Fundamental Physics/Formulas

Force

Force is a physical quantity that interacts with matter to perform a task

Force	Definition	Notation	Formula
Motion Force	Force that sets matter in motion	$F_{a}$	$ma$
Impulse	Force that sets a mass in motion	$F_{p}$	${\frac {p}{t}}$
Opposition force	Force that opposes the interacting force with matter	$F_{-}$	$-F$
Pressure force	Force that acts on surface's area	$F_{A}$	${\frac {F}{A}}$
Friction Force	Force opposes matter movement on a surface
Elastic Force	Force that restores matter equilibrium	$F_{x}$	$-kx$
Circulation Force	Force sets matter in circular motion	$F_{r}$	$\omega r$
Centripetal Force	Force that sets matter moves out of circulation motion	$F_{v}$	$m{\frac {v^{2}}{r}}$
Centripugal Force	Force that sets matter moves in of circulation motion	$F_{\omega }$	$m{\frac {\omega r}{t}}$
Electrostatic Force	Attraction force of 2 different polaity charges	$F_{q}$	$m{\frac {q_{+}q_{-}}{r^{2}}}$
Electromotive Force	Force sets electric charge in motion	$F_{E}$	$qE$
Electromagnetomotive Force	Force that sets moving electric charge to move perpendicular to the intial travel direction	$F_{B}$	$\pm qvB$
Electromagnetic Force	Summation of 2 force Electromotive force and Electromagnetomotive force	$F_{EB}$	$q(E\pm vB)$

Motion

Movement of matter from one place to another place caused by a force

Uniform linear motion

Motion that follows straight path of constant speed for example horizontal uniform linear motion, vertical uniform linear motion,

	Notation	Horizontal uniform linear motion	Vertical uniform linear motion	Inclined uniform linear motion
Distance	s	$s$	$h$	$(v_{o}+a\Delta t)t=s_{o}+at\Delta t=s_{o}+v\Delta t$
Time	t	$t$	$t$	$t$
Speed	v	${\frac {s}{t}}$	${\frac {h}{t}}$	$v_{o}+a\Delta t$
Acceleration	a	${\frac {v}{t}}$	$g={\frac {GM}{h^{2}}}$	${\frac {v-v_{o}}{t-t_{o}}}$
Force	F	$ma$	$mg$	$m{\frac {\|\Delta v}{\Delta t}}$
Work	W	$Fs$	$mgh$	$Ft(v_{o}+a\Delta t)$
power	E	${\frac {W}{t}}$	${\frac {mgh}{t}}$	$F(v_{o}+a\Delta t)$

Non uniform curve motion

Motion that does not follow a straight path

	Notation	Formulas
Distance	s(t)	$\int v(t)dt$
Time	t	$t$
Speed	v(t)	$v(t)$
Acceleration	a(t)	${\frac {d}{dt}}v(t)$
Force	F(t)	$m{\frac {d}{dt}}v(t)$
Work	W(t)	$F\int v(t)dt$
Energy	E(t)	${\frac {F}{t}}\int v(t)dt$

Periodic motion

Motion that keeps repeat itself over a period of time

Circular motion

Motion that follows a circular path

Full circle motion

Motion that completes a circle

	Notation	Formulas
Distance	s	$2\pi r$
Time	t	$t$
Speed	v	${\frac {2\pi r}{t}}=\omega r$
Angular speed	ω	$\omega {\frac {v}{r}}$
Acceleration	a	$vr$
Force	F	$mvr$
Work	W	$mvr2\pi$
Energy	E	${\frac {mvr2\pi }{t}}=pr\omega$

Circle's arc motion

Motion that follows an arc of a circle

	Notation	Formulas
Distance	s	$\theta r$
Time	t	$t$
Speed	v	${\frac {\theta r}{t}}$
Angular speed	ω	$\omega {\frac {v}{r}}$
Acceleration	a	${\frac {\theta r}{t^{2}}}$
Force	F	$m{\frac {\theta r}{t^{2}}}$
Work	W	$p{\frac {\theta r}{t}}$
Energy	E	$p{\frac {\theta r}{t^{2}}}$

Wave

Mathematical formula

A period motion of a sinusoidal motion

	Notation	Formulas
Distance	s	$\lambda$
Time	t	$t$
Speed	v	${\frac {\lambda }{t}}$
Angular Speed	ω	$\lambda f$
Frequency	f	${\frac {1}{t}}$
Wave equation	f"(t)	$-\omega f(t)$
Wave function	f(t)	$ASin\omega t$

Sinusoidal wave

Mathematiccaly sinusoidal can be represent by a wave equation and a wave function

