Classical Mechanics, Electromagnetism, Waves, Thermodynamics, Quantum Mechanics, Relativity, and Laboratory Methods Flashcards

Vocabulary flashcards covering core physics concepts, key formulas, constants, and theoretical principles across classical mechanics, electrodynamics, quantum mechanics, thermodynamics, relativity, and laboratory methods.

Coefficient of static friction for a block on a ramp

The minimum coefficient of static friction $\mu$ required to keep a block in place on a ramp inclined at angle $\theta$ is $\mu = \tan(\theta)$.

Free-body diagram

A diagram where each individual block and only the forces acting on it are drawn, preventing double-counting and ensuring proper application of Newton's third law.

Massless string (GRE definition)

A string that carries the same tension at every point.

Kinematics equations for 2D projectile motion

Equations giving position as a function of time under constant acceleration $g$ acting in the negative y-direction: $x(t) = v_{0x}t + x_0$ and $y(t) = -\frac{1}{2}gt^2 + v_{0y}t + y_0$.

One-dimensional constant acceleration formula

The formula relating initial velocity $v_i$, final velocity $v_f$, constant acceleration $a$, and displacement $\Delta x$: $v_f^2 - v_i^2 = 2a\Delta x$.

Radial acceleration (Uniform circular motion)

The inward radial acceleration experienced by a particle moving in a circular path of radius $r$ with constant speed $v$, given by $a = \frac{v^2}{r}$.

Centripetal force

The net radially inward force required to keep an object of mass $m$ moving in a circular path of radius $r$ at constant speed $v$, given by $F = \frac{mv^2}{r}$.

Conservation of energy

Principle stating that if an object is acted on only by conservative forces, the sum of its kinetic and potential energies is constant along its path.

Conservative force

A force for which the work done is independent of the path taken between starting and ending points, allowing an associated time-independent potential energy.

Translational kinetic energy

Energy of linear motion given by $\frac{1}{2}mv^2$.

Rotational kinetic energy

Energy of rotational motion given by $\frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is angular velocity.

Gravitational potential energy (Earth's surface)

Potential energy given by $mgh$, where $m$ is mass, $g$ is gravitational acceleration, and $h$ is height above a chosen zero reference level.

Spring potential energy

Energy stored in a stretched or compressed spring with spring constant $k$ and displacement $x$, given by $\frac{1}{2}kx^2$.

Potential energy change ($\Delta U$)

The change in potential energy between points $a$ and $b$ for a conservative force $\mathbf{F}$, defined as $\Delta U = -\int_a^b \mathbf{F} \cdot d\mathbf{l}$.

Gravitational force magnitude

The attractive force between two masses $m_1$ and $m_2$ separated by distance $r$, given by $F_{\text{grav}} = \frac{Gm_1m_2}{r^2}$.

Gravitational potential energy (Two point masses)

Potential energy of a mass $m$ at distance $r$ from mass $M$, defined with zero at infinity as $U(r) = -\frac{GmM}{r}$.

Force from potential energy

The relation computing force from a potential energy function $U$, given by $\mathbf{F} = -\nabla U$.

Rolling without slipping condition

The kinematic relationship between linear velocity $v$ and angular velocity $\omega$ for an object of radius $R$ rolling without slipping: $v = R\omega$.

Work-energy theorem

Principle relating initial energy, work done by nonconservative forces $W_{\text{other}}$, and final energy: $E_{\text{initial}} + W_{\text{other}} = E_{\text{final}}$, or $W = \Delta \text{KE}$.

Conservation of linear momentum

Principle stating that linear momentum is always conserved in a system in the absence of external forces.

Elastic collision

A collision in which the total initial and final kinetic energies of the system are equal.

Angular momentum (Point particle)

Vector quantity defined by $\mathbf{L} = \mathbf{r} \times \mathbf{p}$, where $\mathbf{r}$ is the position vector from a reference point and $\mathbf{p}$ is linear momentum.

Angular momentum (Extended body)

Quantity defined as $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is angular velocity.

Torque

Rotational analogue of force, defined by $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$, or scalar relation $\tau = \frac{dL}{dt}$.

Centrifugal force

A fictitious apparent force in a uniformly rotating frame that pushes objects away from the axis of rotation, given by $\mathbf{F}_{\text{centrifugal}} = -m\boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r})$ or magnitude $m\Omega^2 r$.

Coriolis force

A fictitious force appearing in a rotating reference frame acting on a moving object, given by $\mathbf{F}_{\text{Coriolis}} = -2m\boldsymbol{\Omega} \times \mathbf{v}$.

