Physics equations/Equations Sandbox

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This page is a messy desk

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[edit]

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[edit]

TrigonometryTriangle.svg
  • (where A is the angle shown)
InverseTrigFunctions[edit]
  • and defines the arcsine function as the inverse of the sine. Similarly, is called the arctangent, or the inverse tangent, and is called arccosine, or the inverse cosine and so forth. In general, and for any function and its inverse. Complexities occur whenever the inverse is not a true function; for example, since , the inverse is multi-valued:

First year calculus[edit]

CalculusBasic[edit]
  • ,     and the derivative is in the limit that
  • is the chain rule.
  • ,      ,     ,     ,     
  • expresses the fundamental theorem of calculus.

RiemannSum[edit]

Riemann sum convergence.png
  • 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[edit]

  • is the Riemann sum representation of the integral of f(x) from x=a to x=b.


IntegrateFundamentalTheorem[edit]

  • The fundamental theorem of calculus allows us to construct integrals from known derivatives:

Intermediate vector math[edit]

Product of a vector by a scalar[edit]

Multiplication of a vector by a scalar is trivial, eg., . Extension of this concept to multiplication by negative numbers and fractions is trivial.

Dot product[edit]

  • is the dot product between two vectors separated in angle by θ.

CrossProductVisual[edit]

  • is the cross product of and . The cross product, is directed perpendicular to and by the right hand rule.
  • wehre is the angle between vectors and .
  • is also the magnitude of the of the parallelogram defined by the vectors and .
  • if and are either parallel or antiparallel.
  • The unit vectors obey , , and .

CrossProductComponents[edit]

{{#ifeq:CrossProductComponents|CrossProductComponents|


UnitVectors[edit]

  • A unit vector is any vector with unit magnitude equal to one. For any nonzero vector, is a unit vector. An important set of unit vectors is the orthonormal basis associated with Cartesian coordinates:
  • The basis vectors are also written as , so that any vector may be written . Even more elegance is achieved by labeling the directions with integers:

Uniform circular motion and gravity[edit]

UniformCircularMotionDerive[edit]

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 . The distance traveled is

  1.     Define , and
  2.     (rate times time equals distance).
  3.     (definition of acceleration).
  4.     (taking the absolute value of both sides).
  5.     (by similar triangles). Substituting (2) and (4) yields:
  6.     , which leads to , and therefore:
  7.    

NewtonUniversalLawScalar[edit]

  • 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 = g ≈ 9.8 m/s2 is the acceleration of gravity at Earth's surface.

Call with {{Physeq|transcludesection=NewtonUniversalLawScalar}}

NewtonKeplerThirdGeneralized[edit]

  • , 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[edit]

  • ≈ 6.674×10-11 m3·kg−1·s−2 is Newton's universal constant of gravity.
  • ≈ 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[edit]

  • is the generalization of the continuity equation for incompressible fluid flow in three dimensions, where is the outward unit vector and the integral is over the entire surface.

Rotational issues[edit]

Arclength[edit]

Radians Angle Definition.svg
  •   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.


RadianDegreeRevolutionFreqOmegaPeriod[edit]

  • relates the radian, degree, and revolution.
  • is the number of revolutions per second, called frequency.
  • is the number of seconds per revolution, called period. Obviously .
  • is called angular frequency (ω is called omega). Obviously


RotationalUniformAccel[edit]

Radians Angle Definition.svg
  • is the angle (in radians) where s is arclength and r is radius.
  • (or Δθ/Δt), called angular velocity is the rate at which θ changes.
  • (or Δω/Δt), called angular acceleration is the rate at which ω changes.

The equations of uniform angular acceleration are:

  •   (Note that only if the angular acceleration is uniform)


AngularMotionEnergyMomentum[edit]

  • is the kinetic of a rigidly rotating object, where
  • is the moment of inertia, equal to 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 .
  • 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[edit]

GausslawSimple[edit]

  • is Gauss's law for the surface integral of the electric field over any closed surface, and is the total charge inside that surface. The vacuum permittivity is ε0≈ 8.85 × 10−12.
VectorMagneticForce[edit]
  • 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.
  • expresses this result as a cross product.
  • is the force a straight wire segment of length carrying a current, I.
  • expresses thus sum over many segments to model a wire.
  • CALCULUS: In the limit that we have the integral, .
DefineMagneticFieldVector[edit]
Magnetic field element (Biot-Savart Law).svg

{{#ifeq:DefineMagneticFieldVector|DefineMagneticFieldVector|

  • is the contribution to the field due to a short segment of length carrying a current I, where the displacement vector r points from the source point to the field point.