OpenStax College Physics/Equations

http://cnx.org/content/col11406/latest


 * Wright State University Lake Campus/2018-1/Phy1120/Studyguide

Lifted from https://cnx.org/contents/Ax2o07Ul@9.98:pFeekPiU@17/Preface




 * =====Introduction=====




 * =====Kinematics=====

 
 * =====Two-Dimensional Kinematics=====

 
 * =====Dynamics: Force and Newton's Laws of Motion=====




 * =====Further Applications of Newton's Laws: Friction, Drag, and Elasticity=====

 
 * =====Uniform Circular Motion and Gravitation=====




 * =====Work, Energy, and Energy Resources=====




 * =====Linear Momentum and Collisions=====



<section begin=Statics and Torque/>
 * =====Statics and Torque=====

<section end=Statics and Torque/>

<section begin=Rotational Motion and Angular Momentum/>
 * =====Rotational Motion and Angular Momentum=====

<section end=Rotational Motion and Angular Momentum/>

<section begin=Fluid Statics/>
 * =====Fluid Statics=====

<section end=Fluid Statics/>

<section begin=Fluid Dynamics and Its Biological and Medical Applications/>
 * =====Fluid Dynamics and Its Biological and Medical Applications=====

<section end=Fluid Dynamics and Its Biological and Medical Applications/>

<section begin=Temperature, Kinetic Theory, and the Gas Laws/>
 * =====Temperature, Kinetic Theory, and the Gas Laws=====

<section end=Temperature, Kinetic Theory, and the Gas Laws/>

<section begin=Heat and Heat Transfer Methods/>
 * =====Heat and Heat Transfer Methods=====

<section end=Heat and Heat Transfer Methods/>

<section begin=Thermodynamics/>
 * =====Thermodynamics=====

<section end=Thermodynamics/>

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<section begin=Oscillatory Motion and Waves/>

Oscillatory Motion and Waves
<section end=Oscillatory Motion and Waves/>
 * $$F=-kx$$, is Hooke's law, where F is the restoring force, k is the force constant, and x is the displacement from equilibrium.
 * $$PE_{el}=\tfrac 1 2 kx^2$$ is the potential energy stored in the deformation of a spring or elastic system.
 * $$fT=1$$ where f is the frequency and T is period. It is convenient to define omega as, $$\omega\equiv 2\pi/T = \sqrt{k/m}$$, for a spring-mass system. For a simple pendulum, $$\omega = \sqrt{g/L}$$.
 * $$x=X\cos(\omega t),\quad v=-\omega X\sin(\omega t),\quad a=-\omega^2 X \cos(\omega t)$$ describe the position, velocity, and acceleration of simple harmonic motion, respectively.
 * $$f\lambda = v_{w}$$ is the wave velocity, where &lambda; is wavelength.
 * The node is a point of zero motion and the antinode is a point of maximum motion of a standing wave.
 * $$f_B=|f_1-f_2|$$ is the beat frequency when two waves of slightly different frequency are superimposed.

<section begin=Physics of Hearing/>

Physics of Hearing
<section end=Physics of Hearing/>
 * $$v_w=f\lambda\approx(\text{331m/s})\sqrt{T/\text{273K}}$$ is the speed of sound in air, where T is the absolute temperature.
 * $$I=P/A=(\Delta p)^2/(2\rho v_w)$$ is intensity of sound, where P is the power passing through an area A, &Delta;p is the pressure amplitude, and &rho; is the density of the medium.
 * $$\beta \text{(dB)}= 10\log_{10}(I/I_0)$$ is the intensity level in decibels, where I0 = 10-12W/m2 is the threshold level of hearing.
 * $$f_{obs}=f_s\left(\frac{v_w}{v_w\pm v_s}\right)$$ is the observed Doppler shifted frequency for a stationary observer when the source of frequency fs is moving at speed $$v_s$$. The (+/-) sign refers to motion (away/towards) the observer.
 * $$f_{obs}=f_s\left(\frac{v_w\pm v_{obs}}{v_w}\right)$$ is the frequency perceived by an observer moving at speed $$v_s$$ with respect to a stationary source of frequencyfs. The (+/-) sign refers to motion (towards/away) from the source.
 * $$f_n=n\frac{v_w}{4L}$$ is the resonant frequency of the n-th mode of a standing sound wave in a tube that is closed at one end. The mode numbers are $$n=1,3,5...$$.
 * $$f_n=n\frac{v_w}{2L}$$ with $$n=1,2,3...$$ are the resonant modes for a tube open at both ends.
 * $$a=(Z_2-Z_1)^2/(Z_1+Z_2)^2$$ is the intensity reflection coefficient, which is the ratio of the intensity of the reflected wave to that of the incident wave at a boundary between two media. The acoustical impedance is $$Z=\rho v_w$$.