Moment of inertia (Point particle)

Measure of rotational inertia for a point particle of mass $m$ at radius $r$ from the rotation axis, given by $I = mr^2$.

Moment of inertia (Continuous mass distribution)

Integral definition $I = \int r^2 \, dm = \int r^2 \rho \, dV$, where $r$ is the perpendicular distance from the axis of rotation to the mass element.

Parallel axis theorem

Theorem stating that the moment of inertia $I$ about any axis parallel to an axis through the center of mass is $I = I_{\text{CM}} + Mr^2$, where $r$ is the distance between the two axes and $M$ is total mass.

Center of mass (Continuous object)

Position vector defined by $\mathbf{r}_{\text{CM}} = \frac{\int \mathbf{r} \, dm}{M}$.

Lagrangian ($L$)

A scalar function describing a mechanical system, defined as $L(q, \dot{q}, t) = T - U$, where $T$ is kinetic energy and $U$ is potential energy.

Euler-Lagrange equations

Equations of motion derived from the Lagrangian $L$ for each generalized coordinate $q$: $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) = \frac{\partial L}{\partial q}$.

Conjugate momentum ($p_i$)

Momentum corresponding to generalized coordinate $q_i$, defined by $p_i \equiv \frac{\partial L}{\partial \dot{q}_i}$.

Cyclic coordinate

A generalized coordinate $q$ that does not explicitly appear in the Lagrangian ($\frac{\partial L}{\partial q} = 0$), implying its conjugate momentum $p_i$ is conserved.

Hamiltonian ($H$)

Scalar function defined via Legendre transformation as $H(p, q) = \sum_i p_i \dot{q}_i - L$, equal to total energy $T + U$ when $U$ is velocity- and time-independent.

Hamilton's equations of motion

Coupled first-order differential equations of motion: $\dot{p} = -\frac{\partial H}{\partial q}$ and $\dot{q} = \frac{\partial H}{\partial p}$.

Central potential

A potential function $U(r)$ that depends only on the radial distance $r$ between two interacting bodies.

Effective potential (Orbital mechanics)

Equivalent 1D potential for radial motion in a central force field, given by $V(r) = \frac{l^2}{2mr^2} + U(r)$, where $l$ is angular momentum.

Reduced mass ($\mu$)

Effective single-body mass for a two-body system, defined as $\mu = \frac{m_1 m_2}{m_1 + m_2}$.

Kepler's First Law

Statement that planets move in elliptical orbits with the Sun at one focus (or more precisely, orbiting their mutual center of mass).

Kepler's Second Law

Statement that planetary orbits sweep out equal areas in equal times, expressing conservation of angular momentum (areal velocity).

Kepler's Third Law

Statement that the square of the orbital period $T$ is proportional to the cube of the semi-major axis $a$ ($T^2 \propto a^3$).

Hooke's Law

Restoring force equation for a spring displaced by $x$ from equilibrium: $F = -kx$.

Natural angular frequency (Harmonic oscillator)

Angular frequency of oscillation for a mass $m$ on a spring with constant $k$, given by $\omega_0 = \sqrt{\frac{k}{m}}$.

Normal modes

Independent collective modes of oscillation of a system where all parts oscillate at the same fixed normal frequency.

Damped harmonic oscillator equation

Equation of motion $m\ddot{x} + b\dot{x} + kx = 0$, categorized into underdamped, critically damped, and overdamped regimes based on damping parameter $\beta = \frac{b}{2m}$ and natural frequency $\omega_0 = \sqrt{\frac{k}{m}}$.

Resonant frequency (Driven damped oscillator)

The driving frequency that maximizes steady-state amplitude, given by $\omega_R = \sqrt{\omega_0^2 - 2\beta^2}$.

Simple pendulum angular frequency

Small-angle oscillation frequency for a simple pendulum of length $L$, given by $\omega = \sqrt{\frac{g}{L}}$.

Physical pendulum angular frequency

Small-angle oscillation frequency for an extended object of mass $m$, moment of inertia $I$ about pivot, and CM distance $R$, given by $\omega = \sqrt{\frac{mgR}{I}}$.

Equivalent spring constants

For parallel springs, $k_{\text{eq}} = k_1 + k_2$; for series springs, $\frac{1}{k_{\text{eq}}} = \frac{1}{k_1} + \frac{1}{k_2}$.

Bernoulli's principle

Statement that along a streamline of an incompressible fluid, $\frac{v^2}{2} + gz + \frac{p}{\rho} = \text{constant}$.

Buoyant force (Archimedes' principle)

Upward force exerted on a submerged object equal to the weight of the displaced fluid: $F_{\text{buoyant}} = \rho V g$.