Wave equation

f^{''}(t)=-\omega f(t)

Wave function

f(t)=ASin\omega t

Oscillation

Spring's Oscillation

Spring's Oscillation	Symbol	Horizontal	Vertical
Spring's Oscillation equation	$f^{''}(t)$	$m{\frac {d}{dt}}x(t)=-kx(t)$	$m{\frac {d}{dt}}y(t)=-ky(t)$
Spring's wave function	$f(t)$	$x(t)=ASin\omega t$	$y(t)=ASin\omega t$
Angular speed	ω	${\sqrt {\frac {k}{m}}}$	${\sqrt {\frac {k}{m}}}$

Pendulum Oscillation

The differential equation which represents the motion of a simple pendulum is

{\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{\ell }}\sin \theta =0

Eq. 1

where $g$ is acceleration due to gravity, $l$ is the length of the pendulum, and $θ$ is the angular displacement.

Electric Oscillation

Electric's Oscillation	RLC series at equilibrium	RLC series at resonance	LC series at equilibrium	LC series at resonance

Wave equation	${\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d}{dt}}i(t)-\beta i(t)$	$Z_{L}=-Z_{C}.Z_{t}=R$	${\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{T}}$	$Z_{L}=-Z_{C}$
Wave function	$x(t)=A(\alpha )Sin\omega t$	$i(\omega =0)=0$ $i(\omega =\omega _{o})={\frac {v}{R}}$ $i(\omega =0))=0$	$i(t)=ASin\omega t$	$V_{L}=-V_{C}$
	$A(\alpha )=Ae^{-\alpha t}$
	$\omega ={\sqrt {\beta -\alpha }}$	${\sqrt {\frac {1}{T}}}$	${\sqrt {\frac {1}{T}}}$	${\sqrt {\frac {1}{T}}}$
	$\beta ={\frac {1}{T}}$
	$T=LC$	$LC$	$LC$	$LC$
	$\alpha =\beta \gamma$
	$\gamma =RC$

Electromagnetic Oscillation

Electromagnetic's Oscillation	Formula
Wave equation	$\nabla ^{2}E(t)=-\omega E(t)$ $\nabla ^{2}B(t)=-\omega B(t)$
Wave function	$E(t)=ASin\omega t$ $B(t)=ASin\omega t$
Angular speed	$\omega =\lambda f={\sqrt {\frac {1}{T}}}=C$
Time constant	$T=\mu \epsilon$
Wave hape

Momentum

Momentum is defined as motion of a mass at a speed caused by a force

F=ma=m{\frac {v}{t}}

Moment

p=mv=Ft

Momentum of a mass in motion

	Notation	Formulas
Mass	m	$m$
Speed	v	$v$
Moment	p	$mv$
Impulse(Force)	F	${\frac {p}{t}}$
Work	W	$pv$
Energy	E	$pa$

Momentum of a relativistic mass in motion

Momentum refers to movement of a mass at a speed relative to the speed of light

	Notation	Formulas
Mass	m	$m_{0}(\gamma -1)$
Speed	v	$\gamma ={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}$
Momentum	p	$mv$
Force	F	${\frac {p}{t}}$
Work	W	$pv$
Energy	E	$pa$

Momentum of a massless quanta in motion

Momentum refers to movement of a mass at a speed equals to the speed of light

	Notation	Formulas
Speed	v	$\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f$
Work	W	$pv=pC=p\lambda f=hf$
Quanta	h	$p\lambda$
Momentum	p	${\frac {h}{\lambda }}$
Wavelength		${\frac {h}{p}}={\frac {C}{f}}$

Momentum of electric charge

F_{B}=F_{r}

QvB=m{\frac {v^{2}}{r}}

r={\frac {mv}{QB}}

v={\frac {Q}{m}}Br

Momentum of free electron

Absorbing photon, electron becomes free electron travels outward off the atom's circular orbit

hf=hf_{o}+{\frac {1}{2}}mv^{2}

v={\sqrt {{\frac {2}{m}}(hf-hf_{o})}}={\sqrt {{\frac {2}{m}}(nhf_{o})}}

Momentum of a bind electron

Releasing photon, electron becomes free electron travels inward off the circle orbit

nhf=mvr2\pi

r={\frac {1}{2\pi }}{\frac {nhf}{mv}}

v={\frac {1}{2\pi }}{\frac {nhf}{mr}}

n=2\pi r{\frac {mv}{hf}}

Heat

Temperature

Temperature is the measurement of heat's intensity . Temperature is denoted as T measured in degree o

