PGRE — Kahn & Anderson: 334 Equation Recall Cards

Projectile position versus time — For a projectile launched from (x₀, y₀) with velocity components v₀ₓ and v₀ᵧ, what are x(t) and y(t) under uniform downward gravity, neglecting air resistance?

x(t)=v_{0x}t+x_0,\qquad y(t)=-\frac12gt^2+v_{0y}t+y_0

Constant-acceleration relation without time — How are the initial and final velocities, constant acceleration, and displacement related in one-dimensional motion when time is eliminated?

v_f^2-v_i^2=2a\Delta x

Centripetal acceleration — What inward acceleration is required for motion at speed v around a circle of radius r?

a=\frac{v^2}{r}

Centripetal force — What net inward force is required to keep a mass m moving at speed v around a circle of radius r?

F=\frac{mv^2}{r}

Translational kinetic energy — What is the nonrelativistic kinetic energy of a particle of mass m moving at speed v?

\text{Translational kinetic energy: }\frac12mv^2

Rotational kinetic energy — What is the kinetic energy of a rigid body rotating about a fixed axis with moment of inertia I and angular speed ω?

\text{Rotational kinetic energy: }\frac12I\omega^2

Gravitational potential energy near Earth — What gravitational potential energy does a mass m gain when raised a height h in a uniform gravitational field g?

\text{Gravitational potential energy on Earth: }mgh

Elastic potential energy of a spring — What energy is stored in an ideal spring of spring constant k when displaced by x from its unstretched position?

\text{Spring potential energy: }\frac12kx^2

Potential-energy change from conservative work — How is the change in potential energy from point a to point b related to the line integral of a conservative force?

\Delta U=-\int_a^b\mathbf F\cdot d\boldsymbol\ell

Newton's law of gravitation — What is the gravitational force between point masses m₁ and m₂ separated by r, using a unit vector directed toward the attracting mass?

\mathbf F_{\mathrm{grav}}=\frac{Gm_1m_2}{r^2}\hat{\mathbf r}

Force from a potential-energy gradient — How do you obtain the conservative force vector from a position-dependent potential energy U?

\mathbf F=-\nabla U

Rolling without slipping — How are the center-of-mass speed and angular speed related for a wheel of radius R rolling without slipping on a stationary surface?

v=R\omega

Energy balance with additional work — How do you relate initial and final mechanical energy when additional forces do work not already included through potential energy?

E_{\mathrm{initial}}+W_{\mathrm{other}}=E_{\mathrm{final}}

Work–energy theorem — How does the total work done on a particle relate to its change in kinetic energy?

W=\Delta\mathrm{KE}

Work as a line integral — How do you calculate the work done by a position-dependent force along a specified path?

W=\int\mathbf F\cdot d\boldsymbol\ell

Particle angular momentum — How is a particle's angular momentum about an origin expressed using its position vector and linear momentum?

\mathbf L=\mathbf r\times\mathbf p

Rigid-body angular momentum — vector form — For rotation about a principal axis, how is a rigid body's angular momentum vector related to its angular velocity and moment of inertia?

\mathbf L=I\boldsymbol\omega

Torque of a force — How do you calculate the torque about an origin from a force and the position of its point of application?

\boldsymbol\tau=\mathbf r\times\mathbf F

Fixed-axis angular momentum — scalar form — What is the angular momentum about a fixed rotation axis in terms of the moment of inertia about that axis and angular speed?

L=I\omega

Torque and angular-momentum change — How is the net external torque about a fixed inertial axis related to the time evolution of angular momentum about that axis?

\tau=\frac{dL}{dt}

Centrifugal force in a rotating frame — For mass m at perpendicular distance r from an axis rotating at angular speed Ω, what is the centrifugal force magnitude and direction? Check the notes for the printed sign convention.

F_{\mathrm{centrifugal}}=-m\Omega^2r

Coriolis force — What fictitious force acts on a mass moving with velocity v relative to a frame rotating with angular velocity Ω?