Gauss's Law (Electrostatics)

Maxwell equation stating $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$, or in integral form $\oint_S \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{enc}}}{\varepsilon_0}$.

Electric potential ($V$)

Scalar field defined such that $\mathbf{E} = -\nabla V$, or $V(b) - V(a) = -\int_a^b \mathbf{E} \cdot d\mathbf{l}$.

Poisson's and Laplace's equations

Differential equations for electric potential $V$: Poisson's equation is $\nabla^2 V = -\frac{\rho}{\varepsilon_0}$, and Laplace's equation (in empty space) is $\nabla^2 V = 0$.

Electric field of an infinite charged sheet

Uniform electric field produced by a flat sheet with surface charge density $\sigma$: $E = \frac{\sigma}{2\varepsilon_0}$, pointing normal to the sheet.

Electric field of an infinite line charge

Radial electric field at distance $r$ from an infinite line of linear charge density $\lambda$: $E = \frac{\lambda}{2\pi\varepsilon_0 r}$.

Electrostatic boundary conditions

Conditions across a charged surface: parallel electric field is continuous ($E_{\parallel}^{\text{out}} - E_{\parallel}^{\text{in}} = 0$), perpendicular electric field is discontinuous ($E_{\perp}^{\text{out}} - E_{\perp}^{\text{in}} = \frac{\sigma}{\varepsilon_0}$), and potential $V$ is continuous.

Ideal conductor properties

Potential $V$ is constant throughout, $\mathbf{E} = 0$ inside, bulk charge density $\rho = 0$, net charge resides on the surface, and $\mathbf{E}$ just outside is perpendicular to the surface.

Method of images

Technique for solving electrostatic boundary value problems by replacing conducting surfaces with virtual image charges that reproduce the boundary conditions.

Electrostatic field energy

Total energy stored in an electric field: $U_E = \frac{\varepsilon_0}{2} \int |\mathbf{E}|^2 \, d^3r$.

Capacitance ($C$)

Constant of proportionality relating charge and potential difference on conductors: $Q = CV$.

Parallel-plate capacitance

Capacitance of two plates of area $A$ separated by distance $d$: $C = \frac{\varepsilon_0 A}{d}$.

Energy stored in a capacitor

Energy given by $U_C = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}CV^2$.

Ampère's Law (Magnetostatics)

Maxwell equation stating $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$, or in integral form $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}$.

Magnetic vector potential ($\mathbf{A}$)

Vector field defined such that $\nabla \times \mathbf{A} = \mathbf{B}$.

Lorentz force law

Force on a test charge $q$ moving with velocity $\mathbf{v}$ in electric and magnetic fields: $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$.

Biot-Savart law

Formula for magnetic field produced by a current element: $\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\mathbf{l} \times \hat{\mathbf{r}}'}{r'^2}$.

Magnetic field of an infinite straight wire

Field at distance $r$ from an infinite wire carrying current $I$: $B = \frac{\mu_0 I}{2\pi r}$, directed azimuthally.

Magnetic field of a solenoid

Uniform magnetic field inside an infinite solenoid with $n$ turns per unit length carrying current $I$: $B = \mu_0 n I$.

Magnetic field of a toroid

Field inside a toroid with $N$ total turns carrying current $I$ at radius $r$: $B = \frac{\mu_0 N I}{2\pi r}$.

Magnetostatic boundary conditions

Normal magnetic field is continuous ($B_{\perp}^{\text{out}} - B_{\perp}^{\text{in}} = 0$), parallel magnetic field is discontinuous ($B_{\parallel}^{\text{out}} - B_{\parallel}^{\text{in}} = \mu_0 |\mathbf{K} \times \hat{\mathbf{n}}|$).

Magnetostatic field energy

Energy stored in a magnetic field: $U_B = \frac{1}{2\mu_0} \int |\mathbf{B}|^2 \, d^3r$.

Cyclotron radius and frequency

For a charge $q$ with mass $m$ moving at speed $v_{\perp}$ perpendicular to magnetic field $B$: radius $R = \frac{m v_{\perp}}{qB}$ and cyclotron frequency $\Omega = \frac{qB}{m}$.

Faraday's Law of Induction

Maxwell equation stating $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, or in terms of electromotive force $\mathcal{E} = -\frac{d\Phi_B}{dt}$.

Lenz's law

Principle stating that induced currents flow in a direction such that the magnetic field they produce opposes the change in magnetic flux.

Self-inductance ($L$)

Proportionality constant between magnetic flux and current in a circuit ($\Phi_B = LI$), yielding induced emf $\mathcal{E} = -L \frac{dI}{dt}$.

