Fundamentals of Physics

Acceleration is the time derivative of velocity, the second of position.

The constant-acceleration formulas come from integrating this.

Total pressure is constant along a streamline.

Dynamic, hydrostatic, and static pressure trade off.

Archimedes: the buoyant force equals the weight of displaced fluid.

A body floats when this matches its weight.

Ratio of separation speed to approach speed in a collision.

$e = 1$ is perfectly elastic, $e = 0$ perfectly plastic.

Mass per unit volume.

Connects the geometry formulas to mass and weight.

The three direction cosines of a vector are not independent.

They are the components of the unit vector along x, y, z.

Fraction of input power delivered as useful output.

Always below 1 for real machines.

Potential energy near the surface of the Earth.

Only height differences matter; the zero is your choice.

Gauge pressure at depth $h$ in a fluid of density $\rho$.

Independent of the container shape.

Force integrated over time; it equals the change in momentum.

A small force over a long time can match a large brief one.

Translational kinetic energy.

Scales with the square of speed: double the speed, quadruple the energy.

Each side of a triangle over the sine of its opposite angle is constant.

Useful for resolving non-right-angle force triangles.

Momentum is mass times velocity (written $L$ here).

Conserved when no external force acts.

Mass passing a cross section per unit time.

Constant along a pipe for incompressible flow.

Torque is force times the perpendicular lever arm $d$.

Greatest when the force is perpendicular to the arm.

Solid cylinder or disk about its central axis.

Mass far from the axis raises $I$ fast (the $R^2$).

Thin rectangular plate about a central perpendicular axis.

$a$ and $b$ are the side lengths.

Thin rod about its center.

About one end it becomes $\tfrac{1}{3} m L^2$ (parallel-axis).

Solid uniform sphere about a diameter.

A hollow sphere gives $\tfrac{2}{3} m R^2$ instead.

Torque drives angular acceleration, the rotational $F = ma$.

$I$ is the moment of inertia, the rotational analogue of mass.

Pressure is transmitted equally through a fluid.

A small force on a small piston balances a large force on a large one.

Power is the rate of doing work.

The same work in less time means more power.

Instantaneous mechanical power for a force moving at speed $v$.

Use the component of force along the motion.

Horizontal motion of a projectile: constant velocity, no acceleration.

Pair it with the vertical equation to get the trajectory.

Magnitude of the resultant via the law of cosines.

$\alpha$ is the angle of the force triangle opposite the resultant.

Rotational analogue of $\tfrac{1}{2} m v^2$.

A rolling body carries both translational and rotational KE.

Torque times angular velocity.

The rotational twin of $P = F v$.

Hooke’s law for a spring: force per unit stretch.

A stiffer spring has a larger $k$.

Energy stored in a spring stretched by $s$.

It is the area under the $F = k s$ line.

Stress-strain relation in the plastic region.

$n$ is the hardening exponent, $K$ the strength coefficient.

Surface area of a sphere of radius $R$.

Shows up in radiation and flux problems.

Result is perpendicular to both inputs; its length is $|A|\,|B|\sin\theta$.

Order matters: $\vec{A}\times\vec{B} = -\,\vec{B}\times\vec{A}$.

Velocity under constant acceleration.

Slope of the velocity-time line is $a_c$.

Links speed to distance without needing time.

Rearrange for stopping distance or launch speed.

Volume passing a cross section per unit time.

Continuity keeps $v A$ constant: narrow means fast.

Volume of a sphere of radius $R$.

Pairs with density to get mass.

Work is the integral of force over the path.

For a constant force it reduces to $F \cdot s$.

Work to change a spring deflection from $s_0$ to $s_1$.

Equals the change in stored spring energy.

Net work equals the change in kinetic energy.

Positive work speeds a body up, negative slows it.

Average translational kinetic energy per molecule.

Depends only on temperature, not on the gas.

Boltzmann’s constant is the gas constant per molecule.

Connects macroscopic $R$ to microscopic energy.

The maximum efficiency, set only by the two temperatures.

No engine between the same reservoirs can beat it.

Converts Celsius to absolute temperature.

Use Kelvin in all gas-law and energy formulas.

Molar heat capacity of a solid at high temperature.

Six degrees of freedom per atom (kinetic plus potential).

Differential form of the enthalpy.

At constant pressure $dH = dU + p\,dV = dQ$.

Entropy change from reversible heat transfer.

Path-independent because $S$ is a state function.

Heat added to a gas at constant pressure.

