OpenStax College Physics/Equations

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

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Lifted from https://cnx.org/contents/Ax2o07Ul@9.98:pFeekPiU@17/Preface Equations (master) Formulas (master) Equations Formulas Help University Physics V1 University Physics V2 University Physics V3

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=====Introduction===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna *${\vec {F}}=m{\vec {a}}$ Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

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=====Kinematics===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Two-Dimensional Kinematics===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Dynamics: Force and Newton's Laws of Motion===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Further Applications of Newton's Laws: Friction, Drag, and Elasticity===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Uniform Circular Motion and Gravitation===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Work, Energy, and Energy Resources===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Linear Momentum and Collisions===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Statics and Torque===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Rotational Motion and Angular Momentum===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Fluid Statics===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Fluid Dynamics and Its Biological and Medical Applications===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Temperature, Kinetic Theory, and the Gas Laws===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Heat and Heat Transfer Methods===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. =====Thermodynamics===== Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna ${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

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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 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.

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, 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 $Z=\rho v_{w}$.

Electric Charge and Electric Field

The charge of the electron is q_e = −e, where the fundamental charge is e≈1.6 x 10⁻¹⁹ C. (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 ≈ 8.99x10⁹ N m² C⁻².
• ${\vec {F}}=q_{t}{\vec {E}}$ if the force on a test charge q_t 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 q_n is the source charge and r_nt is the distance from the source point to the test charge.

Electric Potential and Electric Field

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

• $1eV\approx 1.6\times 10^{-19}{\text{ Joules}}$
• $E=-\Delta V/\Delta s$ 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 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, $C=\varepsilon _{0}A/d$ for plates of area A separated by a (small) distance d. The permittivity of free space is ε₀ = 8.85×10⁻¹²F/m. 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.
• $E_{cap}={\tfrac {1}{2}}QV={\tfrac {1}{2}}CV^{2}=Q^{2}/(2C)$ is the energy stored in a capacitor.

Electric Current, Resistance, and Ohm's Law

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

• $I=nqAv_{d}$ 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 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 $\rho =\rho _{0}+\alpha \Delta T$, 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 $X(t)=X_{0}\sin(2\pi ft)$, $X_{\text{rms}}={\tfrac {1}{\sqrt {2}}}X_{0}$ where X₀ is the peak value and f is frequency.

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, $R_{eq}^{-1}=R_{1}^{-1}+R_{2}^{-1}+R_{3}^{-1}+\ldots$

• $V_{\text{terminal}}={\text{emf}}-Ir$ 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.

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.)
• $\varepsilon =vB\ell$ 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.

• $\tau =NIAB\sin \theta$ 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 = 4π × 10⁻⁷ T·m/A 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.)

Electromagnetic Induction, AC Circuits, and Electrical Technologies

Magnetic flux is $\Phi =AB\sin \theta$ 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 $emf=-\Delta \Phi /\Delta t$
• 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 ω 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 $I_{s}V_{s}=I_{p}V_{p}$.
• 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 $L=\mu _{0}N^{2}A/\ell$ where $N$ is the number of turns, $A$ is area and $\ell$ is length, where μ₀ = 4π×10⁻⁷
   H/m ≈ 1.2566...×10⁻⁶
  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.

Electromagnetic Waves

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Geometric Optics

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Vision and Optical Instruments

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Wave Optics

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Special Relativity

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Introduction to Quantum Physics

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Atomic Physics

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Radioactivity and Nuclear Physics

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Medical Applications of Nuclear Physics

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Particle Physics

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Frontiers of Physics

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${\vec {F}}=m{\vec {a}}$Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.