Temperature measurements

There are 3 temperature measuring system

Degree Celcius, $^{o}C$
Degree Kevin , $^{o}K$
Degree Fahrenheit , $^{o}F$

Conversion between systems of temperature can be done as shown below

Convert from	to '	Formulas
Degree Fahrenheit	Celsius	°C = (°F – 32) / 1.8
Degree Celsius	Fahrenheit	°F = °C × 1.8 + 32
Degree Fahrenheit	Kelvin	K = (°F – 32) / 1.8 + 273.15

Standard temperatures

Standard temperature	Value
Room temperature	$25^{o}C$
Boiling temperature	$100^{o}C$
Frozen temperature	$0^{o}C$

Heat and matter

Heat and matter interact to create Heat transfer of three phases Heat conduction, Heat convection and Heat radiation

Heat absortion of matter

Matter of dark color absorbs more heat energy than matter of bright color
Matter of thin dimension absorbs more heat energy than matter of thich dimension

For example

Dark and thin clothes dry faster than bright and thick clothes

Heat Transfer

A process of heat interaction with matter through 3 phases of

Heat conduction

Matter change its temperature when it in contact with heat energy

\Delta T=T_{2}-T_{1}

E=mc\Delta T=mc(T_{2}-T_{1})

Heat convection

Matter absorbs heat energy to its maximum level and gives off visible light

E=hf_{o}

f_{o}={\frac {C}{\lambda _{o}}}

Heat radiation

Matter is at its saturation . Matter is no longer absorbs heat energy and use the excess energy to release electron of its atom

E=E_{o}+E_{e}=hf_{o}+{\frac {1}{2}}mv^{2}=hf

v={\sqrt {{\frac {2}{m}}(hf-hf_{o})}}={\sqrt {{\frac {2}{m}}(nhf_{o})}}

Heat flow

Heat flows between 2 objects of different mass follows heat flow rule that heat flows from high temperature to low temperature

Heat energy absorb by mass 1

Q_{1}=m_{1}cT_{1}

Heat energy absorb by mass 2

Q_{2}=m_{2}cT_{2}

Direction of heat flow

T_{1}>T_{2}

Q_{1}-Q_{2}=m_{1}cT_{1}-m_{2}cT_{2}=(m_{1}-m_{2})c(T_{1}-T_{2})

T_{2}>T_{1}

Q_{2}-Q_{1}=m_{2}cT_{2}-m_{1}cT_{1}=(m_{2}-m_{1})c(T_{2}-T_{1})

Light

Measurement speed of visible light

In vacuum	$v=C=299999000m/s$	By Michael Morrison
In air, as electromagnetic radiation	$v=\omega ={\sqrt {\frac {1}{\mu _{o}\epsilon _{o}}}}=C=\lambda f=3\times 10^{8}m/s$	By James Clerk Maxwell
In liquid	$v={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}m/s$	By Lorentz

Visible light

Characteristics

Visible light travels at a constant speed in vacuum and in air which has a value

C=3\times 10^{8}

m/s

Travels as Electromagnetic wave of wavelength

\lambda =400-700nm

m

Of Threshold frequency

f_{o}={\frac {C}{f_{o}}}={\frac {3\times 10^{8}}{400-700nm}}

Hz

Composite colors

Visible light passes through prism decomposes itself into its composites color light of 6 colors

Colors	Wavelength	Angle of refraction
Red
Orange
Yellow
Green
Blue
Violet

Light and matter

Light and matter interacts with each other to create the following effects

Reflection
Refraction
Diffraction
Dispersion
Interference

Sound

Measurement speed of sound

Material medium	Value
In air	$v=$
In water	$v=$
In solid	$v=$

Audible sound

Sound spectrum

Audible sound to human's ears is in the frequency range 20Hz - 20KHz . Sound above 20KHz is called Ultrasound . Sound below 20Hz is called Infrasound

Audible sound wave in air

In air, audible sound travels as wave of thick and thin columns of air

Medium	Speed	Frequency	Wavelength
In air	$v=300m/s$	$f=20Hz-20KHz$	$\lambda ={\frac {300m/s}{20Hz-20KHz}}$

Sound and matter

Sound and matter interacts with each other to create the following effects

Relection
Refraction
Diffraction
Interference

Electricity

Electricity and a straight line conductor

Characteristis	Symbols	Formulas
Voltage	$V$	$V$
Current	$I$	$I$
Resistance	$R$	$R={\frac {V}{I}}$
Conductance	$G$	$G={\frac {I}{V}}={\frac {1}{R}}$
Electromagnet's Field strength	$B$	$B=LI={\frac {N\mu }{l}}I$
Resistance change	$R(T)$	$E(T)=R_{o}+nT$ $E(T)=R_{o}e^{nT}$
Power generated	$E_{V}$	$E_{V}=IV$
Power loss	$E_{R}$	$E_{R}=I^{2}R(T)$
Power transmitted	$E$	$E=E_{E}-E_{R}$