Energy stored in an inductor

Energy stored in the magnetic field of an inductor: $U_L = \frac{1}{2} L I^2$.

Electric dipole moment ($\mathbf{p}$)

Vector defined for two equal and opposite charges $\pm q$ separated by displacement $\mathbf{d}$ as $\mathbf{p} = q\mathbf{d}$, pointing from negative to positive charge.

Electric dipole torque and energy

In external field $\mathbf{E}$: torque $\mathbf{N} = \mathbf{p} \times \mathbf{E}$ and potential energy $U = -\mathbf{p} \cdot \mathbf{E}$.

Magnetic dipole moment ($\mathbf{m}$)

Vector for a planar loop carrying current $I$ with area vector $\mathbf{A}$ defined by $\mathbf{m} = I\mathbf{A}$; torque in field $\mathbf{B}$ is $\mathbf{N} = \mathbf{m} \times \mathbf{B}$ and potential energy is $U = -\mathbf{m} \cdot \mathbf{B}$.

Dielectric constant ($\kappa$)

Factor by which permittivity increases in a material: $\varepsilon = \kappa \varepsilon_0$, increasing capacitance of a parallel-plate capacitor to $C = \kappa \frac{\varepsilon_0 A}{d}$.

Speed of light in vacuum ($c$)

Constant relation defined by $c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$.

Poynting vector ($\mathbf{S}$)

Vector representing energy flux (power per unit area) of an electromagnetic wave: $\mathbf{S} = \frac{1}{\mu_0}(\mathbf{E} \times \mathbf{B})$.

Larmor formula

Total power radiated by a nonrelativistic accelerating point charge $q$ with acceleration $a$: $P = \frac{q^2 a^2}{6\pi \varepsilon_0 c^3}$.

Electric dipole radiation power

Time-averaged power radiated by an oscillating electric dipole $p(t) = p_0 \cos(\omega t)\hat{\mathbf{z}}$: $\langle P \rangle_E = \frac{\mu_0 p_0^2 \omega^4}{12\pi c}$.

Kirchhoff's Rules

Circuit laws: (1) Node rule (charge conservation): $\sum_k I_k = 0$; (2) Loop rule (energy conservation): $\sum_k V_k = 0$.

RC and RL time constants

Characteristic response times: $\tau_{RC} = RC$ for RC circuits, and $\tau_{RL} = \frac{L}{R}$ for RL circuits.

LC circuit resonant frequency

Natural angular frequency of an LC circuit: $\omega_0 = \frac{1}{\sqrt{LC}}$.

One-dimensional wave equation

Differential equation $\frac{\partial^2 f}{\partial t^2} = v^2 \frac{\partial^2 f}{\partial x^2}$, with general solutions $f(x \pm vt)$.

Phase velocity and Group velocity

For a wave with dispersion relation $\omega(k)$, phase velocity is $v_{\text{phase}} = \frac{\omega}{k}$ and group velocity is $v_{\text{group}} = \frac{d\omega}{dk}$.

Malus's Law

Transmitted intensity of polarized light passing through a polarizer at angle $\theta$ relative to its polarization direction: $I = I_0 \cos^2(\theta)$.

Brewster's angle ($\theta_B$)

Angle of incidence at which reflected light is completely polarized perpendicular to the incident plane: $\theta_B = \arctan\left(\frac{n_2}{n_1}\right)$.

Double-slit interference conditions

For slit distance $d$ and angle $\theta$: maxima at $d \sin(\theta) = m\lambda$ and minima at $d \sin(\theta) = \left(m + \frac{1}{2}\right)\lambda$.

Single-slit diffraction minima

Minima occur at $a \sin(\theta) = m\lambda$ for $m = 1, 2, 3, \dots$, where $a$ is slit width.

Optical path length

Equivalent path length in vacuum for light traveling distance $d$ in medium of index $n$: $\text{OPL} = nd$.

Thin-film reflection phase shift rules

Reflection off a boundary with $n_2 > n_1$ undergoes a phase shift of $\pi$; reflection off a boundary with $n_2 < n_1$ undergoes no phase shift.

Rayleigh criterion (Circular aperture)

Resolution limit for a circular aperture of diameter $D$: first minimum occurs at $D \sin(\theta) = 1.22\lambda$.

Bragg diffraction condition

Condition for constructive interference of x-rays scattered from crystal planes separated by distance $d$ at incidence angle $\theta$: $2d \sin(\theta) = n\lambda$.

Thin lens equation

Relation between object distance $s$, image distance $s'$, and focal length $f$: $\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$.