OpenStax College Physics/Equations

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University Physics V1

University Physics V2

University Physics V3

Template:Cot <section begin=Introduction/>

• =====Introduction=====

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• =====Kinematics=====

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• =====Two-Dimensional Kinematics=====

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

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<section begin=Further Applications of Newton's Laws: Friction, Drag, and Elasticity/>

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

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• =====Uniform Circular Motion and Gravitation=====

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<section begin=Work, Energy, and Energy Resources/>

• =====Work, Energy, and Energy Resources=====

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<section begin=Linear Momentum and Collisions/>

• =====Linear Momentum and Collisions=====

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<section begin=Statics and Torque/>

• =====Statics and Torque=====

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<section begin=Rotational Motion and Angular Momentum/>

• =====Rotational Motion and Angular Momentum=====

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<section begin=Fluid Statics/>

• =====Fluid Statics=====

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

• =====Fluid Dynamics and Its Biological and Medical Applications=====

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

• =====Temperature, Kinetic Theory, and the Gas Laws=====

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<section begin=Heat and Heat Transfer Methods/>

• =====Heat and Heat Transfer Methods=====

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<section begin=Thermodynamics/>

• =====Thermodynamics=====

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<section begin=Oscillatory Motion and Waves/>

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.

• $P{E}_{el}=\frac{1}{2}k{x}^2$ 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, $\omega =\sqrt{g/L}$.
• $x=X\cos (\omega t),v=-\omega X\sin (\omega t),a=-{\omega }^{2}X\cos (\omega t)$ describe the position, velocity, and acceleration of simple harmonic motion, respectively.
• $f\lambda =v_w$ is the wave velocity, where λ 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 end=Oscillatory Motion and Waves/>

<section begin=Physics of Hearing/>

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.

• $I=P/A=(\mathrm{\Delta }p{)}^2/(2\rho v_w)$ is intensity of sound, where P is the power passing through an area A, Δp is the pressure amplitude, and ρ 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 f_s 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 frequencyf_s. 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 end=Physics of Hearing/>

<section begin=Electric Charge and Electric Field/>

Electric Charge and Electric Field

The charge of the electron is q_e = −e, where the fundamental charge is Template:Nowrap begine≈1.6 x 10⁻¹⁹ CTemplate:Nowrap end. (C≡Coulombs is the SI unit of charge).

• $F=k\frac{|q_1{q}_2|}{r^2}$ is Coulomb's law for the force between two point charges q₁ and q₂ separated by a distance r. Coulomb's constant is k ≈ Template:Nowrap begin8.99x10⁹ N m² C⁻².Template:Nowrap end
• $\overrightarrow{F}=q_t\overrightarrow{E}$ if the force on a test charge q_t due to the electric field $\overrightarrow{E}=\mathrm{\Sigma }{\overrightarrow{E}}_j$ at the location of the test charge. The vector sum is over $n=1\to N$ source charges, and the magnitude of each contribution to the field is $|{\overrightarrow{E}}_n|=k{q}_n/r_{nt}^2$ where q_n is the source charge and r_nt is the distance from the source point to the test charge.

<section end=Electric Charge and Electric Field/>

<section begin=Electric Potential and Electric Field/>

Electric Potential and Electric Field

• $\mathrm{\Delta }\text{PE}=q\mathrm{\Delta }V=q(V_B-V_A)$ is the potential energy associated with moving a charge q from a point where the electric potential is V_A to a point where it is V_B.

• $1eV\approx 1.6\times 10^{-19}\text{ Joules}$
• $E=-\mathrm{\Delta }V/\mathrm{\Delta }s$ is the component of the electric field parallel to a small displacement $\mathrm{\Delta }\overrightarrow{s}$ (the electric field points from high to low voltage, and ΔV=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, $C={\epsilon }_{0}A/d$ for plates of area A separated by a (small) distance d. The permittivity of free space is ε₀ = Template:Nowrap. If a dielectric fills the gap, replace ε₀ by κε₀
• $\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\dots$ is the total capacitance in series.
• $C_p=C_1+C_2+C_3+\dots$ is the total capacitance in parallel.
• $E_{cap}=\frac{1}{2}QV=\frac{1}{2}C{V}^2=Q^2/(2C)$ is the energy stored in a capacitor.

<section end=Electric Potential and Electric Field/> <section begin=Electric Current, Resistance, and Ohm's Law/>

Electric Current, Resistance, and Ohm's Law

The electric current, in amperes (A≡C/s) is the rate at which charge Q flows: $I=\mathrm{\Delta }Q/\mathrm{\Delta }t$

• $I=nqA{v}_d$ 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 v_d.
• $V=IR$ is w:Ohm's law, where 1Ω=1V/A. $R=\rho L/A$ where ρ is resistivity, L is length, and A is area. For changes in temperature T, both R and ρ typically increase linearly with ΔT, as $\rho ={\rho }_0+\alpha \mathrm{\Delta }T$, where α is the temperature coefficient of resistivity.
• $P=IV=I^{2}R=V^2/R$ is electrical power (energy divided by time). For alternating current, the average power is $P_{\text{ave}}=I_{\text{rms}}{V}_{\text{rms}}=I_{\text{rms}}^{2}R=V_{\text{rms}}^2/R$, where rms denotes root mean square). If $X(t)=X_0\sin (2\pi ft)$, $X_{\text{rms}}=\frac{1}{\sqrt{2}}{X}_0$ where X₀ is the peak value and f is frequency.