\mathbf F_{\mathrm{Coriolis}}=-2m\boldsymbol\Omega\times\mathbf v

Moment of inertia of a point mass — What is the moment of inertia of a point mass m at perpendicular distance r from a rotation axis?

I=mr^2

Moment of inertia of a continuous body — How do you calculate a continuous body's moment of inertia by integrating over its mass, using perpendicular distance from the axis?

I=\int r^2\,dm

Parallel-axis theorem — How does the moment of inertia about an axis change when that axis is displaced a distance r parallel to an axis through the center of mass?

I=I_{\mathrm{CM}}+Mr^2

Center of mass — continuous distribution — How do you calculate the center-of-mass position vector of a continuous mass distribution with total mass M?

\mathbf r_{\mathrm{CM}}=\frac{\int\mathbf r\,dm}{M}

Center of mass — discrete particles — How do you calculate the center-of-mass position vector for particles with masses mᵢ at positions rᵢ?

\mathbf r_{\mathrm{CM}}=\frac{\sum_i\mathbf r_i m_i}{M}

Mechanical Lagrangian — How is the usual mechanical Lagrangian constructed from kinetic and potential energy?

L(q,\dot q,t)=T-U

Euler–Lagrange equation — What differential equation determines a generalized coordinate q from a Lagrangian depending on q, its velocity, and time?

\frac{d}{dt}\frac{\partial L}{\partial\dot q}=\frac{\partial L}{\partial q}

Conjugate momentum — How do you obtain the momentum conjugate to a generalized coordinate from the Lagrangian?

p_i\equiv\frac{\partial L}{\partial\dot q}\quad\text{(momentum conjugate to }q\text{)}

Hamiltonian as a Legendre transform — How do you construct the Hamiltonian from the Lagrangian, generalized velocities, and conjugate momenta?

H(p,q)=\sum_i p_i\dot q_i-L

Hamiltonian and mechanical energy — For a standard quadratic kinetic energy and a potential independent of velocities and time, how is the Hamiltonian related to kinetic and potential energy?

H=T+U\quad\text{(if }U\text{ does not depend explicitly on velocities or time)}

Hamilton's equations — How do the partial derivatives of the Hamiltonian determine the time derivatives of a generalized coordinate and its conjugate momentum?

\dot p=-\frac{\partial H}{\partial q},\qquad\dot q=\frac{\partial H}{\partial p}

Central-force Lagrangian in polar coordinates — What is the planar Lagrangian for a mass m moving in a central potential U(r), written using r and φ?

L=\frac12m\dot r^2+\frac12mr^2\dot\phi^2-U(r)

Angular momentum in polar coordinates — What conserved angular momentum corresponds to the polar angle φ for a particle in a central potential?

l=mr^2\dot\phi

Effective radial potential — For central-force motion with conserved angular momentum l, what effective potential governs the radial coordinate?

V(r)=\frac{l^2}{2mr^2}+U(r)

Reduced mass — What effective mass describes the relative motion of two particles of masses m₁ and m₂ after separating out center-of-mass motion?

\mu=\frac{m_1m_2}{m_1+m_2}

Energy of central-force motion — How do you express the total energy in terms of radial velocity, conserved angular momentum l, and a central potential U(r)?

E=T+V=\frac12m\dot r^2+\frac{l^2}{2mr^2}+U(r)

Equation of motion for a mass on a spring — What equation of motion follows from Newton's second law for a mass m displaced by x on an ideal spring with constant k?

F=m\ddot x=-kx

Mass–spring angular frequency — What is the natural angular frequency of an undamped mass m attached to an ideal spring with constant k?

\omega=\sqrt{\frac{k}{m}}

Complex representation of harmonic motion — How can you express a simple harmonic displacement using a complex amplitude A and angular frequency ω, taking the real part for the physical motion?

x(t)=Ae^{i\omega t}

Normal-mode ansatz — How do you express each generalized coordinate qₖ in a normal mode with amplitude aₖ and a shared angular frequency ω?

q_k(t)=a_ke^{i\omega t}

Normal-mode frequency condition — What determinant condition determines normal-mode frequencies from the stiffness matrix Aⱼₖ and mass matrix mⱼₖ?