Electromagnet

For a straight line conductor		$B=LI={\frac {\mu }{2\pi r}}I$
For a circular loop made from straight line conductor		$B=LI={\frac {\mu }{2r}}I$
For a coil of N circular loops made from straight line conductor		$B=LI={\frac {N\mu }{l}}I$

Electric Oscillation

Electric's Oscillation	RLC series at equilibrium	RLC series at resonance	LC series at equilibrium	LC series at resonance

Wave equation	${\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d}{dt}}i(t)-\beta i(t)$	$Z_{L}=-Z_{C}.Z_{t}=R$	${\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{T}}$	$Z_{L}=-Z_{C}$
Wave function	$x(t)=A(\alpha )Sin\omega t$	$i(\omega =0)=0$ $i(\omega =\omega _{o})={\frac {v}{R}}$ $i(\omega =0))=0$	$i(t)=ASin\omega t$	$V_{L}=-V_{C}$
	$A(\alpha )=Ae^{-\alpha t}$
	$\omega ={\sqrt {\beta -\alpha }}$	${\sqrt {\frac {1}{T}}}$	${\sqrt {\frac {1}{T}}}$	${\sqrt {\frac {1}{T}}}$
	$\beta ={\frac {1}{T}}$
	$T=LC$	$LC$	$LC$	$LC$
	$\alpha =\beta \gamma$
	$\gamma =RC$

Electromagnetism

Electric current interacts with magnetic material to generate Magnetic field

Electromagnetic Field

Electromagnetic Field	Definition	symbol	formula
Straight line conductor	The magnetic field is made up of circular magnetic circles rotate counterclockwise or clockwise direction	$B$	$B=LI={\frac {\mu }{2\pi r}}I$
Circular loop conductor	The magnetic field is made up of circular magnetic circle around a point charge that moves around the circular loop	$B$	$B=LI={\frac {\mu }{2r}}I$
Coil of N circular loop conductor	The magnetic field is made up of elliptic magnetic lines running from North pole [N] to South pole [S] With North pole [N] corresponds to positive polarity (+) and South pole [S] corresponds to negative polarity (-)	$B$	$B=LI={\frac {N\mu }{l}}I$ $\phi =-NB=-NLI$ $H={\frac {B}{\mu }}={\frac {\phi }{N\mu }}$

Electromagnetic Induction

For a Faraday's coil of N circular loops . The magnetic field is made up of elliptic magnetic lines running from North pole [N] to South pole [S]

Magnetic Potential Difference	$V_{L}$	${\frac {dB}{dt}}=L{\frac {dI}{dt}}$
Induced Magnetic Voltage	$\epsilon$	$-{\frac {d\phi }{dt}}=-N{\frac {dB}{dt}}=-NL{\frac {dI}{dt}}$

Electromagnetization

Process of generating permanent electromagnet from a magnetic material placed in the turns of the magnetic coil

B=LI={\frac {N\mu }{l}}I

H={\frac {B}{\mu }}={\frac {\phi }{N\mu }}

Maxwell's Electromagnetization Vector Equation

\nabla \cdot D=\rho

\nabla \times E=-\nabla B

\nabla \cdot B=0

\nabla \times H=J+\nabla B

Electromagnetic Wave Oscillation

Electromagnetic Wave Vectore Equation	$\nabla \cdot E=0$ $\nabla \times E={\frac {1}{T}}E$ $\nabla \cdot B=0$ $\nabla \times B={\frac {1}{T}}B$ $T=\mu \epsilon$
Electromagnetic Wave Equation	$\nabla ^{2}E=-\omega E$ $\nabla ^{2}B=-\omega B$
Electromagnetic Wave Function	$E=ASin\omega t$ $B=ASin\omega t$ $\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f$
Electromagnetic Wave Radiation	$v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f$ $E=pv=pC=p\lambda f=hf$ $h=p\lambda$