<section begin=Electric Charge and Electric Field/>

Electric Charge and Electric Field
The charge of the electron is qe = &minus;e, where the fundamental charge is e&asymp;1.6 x 10&minus;19 C. (C&equiv;Coulombs is the SI unit of charge). <section end=Electric Charge and Electric Field/>
 * $$F=k\frac{|q_1q_2|}{r^2}$$ is Coulomb's law for the force between two point charges q1 and q2 separated by a distance r. Coulomb's constant is k &asymp; 8.99x109 N m2 C&minus;2.
 * $$\vec F = q_t\vec E$$ if the force on a test charge qt due to the electric field $$\vec E=\Sigma \vec E_j$$ at the location of the test charge. The vector sum is over $$n=1\rightarrow N$$ source charges, and the magnitude of each contribution to the field is $$|\vec E_n| = kq_n/r_{nt}^2$$ where qn is the source charge and rnt is the distance from the source point to the test charge.

<section begin=Electric Potential and Electric Field/>

Electric Potential and Electric Field
<section end=Electric Potential and Electric Field/> <section begin=Electric Current, Resistance, and Ohm's Law/>
 * $$\Delta\text{PE}=q\Delta V = q(V_B-V_A)$$ is the potential energy associated with moving a charge q from a point where the electric potential is VA to a point where it is VB.
 * $$1eV\approx 1.6\times 10^{-19}\text{ Joules}$$
 * $$E = -\Delta V/\Delta s$$ is the component of the electric field parallel to a small displacement $$\Delta \vec{s}$$ (the electric field points from high to low voltage, and &Delta;V=Ed if one moves a distance d along a uniform electric field E.)
 * $$V=kQ/r$$ is the electric potential at a distance r from a point charge Q.
 * $$Q=CV$$ is the charge stored by a capacitor of capacitance C with a voltage drop V across its terminals. For a parallel plate capacitor, $$C=\varepsilon_0 A/d$$ for plates of area A separated by a (small) distance d.  The permittivity of free space is &epsilon;0 = 8.85&times;10&minus;12F/m.  If a dielectric fills the gap, replace &epsilon;0 by &kappa;&epsilon;0
 * $$\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\dots$$ is the total capacitance in series.
 * $$C_p=C_1+C_2+C_3+\dots$$ is the total capacitance in parallel.
 * $$E_{cap}=\tfrac 1 2 QV =\tfrac 1 2 CV^2 =Q^2/(2C)$$ is the energy stored in a capacitor.

Electric Current, Resistance, and Ohm's Law
The electric current, in amperes (A&equiv;C/s) is the rate at which charge Q flows:  $$I=\Delta Q/\Delta t$$ <section end=Electric Current, Resistance, and Ohm's Law/>
 * $$I=nqAv_d$$ relates the current through the wire to the density of carriers n, charge of each carrier q, cross-sectional area A, and carrier drift velocity vd.
 * $$V=IR$$ is Ohm's law, where 1&Omega;=1V/A. $$R=\rho L/A$$ where &rho; is resistivity, L is length, and A is area. For changes in temperature T, both R and &rho; typically increase linearly with &Delta;T, as $$\rho=\rho_0 + \alpha\Delta T$$, where &alpha; is the temperature coefficient of resistivity.
 * $$P=IV=I^2R=V^2/R$$ is electrical power (energy divided by time). For alternating current, the average power is $$P_\text{ave}=I_\text{rms}V_\text{rms}=I_\text{rms}^2R=V_\text{rms}^2/R$$, where rms  denotes root mean square). If $$X(t)=X_0\sin(2\pi ft)$$, $$X_\text{rms}=\tfrac{1}{\sqrt 2} X_0$$ where X0 is the peak value and f is frequency.

<section begin=Circuits and DC Instruments/>

Circuits and DC Instruments
If resistors are in series, the equivalent resistance is $$R_s=R_1+R_2+R_3+\ldots$$. If they are in parallel, $$R_{eq}^{-1} = R_1^{-1}+R_2^{-1}+R_3^{-1}+\ldots$$ <section end=Circuits and DC Instruments/>
 * $$V_\text{terminal} = \text{emf} -Ir$$ relates the terminal voltage to current I, internal resistance r and emf is the current, and r is the internal resistance.
 * Kirchoff's node rule is $$\Sigma I_\text{out}=\Sigma I_\text{in}$$. The loop rules is $$\Sigma\Delta V = 0$$.  The voltage drop $$\Delta V > 0$$ if the path crosses through a voltage source from the negative to the positive terminals, or if it travels opposite the current when passing through a resistor.
 * $$\tau = RC$$ is the decay constant for a resistor and capacitor.
 * While charging, $$V=\text{emf}\,(1-e^{-t/RC})$$, so that if the voltage is $$V$$, it will rise by $$0.631(\text{emf}-V)$$ in the next RC time.
 * While discharging, the voltage will drop from $$V$$ to $$0.368V$$ in one RC time.