Larger than the constant-volume case: the gas also does work.

Heat added to a gas held at constant volume.

All of it goes into internal energy; none into work.

Constant-volume molar heat capacity of a diatomic gas.

Adds two rotational degrees to the three translational.

Constant-volume molar heat capacity of a monatomic ideal gas.

Three translational degrees of freedom, each worth $\tfrac{1}{2} R$.

Heat conduction rate through a slab of thickness $L$.

$k$ is the thermal conductivity.

Rate of heat transfer by convection.

$h$ is the convective coefficient, set by the flow.

Net radiated power from a surface (Stefan-Boltzmann).

The $T^4$ makes radiation dominate at high temperatures.

Net work out per unit heat drawn from the hot reservoir.

The minus sign follows the work-on-the-gas convention.

The ideal gas law in molar form.

Temperature must be in Kelvin.

The gas law written per molecule with Boltzmann’s constant.

$N$ is the number of molecules, not moles.

Entropy change for heat $Q_{rev}$ at constant temperature.

Used for phase changes and isothermal steps.

Heat for a phase change at constant temperature.

$L$ is the latent heat of fusion or vaporization.

Fractional length change with temperature.

$\alpha$ is the linear expansion coefficient.

The molar heat capacities differ by the gas constant.

The extra $R$ is the work of expanding at constant pressure.

Average distance between molecular collisions.

Larger molecules or a denser gas shorten it.

Moles from sample mass and molar mass $M$.

Lets you put a mass into the ideal gas law.

Moles from molecule count via Avogadro’s number.

Bridges the molar and per-molecule gas laws.

Root-mean-square speed of ideal-gas molecules.

Lighter gases and higher temperatures move faster.

Heat to change a substance’s temperature.

$c$ is the specific heat capacity.

Fractional volume change with temperature.

For isotropic solids $\beta \approx 3\alpha$.

Work done on a gas as its volume changes.

Expansion ($dV > 0$) means the gas does work, so this is negative.

Charge stored per volt: the definition of capacitance.

Measured in farads.

Capacitor current leads its voltage by 90 degrees.

Current peaks while the voltage is zero.

Charge decays exponentially as a capacitor discharges.

After one time constant $RC$, about 37% remains.

Voltage across a capacitor in AC.

Lags the current by a quarter cycle.

Capacitors in parallel add directly.

They share the same voltage.

Reciprocals add for capacitors in series.

The combination is smaller than any single one.

The Coulomb constant in terms of the vacuum permittivity.

About $8.99 \times 10^{9}\ \mathrm{N\,m^2/C^2}$.

Force between two point charges.

Attractive for opposite signs, repulsive for like.

Current as the flux of current density through a cross section.

Use it when current is spread unevenly.

Energy of a dipole in a field.

Lowest when the dipole lines up with the field.

Current is charge flow per unit time.

One ampere is one coulomb per second.

Charge times separation, pointing from $-$ to $+$.

Sets how a dipole feels a field and makes one.

Electric flux through a surface.

Counts the field lines crossing the area.

Energy stored in a charged capacitor.

Held in the electric field between the plates.

Energy stored in an inductor’s magnetic field.

Mirrors the capacitor’s $\tfrac{1}{2} C V^2$.

Field inside a solenoid of $N$ turns over length $l$.

Nearly uniform along the axis.

Field on the axis of a dipole, far away.

Falls off as $1/z^3$, faster than a point charge.

Magnetic field around a long straight wire.

Circles the wire, weakening with distance.

Force on a moving charge near a current-carrying wire.

Combines the wire’s field with the Lorentz force.

Force on a charge in an electric field.

This defines the field as force per unit charge.

Force on a current-carrying wire element in a field.

Integrate along the wire for the total force.

Force on a straight wire of length $L$ in a uniform field.

Maximum when the wire is perpendicular to $B$.

Enclosed charge sets the total electric flux.

Field lines start and end on charge.

Inductor current lags its voltage by 90 degrees.

The opposite phase shift to a capacitor.

Voltage across an inductor opposes a change in current.

Inductors resist sudden jumps in current.

Voltage across an inductor in AC.

Leads the current by a quarter cycle.

Magnetic force on a moving charge.

Perpendicular to velocity and field, so it does no work.

Magnetic flux through a surface.

A changing flux is what induces an emf.

Flux linked in one coil due to current in another.

$M$ is the mutual inductance.

Emf induced in coil 2 by a changing current in coil 1.

