# Physics equations/Equations Sandbox

Never transclude out of this sandbox. Instead cut/paste or copy/paste/modify the material into Physics equations/Equations or one of its subpages.

This Sandbox is for material that might later be incorporated into other equation sheets. It's a bit disorganized and will probably stay that way (messy desk).

#### LargeMetrixPrefixes

Large Metric Prefixes are hidden
Text Symbol Factor Exponential
tera T 1000000000000 E12
giga G 1000000000 E9
mega M 1000000 E6
kilo k 1000 E3
hecto h 100 E2
deca da 10 E1
(none) (none) 1 E0
deci d 0.1 E−1
centi c 0.01 E−2
milli m 0.001 E−3
micro μ 0.000001 E−6
nano n 0.000000001 E−9
pico p 0.000000000001 E−12
fempto f 0.000000000000001 E−15

## 00 Mathematics

• ${\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{\,c\,}}\,.}$ (where A is the angle shown)
• ${\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{\,c\,}}\,.}$
• ${\displaystyle \tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{\,b\,}}={\frac {\sin A}{\cos A}}\,.}$
##### InverseTrigFunctions
• ${\displaystyle \sin \left(\sin ^{-1}(z)\right)=z}$ and ${\displaystyle sin^{-1}\left(\sin \theta \right)=\theta }$ defines the arcsine function as the inverse of the sine. Similarly, ${\displaystyle \tan ^{-1}}$ is called the arctangent, or the inverse tangent, and ${\displaystyle \cos ^{-1}}$ is called arccosine, or the inverse cosine and so forth. In general, ${\displaystyle f(f^{-1}(y))=y}$ and ${\displaystyle f^{-1}((f(x))=x}$ for any function and its inverse. Complexities occur whenever the inverse is not a true function; for example, since ${\displaystyle \tan(\theta )=\tan(\theta +\pi )}$, the inverse is multi-valued: ${\displaystyle \tan ^{-1}(\tan \theta )=\theta \;or\;\theta +\pi \,.}$

### First year calculus

##### CalculusBasic
• ${\displaystyle \Delta f=f\left(x+\Delta x\right)-f(x)}$,     and the derivative is ${\displaystyle {\frac {\Delta f}{\Delta x}}\rightarrow {\frac {df}{dx}}}$ in the limit that ${\displaystyle \Delta x\rightarrow 0}$
• ${\displaystyle {\frac {d}{dx}}f\left(g(x)\right)={\frac {df}{dg}}{\frac {dg}{dx}}}$ is the chain rule.
• ${\displaystyle {\frac {d}{dx}}Ax^{p}=(p-1)Ax^{p-1}}$,      ${\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{x}}}$,     ${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}$,     ${\displaystyle {\frac {d}{dx}}\sin x=\cos x}$,     ${\displaystyle {\frac {d}{dx}}\cos x=-\sin x}$
• ${\displaystyle \int {\frac {df}{dx}}dx=f(x)+c}$ expresses the fundamental theorem of calculus.

#### RiemannSum

• ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx \sum f(x_{j})\Delta x_{j}}$ is the Riemann sum representation of the integral of f(x) from x=a to x=b. It is the area under the curve, with contributions from f(x)<0 being negative (if a>b). The sum equals the integral in the limit that the widths of all the intervals vanish (Δxj→0).

#### RiemannSumShort

• ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx \sum f(x_{j})\Delta x_{j}}$ is the Riemann sum representation of the integral of f(x) from x=a to x=b.

#### IntegrateFundamentalTheorem

• The fundamental theorem of calculus allows us to construct integrals from known derivatives:
• ${\displaystyle \int _{a}^{b}Ax^{n}\mathrm {d} x=\left\{{\frac {A}{n+1}}b^{n+1}-{\frac {A}{n+1}}a^{n+1}\right\}}$
• ${\displaystyle \int _{a}^{b}Ax^{-1}\mathrm {d} x=A\ln b-A\ln a=A\ln {\frac {b}{a}}}$

### Intermediate vector math

#### Product of a vector by a scalar

Multiplication of a vector by a scalar is trivial, eg., ${\displaystyle {\vec {A}}+{\vec {A}}\equiv 2{\vec {A}}}$. Extension of this concept to multiplication by negative numbers and fractions is trivial.

#### Dot product

• ${\displaystyle {\vec {A}}\cdot {\vec {B}}=AB\cos \theta =A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}\,}$ is the dot product between two vectors separated in angle by θ.

#### CrossProductVisual

• ${\displaystyle {\vec {A}}\times {\vec {B}}={\vec {C}}}$ is the cross product of ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$. The cross product, ${\displaystyle {\vec {C}}}$ is directed perpendicular to ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$ by the right hand rule.
• ${\displaystyle |{\vec {A}}\times {\vec {B}}|=C=|AB\sin \theta |}$ wehre ${\displaystyle \theta }$ is the angle between vectors ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$.
• ${\displaystyle |{\vec {A}}\times {\vec {B}}|=C}$ is also the magnitude of the of the parallelogram defined by the vectors ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$.
• ${\displaystyle {\vec {A}}\times {\vec {B}}=0}$ if ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$ are either parallel or antiparallel.
• The unit vectors obey ${\displaystyle {\hat {x}}\times {\hat {y}}={\hat {z}}}$, ${\displaystyle {\hat {y}}\times {\hat {z}}={\hat {x}}}$, and ${\displaystyle {\hat {z}}\times {\hat {x}}={\hat {y}}}$.