Electromagnetic wave radiation

Electromagnetic wave radiation is generated from Electromagnetic wave propagates at speed of visible light

v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f

E=pv=pC=p\lambda f=hf

h=p\lambda

Electromagnetic Wave Radiation States

Radiant Photon

Electromagnetic Wave Radiation of Radian Photon just like visible light perceive by human eyes

v=\omega _{o}={\sqrt {\frac {1}{\mu _{o}\epsilon _{o}}}}=C=\lambda _{o}f_{o}

E=pv=pC=p\lambda _{o}f=hf_{o}

h=p\lambda _{o}

Non radiant photon

Electromagnetic Wave Radiation of Non radiant photon that can free electron off matter's atom

v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f

E=pv=pC=p\lambda f=hf

h=p\lambda

Heinseinberg's Uncertainty Principle

Photon can only exist in one state at a time

\Delta p\Delta \lambda ={\frac {1}{2}}{\frac {h}{2\pi }}={\frac {\hbar }{2}}

\hbar ={\frac {h}{2\pi }}

Quantization

Photon is energy of a quantity that process no mass known as Quanta travels at speed of light

v=C=\lambda f

W=pv=pC=p\lambda f=\lambda f=hf

h=p\lambda

Quanta's Wave-Particle Duality

Quanta processes Wave-Particle Duality . Sometimes, behave like wave of wavelength λ . Sometimes, behave like particle of a momentum p

Wave like .

\lambda ={\frac {h}{p}}

Particle like .

p={\frac {h}{\lambda }}={\frac {C}{f}}

Photon cannot exist in 2 states at the same time . The chances of finding photon at any one state is one half . This is the uncertainty principle proposed by schroduinger which can be expressed mathematicaaly as

\Delta p\Delta \lambda ={\frac {1}{2}}{\frac {h}{2\pi }}={\frac {h}{4\pi }}={\frac {\hbar }{2}}

Photon's radiation spectrum

Photon has a spectrumand are found in the frequenct bands below

VF

UVF

X

γ

Quanta

Mathematical formula

h=p\lambda ={\frac {E}{C}}\lambda

Wave-Particle duality

Quanta processes Wave-Particle duality

Some time Quanta behaves as a particle of a momentum

p={\frac {h}{\lambda }}

Some other time Quanta behaves as a wave of wavelength

\lambda ={\frac {h}{p}}={\frac {C}{f}}

Relativity

Relativistic mass in motion

	Symbol	Mathematical formula
Speed	$v$	${\sqrt {1-{\frac {v^{2}}{C^{2}}}}}$
Mass	$m$	$m=m_{o}(v-1)$
Moment	$p$	$mv$
Energy	$E$	$pv$

Massless quanta in motion

	Symbol	Mathematical formula
Speed	$v$	$C=\lambda f$
Energy	$E$	$pv=pC=p\lambda f=hf$
Quanta	$h$	$h=p\lambda$
Moment	$p$	${\frac {h}{\lambda }}$
Wave length	$\lambda$	${\frac {C}{f}}={\frac {h}{p}}$

Mass change

At speed relative to speed of visible light

v=\gamma ={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}

.

M=m_{o}(\gamma -1)

At speed equals to speed of visible light

v=C=\lambda f

.

h=p\lambda ={\frac {E}{C}}\lambda

Einstein's relativity theory

Newton's motion's speed

When matter travels at any speed less than speed of visible light, matter does not change its mass

Speed of motion	Mathematical formula of speed	Mathematical formula of mass
At any speed less than speed of visible light	$v=v$	$m=m$

Einstein's motion's speed

When matter travels at a speed relative to or equal to the speed of visible light . Matter changes its mass

Speed of motion	Mathematical formula of speed	Mathematical formula of mass
At speed relative to speed of visible light	$v={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}$	$m=m_{o}(v-1)$
At speed equals to speed of visible light	$v=C=\lambda f$	$h=p\lambda ={\frac {E}{C}}\lambda$

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Non uniform curve motion

Mathematical formula

Sinusoidal wave

Spring's Oscillation

Temperature measurements

Standard temperatures

Heat and matter

Heat absortion of matter

Heat Transfer

Heat conduction

Heat convection

Heat radiation

Heat flow

Measurement speed of visible light

Visible light

Characteristics

Composite colors

Light and matter

Measurement speed of sound

Audible sound

Sound spectrum

Audible sound wave in air

Sound and matter

Electricity and a straight line conductor

Electromagnetic Field

Electromagnetic Wave Oscillation

Electromagnetic Wave Radiation States

Radiant Photon

Non radiant photon

Heinseinberg's Uncertainty Principle

Quantization

Quanta's Wave-Particle Duality

Electromagnetic radiation and matter

Photon's characteristics

Photon's states

Photon's radiation spectrum

Mathematical formula

Wave-Particle duality

Relativistic mass in motion

Massless quanta in motion

Mass change

Newton's motion's speed

Einstein's motion's speed