The minus sign is Lenz’s law.

Voltage across a resistor is current times resistance.

Linear for ohmic materials.

Capacitance of a parallel-plate capacitor.

Bigger plates or a thinner gap store more charge.

Voltage as work per charge against the field.

The field points from high to low potential.

Energy of a charge at potential $V$.

Voltage is potential energy per unit charge.

Energy of a charge moved a distance $s$ in a uniform field.

The electrical analogue of $mgh$.

Potential energy of a pair of point charges.

Positive for like charges, which want to fly apart.

Power dissipated as heat in a resistor.

Equivalent to $V^2/R$ or $VI$.

Time constant of an RC circuit.

Sets how fast it charges or discharges.

Resistance from resistivity, length, and cross section.

Long thin wires resist more.

Resistor current in AC.

No phase shift relative to its voltage.

Voltage across a resistor in AC.

In phase with the current through it.

Reciprocals add for resistors in parallel.

The combination is smaller than the smallest one.

Resistances in series add.

They carry the same current.

Resonant angular frequency of an LC or RLC circuit.

Where inductive and capacitive reactances are equal.

Current rising toward its steady value in an RL circuit.

Reaches about 63% after one time constant.

Time constant of an RL circuit.

Larger inductance or smaller resistance slows the change.

Flux a coil links through its own current.

$L$ is the self-inductance.

Voltage ratio equals the turns ratio.

More secondary turns step the voltage up.

Angular frequency in radians per second.

One full cycle is $2\pi$ radians.

The damping coefficient at the boundary case.

Below it the system oscillates, above it it just creeps back.

Oscillation frequency of a damped system.

Damping $\xi$ lowers it below the natural frequency.

Displacement of an underdamped oscillator.

The amplitude decays exponentially inside the cosine.

Shifted frequency when source or detector moves.

Approaching raises the pitch, receding lowers it.

Phase lag of the response behind the drive.

Passes through 90 degrees at resonance.

Frequency is the reciprocal of the period.

Hertz means cycles per second.

Period of a mass-spring oscillator.

Independent of amplitude for SHM.

Phase accumulated after time $t$.

Sets where in the cycle the motion is.

Undamped simple harmonic motion.

$A$ is the amplitude, $\phi$ the starting phase.

A sinusoidal wave moving in the $+x$ direction.

$y_m$ is the amplitude, $k$ the wave number.

The spatial analogue of angular frequency.

One wavelength is $2\pi$ in phase.

Bulk modulus from Young’s modulus and Poisson’s ratio $v$.

Resistance to uniform compression.

Bright fringes: path difference is a whole number of wavelengths.

$j$ is the order of the fringe.

Dark fringes: path difference is a half-integer of wavelengths.

The waves arrive exactly out of phase.

Intensity spreading from a point source.

Inverse-square: double the distance, quarter the intensity.

Focal length from the lens shape and material.

Flatter surfaces (larger $R$) give a longer focal length.

Extra path between two slits at angle $\theta$.

It drives the interference pattern.

Phase difference produced by a path difference.

One wavelength of path is $2\pi$ of phase.

Sound pressure amplitude from the displacement amplitude $y_m$.

Higher frequency means more pressure for the same displacement.

Refraction at a boundary between two media.

Light bends toward the normal entering a denser medium.

Sound intensity from the displacement amplitude.

Grows with the square of both frequency and amplitude.

Sound level in decibels relative to a reference $I_0$.

Every 10 dB is a tenfold jump in intensity.

Pressure variation of a traveling sound wave.

$p_m$ is the pressure amplitude.

Light slows by the refractive index $n$ in a medium.

Since $n \ge 1$, light is fastest in vacuum.

Speed of sound from bulk modulus and density.

Stiffer, lighter media carry sound faster.

Wavelength shrinks by $n$ inside a medium.

Frequency stays the same; speed and wavelength drop.

Activity decays exponentially with the same lifetime.

It falls in step with the number of nuclei.

Energy equivalent of one atomic mass unit.

Handy for nuclear binding-energy sums.

Frequency and wavelength of light are inversely linked.

Their product is the speed of light.

Half-life is the mean lifetime times $\ln 2$.

The time for half the sample to decay.

A mass defect converts to energy in nuclear reactions.

The source of fission and fusion energy.

Initial decay rate from the nucleus count and lifetime $\tau$.

Activity is decays per second (becquerel).

Number of undecayed nuclei over time.

After one mean lifetime $\tau$, a fraction $1/e$ remains.