\det(A_{jk}-\omega^2m_{jk})=0

Linearly damped oscillator equation — What is the unforced equation of motion for a mass–spring oscillator with mass m, spring constant k, and linear drag coefficient b?

m\ddot x+b\dot x+kx=0

Displacement resonance frequency — For a driven oscillator with natural frequency ω₀ and damping parameter β, what driving frequency maximizes the displacement amplitude when a nonzero-frequency peak exists?

\omega_R=\sqrt{\omega_0^2-2\beta^2}

Small-angle pendulum equation of motion — For a simple pendulum of length L and bob mass m, what linearized equation governs a small tangential displacement x?

m\ddot x=-\frac{mgx}{L}

Simple-pendulum angular frequency — What is the angular frequency of small oscillations of a simple pendulum of length L in gravity g?

\omega=\sqrt{\frac{g}{L}}

Physical-pendulum angular frequency — What is the small-oscillation angular frequency of a rigid pendulum of mass m, pivot-to-center-of-mass distance R, and moment of inertia I about the pivot?

\omega=\sqrt{\frac{mgR}{I}}

Bernoulli's constant — What combination of speed, height, pressure, and density stays constant along a streamline in steady, incompressible, inviscid flow?

\frac{v^2}{2}+gz+\frac{p}{\rho}=\text{constant}

Continuity for incompressible flow — How are the flow speeds and cross-sectional areas related at two sections of a tube carrying steady incompressible flow?

v_1A_1=v_2A_2

Bernoulli's equation between two points — How are speed, height, and pressure at two points on a streamline related for steady, incompressible, inviscid flow?

\frac{v_1^2}{2}+gz_1+\frac{p_1}{\rho}=\frac{v_2^2}{2}+gz_2+\frac{p_2}{\rho}

Gauss's law — electrostatic differential form — How is the divergence of the electrostatic field related to the local charge density in vacuum?

\nabla\cdot\mathbf E=\frac{\rho}{\epsilon_0}

Curl-free electrostatic field — What is the curl of the electric field in electrostatics?

\nabla\times\mathbf E=0\quad\text{(electrostatics)}

Electric force on a charge — What force does an electric field E exert on a charge q?

\mathbf F_E=q\mathbf E

Electric field from potential — How do you obtain the electrostatic field vector from the scalar electric potential V?

\mathbf E=-\nabla V

Electric potential from a field integral — Taking V(a) as zero, how do you calculate the potential at b from the electric field along a path from a to b?

V(b)=-\int_a^b\mathbf E\cdot d\boldsymbol\ell

Poisson's equation for electric potential — What differential equation relates electrostatic potential to charge density in vacuum?

\nabla^2V=-\frac{\rho}{\epsilon_0}

Laplace's equation for electric potential — What differential equation does electrostatic potential satisfy in a charge-free region?

\nabla^2V=0\quad\text{(empty space)}

Potential of a continuous charge distribution — How do you calculate the potential at an observation point r from a localized volume charge density, with potential zero at infinity?

V(\mathbf r)=\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\mathbf r')}{|\mathbf r-\mathbf r'|}\,d^3r'

Gauss's law — integral form — How is the electric flux through a closed surface related to its enclosed charge in vacuum?

\oint_S\mathbf E(\mathbf r)\cdot d\mathbf S=\frac{Q_{\mathrm{enc}}}{\epsilon_0}

Electrostatic circulation around a closed loop — What is the line integral of an electrostatic field around any closed contour?

\oint_C\mathbf E(\mathbf r)\cdot d\boldsymbol\ell=0\quad\text{(electrostatics)}

Electric field of a point charge — What electric field does a point charge q at the origin produce at a point a distance r away in vacuum?