<section begin=Magnetism/>

Magnetism
The SI unit for magnetic field is the tesla: 1T=1&middot;N &middot;C&minus;1(m/s)&minus;1 = 1N&middot;A&minus;1m&minus;1,The magnetic force on a moving charge q in the presence of a magnetic field B is $$F=qvB\sin\theta$$ where v is speed and &theta; is the angle between the velocity and magnetic field. The direction of the force is given by the cross product version of the right-hand-rule (RHR-1). <section end=Magnetism/> <section begin=Electromagnetic Induction, AC Circuits, and Electrical Technologie/>
 * $$r=mv/qB$$ is the orbital radius of a charged particle in a magnetic field (m,v,q are mass, speed, and charge, respectively.)
 * $$\varepsilon=vB\ell$$ is the Hall emf across a distance &#8467;, for moving charged particles (v B, &#8467; are mutually perpendicular vectors.)
 * $$F=I\ell B\sin\theta$$ is the force on a wire of length &#8467;, current I, in the presence of uniform magnetic field. The direction follows RHR-1.
 * $$\tau=NIAB\sin\theta$$ is the torque on N turns of wire around an area A, where &theta; is the angle between the uniform magnetic field and the perpendicular to the loop.
 * $$B=\frac{\mu_0I}{2\pi r}$$ is the magnetic field at a distance r from a long straight wire, where &mu;0 = 4&pi; &times; 10&minus;7 T&middot;m/A is the permeability of free space.
 * $$B=\frac{\mu_0I}{2R}$$ is the magnetic field at the center of a loop of radius R.
 * $$B=\mu_0nI$$ is the magnetic field inside a long, thin solenoid, where n=N/&#8467; is the number of turns per unit length.
 * Parallel wires attract (repel) if the currents are parallel (antiparallel), with $$F/\ell=\mu_0\frac{I_1I_2}{2\pi r}$$.
 * The net force on any charged particle is zero if $$v=E/B$$ and the velocity, magnetic, and electric fields are mutually perpendicular (see Wein filter.)

Electromagnetic Induction, AC Circuits, and Electrical Technologies
Magnetic flux is $$\Phi = AB\sin\theta$$ where A is area and &theta; is the angle between the magnetic field, B, and the normal to the area.
 * If a coil has N turns the induced emf is $$emf=-\Delta\Phi/\Delta t$$
 * The motional $$emf=vB\ell$$ if a wire of length $$\ell$$, its velocity and the magnetic field are mutually perpendicular.
 * If a coil is rotating at angular speed &omega; in a magnetic field, $$emf=NAB\omega\sin\omega t$$, with the peak $$emf_0=NAB\omega$$
 * In an ideal transformer, the primary and secondary voltages and currents are related by $$\frac{V_s}{V_p}=\frac{N_s}{N_p}$$ and $$I_sV_s=I_pV_p$$.
 * Mutual induction $$emf_2=-M\frac {\Delta I_1}{\Delta t},\;emf_1=-M\frac {\Delta I_2}{\Delta t}$$
 * Self inductance $$L=\frac{\Delta\Phi}{\Delta I}$$ where N is the number of turns and $$I$$ is the current in one turn.
 * Self inductance of a long thin solenoid $$emf=-L\frac {\Delta I}{\Delta t}$$, and $$L=\mu_0N^2A/\ell$$ where $$N$$ is the number of turns, $$A$$ is area and $$\ell$$ is length, where μ0 = $4 H/m$ ≈ $1.257$ N/A2 or T⋅m/A or Wb/(A⋅m) or V⋅s/(A⋅m).
 * $$E_{ind}=\tfrac 1 2 LI^2$$ is the energy stored in an inductor.

<section end=Electromagnetic Induction, AC Circuits, and Electrical Technologies/>

<section begin=Electromagnetic Waves/>

Electromagnetic Waves
<section end=Electromagnetic Waves/>

<section begin=Geometric Optics/>

Geometric Optics
<section end=Geometric Optics/>

<section begin=Vision and Optical Instruments/>

Vision and Optical Instruments
<section end=Vision and Optical Instruments/>

<section begin=Wave Optics/>

Wave Optics
<section end=Wave Optics/>

<section begin=Special Relativity/>

Special Relativity
<section end=Special Relativity/>

<section begin=Introduction to Quantum Physics/>

Introduction to Quantum Physics
<section end=Introduction to Quantum Physics/>

<section begin=Atomic Physics/>

Atomic Physics
<section end=Atomic Physics/>

<section begin=Radioactivity and Nuclear Physics/>

Radioactivity and Nuclear Physics
<section end=Radioactivity and Nuclear Physics/>

<section begin=Medical Applications of Nuclear Physics/>

Medical Applications of Nuclear Physics
<section end=Medical Applications of Nuclear Physics/>

<section begin=Particle Physics/>

Particle Physics
<section end=Particle Physics/>

<section begin=Frontiers of Physics/>

Frontiers of Physics
<section end=Frontiers of Physics/>