#### CrossProductComponents

{{#ifeq:CrossProductComponents|CrossProductComponents| {\displaystyle {\begin{aligned}{\vec {A}}\times {\vec {B}}\Leftrightarrow &C_{x}&=&\ A_{y}B_{z}-A_{x}B_{y}\ {\text{and}}\\&C_{y}&=&\ A_{z}B_{x}-A_{x}B_{z}\ {\text{and}}\\&C_{z}&=&\ A_{x}B_{y}-A_{y}B_{x}\ .\end{aligned}}}

#### UnitVectors

• A unit vector is any vector with unit magnitude equal to one. For any nonzero vector, ${\displaystyle {\hat {V}}={\vec {V}}/V}$ is a unit vector. An important set of unit vectors is the orthonormal basis associated with Cartesian coordinates:
• ${\displaystyle \mathbf {\hat {i}} \cdot \mathbf {\hat {i}} =\mathbf {\hat {j}} \cdot \mathbf {\hat {j}} =\mathbf {\hat {k}} \cdot \mathbf {\hat {k}} =1}$
• ${\displaystyle \mathbf {\hat {j}} \cdot \mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {\hat {i}} =\mathbf {\hat {j}} \cdot \mathbf {\hat {k}} =0}$
• The basis vectors ${\displaystyle (\mathbf {\hat {i}} ,\mathbf {\hat {j}} ,\mathbf {\hat {k}} )}$ are also written as ${\displaystyle ({\hat {x}},{\hat {y}},{\hat {z}})}$, so that any vector may be written ${\displaystyle {\vec {A}}=A_{x}{\hat {x}}+A_{y}{\hat {y}}+A_{z}{\hat {z}}}$. Even more elegance is achieved by labeling the directions with integers: ${\displaystyle {\vec {A}}=A_{1}{\hat {e_{1}}}+A_{2}{\hat {e_{2}}}+A_{3}{\hat {e_{3}}}}$ ${\displaystyle =\Sigma A_{j}{\hat {e_{j}}}\,.}$

### Uniform circular motion and gravity

#### UniformCircularMotionDerive

uniform circular motion (here the Latin d was used instead of the Greek Δ

The figure depicts a change in the position and velocity of a particle during a brief time interval ${\displaystyle \Delta t}$. The distance traveled is

1.     Define ${\displaystyle \Delta \ell =|{\vec {r}}_{2}-{\vec {r}}_{1}|}$, and ${\displaystyle \Delta v=|{\vec {v}}_{2}-{\vec {v}}_{1}|}$
2.     ${\displaystyle \Delta \ell =v\Delta t}$ (rate times time equals distance).
3.     ${\displaystyle \Delta {\vec {v}}={\vec {a}}\Delta t}$ (definition of acceleration).
4.     ${\displaystyle \Delta v=a\Delta t}$ (taking the absolute value of both sides).
5.     ${\displaystyle {\frac {\Delta v}{v}}={\frac {\Delta \ell }{r}}}$ (by similar triangles). Substituting (2) and (4) yields:
6.     ${\displaystyle {\frac {a\Delta t}{v}}={\frac {v\Delta t}{r}}}$, which leads to ${\displaystyle {\frac {a}{v}}={\frac {v}{r}}}$, and therefore:
7.     ${\displaystyle a={\frac {v^{2}}{r}}}$

#### NewtonUniversalLawScalar

• ${\displaystyle F=G{\frac {mM}{r^{2}}}=mg^{*}}$ is the force of gravity between two objects, where the universal constant of gravity is G ≈ 6.674 × 10-11 m3·kg−1·s−2. If, M =M ≈ 5.97 × 1024 kg, and R =R ≈ 6.37 × 106 kg, then ${\displaystyle g^{*}}$ = g ≈ 9.8 m/s2 is the acceleration of gravity at Earth's surface.

Call with {{Physeq|transcludesection=NewtonUniversalLawScalar}}

#### NewtonKeplerThirdGeneralized

• ${\displaystyle a^{3}={\frac {(M+m)G}{4\pi ^{2}}}T^{2}}$, 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: a = ½(rmin+rmax) where rmin and rmax are the minimum and maximum separations between the moving bodies, respectively.

Call with {{Physeq|transcludesection=NewtonKeplerThirdGeneralized}}

#### FundamentalConstantsGravity

• ${\displaystyle G\,}$ ≈ 6.674×10-11 m3·kg−1·s−2 is Newton's universal constant of gravity.
• ${\displaystyle g={\frac {GM_{\oplus }}{R_{\oplus }^{2}}}}$≈ 9.8 m·s-2 where M and R are Earth's mass and radius, respectively. (g is called the acceleration of gravity).

taken from Physical constants

### calculus continuity

• ${\displaystyle A_{1}v_{1}=A_{2}v_{2}=0\rightarrow \oint {\vec {v}}\cdot {\hat {n}}dA=0\rightarrow \nabla \cdot {\vec {v}}=0\;}$ is the generalization of the continuity equation for incompressible fluid flow in three dimensions, where ${\displaystyle {\hat {n}}}$ is the outward unit vector and the integral is over the entire surface.