\mathbf E(\mathbf r)=\frac{q}{4\pi\epsilon_0r^2}\hat{\mathbf r}

Potential of a point charge — What is the potential a distance r from a point charge q in vacuum, taking zero potential at infinity?

V(r)=\frac{q}{4\pi\epsilon_0r}

Electric field of an infinite charged sheet — What field does a single infinite sheet with uniform surface charge density σ produce on either side in vacuum, with the normal directed away from the sheet?

\mathbf E=\frac{\sigma}{2\epsilon_0}\hat{\mathbf n}

Tangential electrostatic boundary condition — How does the component of the electrostatic field parallel to a surface change when you cross that surface?

\mathbf E_{\mathrm{out}}^{\parallel}-\mathbf E_{\mathrm{in}}^{\parallel}=0

Normal electric-field boundary condition — What jump in the normal electric-field component occurs across a sheet of surface charge density σ in vacuum, using the normal from inside to outside?

E_{\mathrm{out}}^{\perp}-E_{\mathrm{in}}^{\perp}=\frac{\sigma}{\epsilon_0}

Electrostatic energy of discrete charges — How do you calculate the interaction energy of point charges qᵢ from the potential at each charge due to all the other charges?

W=\frac12\sum_{i=1}^n q_iV(\mathbf r_i)

Electrostatic energy from charge and potential — How do you express the electrostatic energy of a continuous charge distribution as a volume integral involving charge density and potential?

W=\frac12\int\rho(\mathbf r)V(\mathbf r)\,d^3r

Energy stored in an electric field — How do you express the electrostatic energy in vacuum as an integral over the electric field throughout space, assuming the surface term vanishes?

U_E=\frac{\epsilon_0}{2}\int|\mathbf E|^2\,d^3r

Definition of capacitance — How are the charge magnitude on a capacitor plate, its capacitance, and the voltage difference between the plates related?

Q=CV

Vacuum parallel-plate capacitance — What is the capacitance of parallel plates of area A separated by distance d in vacuum, neglecting edge effects?

C=\frac{\epsilon_0A}{d}\quad\text{(parallel-plate capacitor)}

Energy stored in a capacitor — What energy is stored in a capacitor, expressed once using its charge and capacitance and once using its voltage and capacitance?

U_C=\frac12\frac{Q^2}{C}=\frac12CV^2

Gauss's law for magnetism — magnetostatic differential form — What does the absence of magnetic monopoles imply for the divergence of a static magnetic field?

\nabla\cdot\mathbf B=0

Ampère's law — differential magnetostatic form — How is the curl of the magnetic field related to the current density for steady currents in vacuum?

\nabla\times\mathbf B=\mu_0\mathbf J\quad\text{(magnetostatics)}

Gauss's law for magnetism — integral form — What is the net magnetic flux through any closed surface?

\oint_S\mathbf B\cdot d\mathbf S=0

Ampère's law — integral magnetostatic form — How is the circulation of a static magnetic field around a closed contour related to the current passing through a spanning surface?

\oint_C\mathbf B\cdot d\boldsymbol\ell=\mu_0I_{\mathrm{enc}}\quad\text{(magnetostatics)}

Magnetic field from vector potential — How do you obtain the magnetic field from the magnetic vector potential A?

\nabla\times\mathbf A=\mathbf B

Magnetic Lorentz force — What magnetic force acts on a charge q moving with velocity v through a magnetic field B?

\mathbf F_B=q\mathbf v\times\mathbf B

Magnetic force on a current element — What magnetic force acts on a short wire element carrying current I, with its directed length element pointing along the current?

d\mathbf F_B=I\,d\boldsymbol\ell\times\mathbf B

Biot–Savart law for a thin wire — How do you calculate the magnetic field at a point from a thin wire carrying steady current I, using the displacement from each source element to the observation point?

