\varpi}</math> for a semi-classical gas with <math>\varpi</math> [[w:degrees of freedom (physics and chemistry)|degrees of freedom]]. For a diatomic gas such as Nitrogen, {{nowrap begin}}&gamma; = 1.4.{{nowrap end}}
<section end=SpeedSound/>
}}</onlyinclude>
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==References==
*No such template exists in Physeq1
{{hidden|click to see references|<references/>}}

<noinclude>
[[Category:Template:Physeq]]
[[Category:Physics equations/Equations/Transcluded]]
[[Category:User:Guy vandegrift/Trash/Physeq]]
</noinclude>