## Rotational issues

#### Arclength

• ${\displaystyle s=r\theta _{\mathrm {rad} }\approx r{\frac {\theta _{\mathrm {deg} }}{57.3}}\,}$   is the 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.

• ${\displaystyle 2\pi \;rad=360\;deg=1\;rev}$ relates the radian, degree, and revolution.
• ${\displaystyle f={\frac {\#\,{\text{revs}}}{\#\,{\text{secs}}}}}$ is the number of revolutions per second, called frequency.
• ${\displaystyle T={\frac {\#\,{\text{secs}}}{\#\,{\text{revs}}}}}$ is the number of seconds per revolution, called period. Obviously ${\displaystyle fT=1}$.
• ${\displaystyle \omega ={\frac {\Delta \theta }{\Delta t}}}$ is called angular frequency (ω is called omega). Obviously ${\displaystyle \omega T=2\pi }$

#### RotationalUniformAccel

• ${\displaystyle \theta ={\frac {s}{r}}}$ is the angle (in radians) where s is arclength and r is radius.
• ${\displaystyle \omega ={\frac {d\theta }{dt}}}$ (or Δθ/Δt), called angular velocity is the rate at which θ changes.
• ${\displaystyle \alpha ={\frac {d\omega }{dt}}}$ (or Δω/Δt), called angular acceleration is the rate at which ω changes.

The equations of uniform angular acceleration are:

• ${\displaystyle \theta (t)=\theta _{0}+\omega _{0}t+{\frac {1}{2}}\alpha t^{2}}$
• ${\displaystyle \omega =\omega _{0}+\alpha t}$
• ${\displaystyle \omega ^{2}=\omega _{0}^{2}+2\alpha \left(\theta -\theta _{0}\right)}$
• ${\displaystyle \theta -\theta _{0}={\frac {\omega _{0}+\omega }{2}}={\bar {\omega }}t}$   (Note that ${\displaystyle \omega _{\mathrm {ave} }={\bar {\omega }}={\frac {\omega _{0}+\omega }{2}}}$ only if the angular acceleration is uniform)

#### AngularMotionEnergyMomentum

• ${\displaystyle KE_{rot}={\frac {1}{2}}\sum m_{n}v_{n}^{2}={\frac {1}{2}}\sum m_{n}(\omega r_{n})^{2}={\frac {1}{2}}I\omega ^{2}}$ is the kinetic of a rigidly rotating object, where
• ${\displaystyle I=\sum m_{n}r_{n}^{2}}$ is the moment of inertia, equal to ${\displaystyle MR^{2}}$ 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 ${\displaystyle {\frac {1}{2}}MR^{2}}$.
• The generalization of F=ma for rotational motion through a fixed axis is τ = Iα , where τ (called tau) is torque. If the force is perpendicular to r, then τ = r F
• The total angular momentum, Lnet = Σ Iω is conserved if no net external torque is acting on a system.

## Field theories

#### GausslawSimple

• ${\displaystyle \varepsilon _{0}\int {\vec {E}}\cdot {\vec {dA}}=Q_{encl}}$ is Gauss's law for the surface integral of the electric field over any closed surface, and ${\displaystyle Q_{encl}}$ is the total charge inside that surface. The vacuum permittivity is ε0≈ 8.85 × 10−12.
##### VectorMagneticForce
• ${\displaystyle F=qv\sin \theta \ }$ is the force on a particle with charge q moving at velocity v with in the presence of a magnetic field B. The angle between velocity and magnetic field is θ and the force is perpeduclar to both velocity and magnetic field by the right hand rule.
• ${\displaystyle {\vec {F}}=q{\vec {v}}\times {\vec {B}}}$ expresses this result as a cross product.
• ${\displaystyle \Delta {\vec {F}}=I\Delta {\vec {\ell }}\times {\vec {B}}}$ is the force a straight wire segment of length ${\displaystyle \Delta \ell }$ carrying a current, I.
• ${\displaystyle {\vec {F}}=I\sum \Delta \ell \times {\vec {B}}\ }$ expresses thus sum over many segments to model a wire.
• CALCULUS: In the limit that ${\displaystyle \Delta \ell \rightarrow 0}$ we have the integral, ${\displaystyle {\vec {F}}=I\oint d\ell \times {\vec {B}}}$.
##### DefineMagneticFieldVector

{{#ifeq:DefineMagneticFieldVector|DefineMagneticFieldVector|

• ${\displaystyle \Delta {\vec {B}}={\frac {\mu _{0}}{4\pi }}{\frac {I\Delta {\vec {\ell }}\times {\hat {r}}}{|r|^{2}}}}$ is the contribution to the field due to a short segment of length ${\displaystyle \Delta \ell }$ carrying a current I, where the displacement vector r points from the source point to the field point.