\mathbf B(\mathbf r)=\frac{\mu_0I}{4\pi}\int\frac{d\boldsymbol\ell\times\hat{\mathbf r}'}{r'^2}

Magnetic field of a long straight wire — What is the magnitude and direction of the magnetic field a perpendicular distance r from an infinitely long straight wire carrying current I in vacuum?

|\mathbf B|(2\pi r)=\mu_0I\;\Longrightarrow\;\mathbf B=\frac{\mu_0I}{2\pi r}\hat{\boldsymbol\phi}

Magnetic field inside a long solenoid — What is the magnetic-field magnitude deep inside an ideal long air-core solenoid with n turns per unit length carrying current I?

B=\mu_0nI\quad\text{(solenoid)}

Magnetic field inside a toroid — What is the magnetic-field magnitude at radius r inside an ideal air-core toroid with N turns carrying current I?

B=\frac{\mu_0NI}{2\pi r}\quad\text{(toroid)}

Normal magnetic-field boundary condition — How does the component of the magnetic field normal to an interface change across that interface?

B_{\mathrm{out}}^{\perp}-B_{\mathrm{in}}^{\perp}=0

Tangential magnetic-field boundary condition — Across a surface current K in vacuum, what is the jump in the tangential magnetic field, using a unit normal directed from inside to outside?

\mathbf B_{\mathrm{out}}^{\parallel}-\mathbf B_{\mathrm{in}}^{\parallel}=\mu_0\mathbf K\times\hat{\mathbf n}

Energy stored in a magnetic field — How do you calculate the total energy stored in a magnetic field in vacuum as a volume integral?

U_B=\frac{1}{2\mu_0}\int|\mathbf B|^2\,d^3r

Cyclotron radius — What is the radius of nonrelativistic circular motion for a particle of mass m and positive charge q moving perpendicular to a uniform field B at speed v?

R=\frac{mv}{qB}

Cyclotron angular frequency — What is the nonrelativistic cyclotron angular frequency of a particle with positive charge q and mass m in a uniform magnetic field B?

\omega=\frac{qB}{m}

Gauss's law — full Maxwell system — Even when fields vary in time, how is the divergence of the electric field related to charge density in vacuum?

\nabla\cdot\mathbf E=\frac{\rho}{\epsilon_0}

Gauss's law for magnetism — full Maxwell system — Even when fields vary in time, what condition does the magnetic field's divergence satisfy?

\nabla\cdot\mathbf B=0

Faraday's law — differential form — How does a time-changing magnetic field determine the curl of the electric field?

\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}

Ampère–Maxwell law — differential form — How is the curl of the magnetic field related to both current density and a time-changing electric field in vacuum?

\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\frac{\partial\mathbf E}{\partial t}

Faraday's law — magnetic flux and induced EMF — How is the induced EMF around a loop related to the time derivative of magnetic flux through it, including the sign?

\mathcal E=-\frac{d\Phi_B}{dt}

Definition of mutual inductance — How is the magnetic flux linkage through circuit 2 due to current in circuit 1 related to their mutual inductance?

\Phi_{21}=M_{12}I_1

Definition of self-inductance — How is a circuit's own magnetic flux linkage related to its current and self-inductance?

\Phi_B=LI

Self-induced EMF — What EMF is induced in a coil of constant inductance L when its current changes with time?

\mathcal E=-L\frac{dI}{dt}

Inductance of a long solenoid — What is the inductance of a long air-core solenoid with N turns, cross-sectional area A, and length l, neglecting fringe fields?

L=\frac{\mu_0N^2A}{l}\quad\text{(solenoid)}

Energy stored in an inductor — How much energy is stored in an ideal inductor of inductance L carrying current I?

U_L=\frac12LI^2

Electric dipole moment of two charges — What is the electric dipole moment of charges +q and −q at positions r₁ and r₂, and how does it relate to the displacement from the negative charge to the positive charge?

\mathbf p=q\mathbf r_1-q\mathbf r_2=q\mathbf d

Electric dipole moment of a continuous distribution — How do you calculate the electric dipole moment of a continuous charge density relative to a chosen origin?

\mathbf p=\int\mathbf r\rho(\mathbf r)\,d^3r