<section end=Electric Current, Resistance, and Ohm's Law/>

<section begin=Circuits and DC Instruments/>

Circuits and DC Instruments

If resistors are in series, the equivalent resistance is $R_s=R_1+R_2+R_3+\dots$. If they are in parallel, $R_{eq}^{-1}=R_1^{-1}+R_2^{-1}+R_3^{-1}+\dots$

• $V_{\text{terminal}}=\text{emf}-Ir$ 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 $\mathrm{\Sigma }{I}_{\text{out}}=\mathrm{\Sigma }{I}_{\text{in}}$. The loop rules is $\mathrm{\Sigma }\mathrm{\Delta }V=0$. The voltage drop $\mathrm{\Delta }V>0$ 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 end=Circuits and DC Instruments/>

<section begin=Magnetism/>

Magnetism

RHR-1

The SI unit for magnetic field is the tesla: 1T=1·N ·C⁻¹(m/s)⁻¹ = 1N·A⁻¹m⁻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 θ 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).

RHR-2

• $r=mv/qB$ is the orbital radius of a charged particle in a magnetic field (m,v,q are mass, speed, and charge, respectively.)
• $\epsilon =vB\ell$ is the Hall emf across a distance ℓ, for moving charged particles (v B, ℓ are mutually perpendicular vectors.)
• $F=I\ell B\sin \theta$ is the force on a wire of length ℓ, current I, in the presence of uniform magnetic field. The direction follows RHR-1.

<section end=Magnetism/>

• $\tau =NIAB\sin \theta$ is the torque on N turns of wire around an area A, where θ is the angle between the uniform magnetic field and the perpendicular to the loop.
• $B=\frac{{\mu }_{0}I}{2\pi r}$ is the magnetic field at a distance r from a long straight wire, where μ₀ = Template:Nowrap is the permeability of free space.
• $B=\frac{{\mu }_{0}I}{2R}$ is the magnetic field at the center of a loop of radius R.
• $B={\mu }_{0}nI$ is the magnetic field inside a long, thin solenoid, where n=N/ℓ is the number of turns per unit length.
• Parallel wires attract (repel) if the currents are parallel (antiparallel), with $F/\ell ={\mu }_0\frac{I_1{I}_2}{2\pi r}$.
• 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.)

<section begin=Electromagnetic Induction, AC Circuits, and Electrical Technologie/>

Electromagnetic Induction, AC Circuits, and Electrical Technologies

Magnetic flux is $\mathrm{\Phi }=AB\sin \theta$ where A is area and θ is the angle between the magnetic field, B, and the normal to the area.

• If a coil has N turns the induced emf is $emf=-\mathrm{\Delta }\mathrm{\Phi }/\mathrm{\Delta }t$
• 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 ω in a magnetic field, $emf=NAB\omega \sin \omega t$, with the peak $em{f}_0=NAB\omega$
• 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 $I_s{V}_s=I_p{V}_p$.
• Mutual induction $em{f}_2=-M\frac{\mathrm{\Delta }{I}_1}{\mathrm{\Delta }t},em{f}_1=-M\frac{\mathrm{\Delta }{I}_2}{\mathrm{\Delta }t}$
• Self inductance $L=\frac{\mathrm{\Delta }\mathrm{\Phi }}{\mathrm{\Delta }I}$ where N is the number of turns and $I$ is the current in one turn.
• Self inductance of a long thin solenoid $emf=-L\frac{\mathrm{\Delta }I}{\mathrm{\Delta }t}$, and $L={\mu }_0{N}^{2}A/\ell$ where $N$ is the number of turns, $A$ is area and $\ell$ is length, where μ₀ = Template:Val ≈ Template:Val N/A² or T⋅m/A or Wb/(A⋅m) or V⋅s/(A⋅m).
• $E_{ind}=\frac{1}{2}L{I}^2$ is the energy stored in an inductor.

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

<section begin=Electromagnetic Waves/>

Electromagnetic Waves

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<section begin=Geometric Optics/>

Geometric Optics

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<section begin=Vision and Optical Instruments/>

Vision and Optical Instruments

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<section begin=Wave Optics/>

Wave Optics

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<section begin=Special Relativity/>

Special Relativity

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<section begin=Introduction to Quantum Physics/>

Introduction to Quantum Physics

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<section begin=Atomic Physics/>

Atomic Physics

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<section begin=Radioactivity and Nuclear Physics/>

Radioactivity and Nuclear Physics

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<section begin=Medical Applications of Nuclear Physics/>

Medical Applications of Nuclear Physics

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<section begin=Particle Physics/>

Particle Physics

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<section begin=Frontiers of Physics/>

Frontiers of Physics

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