Formula solver index — formula.expert

Angular Velocity from Period

\omega = \frac{2\pi}{T}

MechanicsWaves & OscillationsPhysicsOne full revolution is 2π radians, so angular velocity is 2π divided by the period.

Wave Speed (v = fλ)

v = f \lambda

Waves & OscillationsPhysicsThe universal wave equation: a wave's speed equals its frequency times its wavelength.

Period-Frequency Relation

T = \frac{1}{f}

Waves & OscillationsPhysicsPeriod and frequency are reciprocals: seconds per cycle versus cycles per second.

Simple Pendulum Period

T = 2\pi \sqrt{\frac{L}{g}}

Waves & OscillationsMechanicsPhysicsPeriod of a simple pendulum swinging through small angles, with g = 9.80665 m/s² (standard gravity).

Fundamental Frequency of a String

f = \frac{v}{2L}

Waves & OscillationsPhysicsLowest standing-wave frequency of a string fixed at both ends.

Period of a Spring-Mass Oscillator

T = 2\pi \sqrt{\tfrac{m}{k}}

Waves & OscillationsMechanicsPhysicsPeriod of a mass bouncing on a spring, set only by the mass and the spring stiffness.

Sound Intensity (I = P/A)

I = \frac{P}{A}

Waves & OscillationsPhysicsSound intensity is the acoustic power passing through each square meter of surface.

Inverse-Square Law for Sound

I = \frac{P}{4\pi r^{2}}

Waves & OscillationsPhysicsA point source's intensity falls with the square of distance as its power spreads over an expanding sphere.

Decibel Sound Level

\beta = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

Waves & OscillationsPhysicsSound level in decibels compares an intensity to the threshold of hearing, I₀ = 10⁻¹² W/m².

Doppler Effect (Approaching Source)

f' = \frac{f v}{v - v_s}

Waves & OscillationsPhysicsObserved frequency rises when a sound source approaches, as each wavefront is emitted closer to the listener.

Doppler Effect (Approaching Observer)

f' = \frac{f (v + v_o)}{v}

Waves & OscillationsPhysicsA listener moving toward a stationary source meets wavefronts more often and hears a higher frequency.

Wave Speed on a String

v = \sqrt{\frac{F}{\mu}}

Waves & OscillationsPhysicsWaves travel faster on a tighter, lighter string: speed is the square root of tension over linear density.

Fundamental of a Closed Pipe

f = \frac{v}{4L}

Waves & OscillationsPhysicsA pipe closed at one end resonates with a quarter wavelength inside, giving a fundamental of v/4L.

Harmonic Frequencies

f_n = n f_1

Waves & OscillationsPhysicsThe nth harmonic of a vibrating system is n times its fundamental frequency.

Beat Frequency

f_{\text{beat}} = f_1 - f_2

Waves & OscillationsPhysicsTwo nearby tones interfere to produce a loudness pulse at their difference frequency, with f₁ the higher of the pair.

Speed of Sound in Air

v = 331.3 + 0.606\, T_C

Waves & OscillationsPhysicsThe speed of sound in dry air grows about 0.6 m/s for every degree Celsius above freezing.

Double-Slit Fringe Spacing

\Delta y = \frac{\lambda L}{d}

OpticsWaves & OscillationsPhysicsSpacing between adjacent bright fringes on a screen a distance L behind two slits d apart.

Diffraction Grating Equation

m \lambda = d \sin\theta

OpticsWaves & OscillationsPhysicsBright beams leave a grating at angles where the path difference d sin θ is a whole number of wavelengths.

Thin-Film Constructive Interference (Bright Reflection)

2 n t = \left(m + \tfrac{1}{2}\right)\lambda

OpticsWaves & OscillationsPhysicsBright-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a half-odd number of wavelengths.

Thin-Film Destructive Interference (Dark Reflection)

2 n t = m \lambda

OpticsWaves & OscillationsPhysicsDark-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a whole number of wavelengths.

LC Resonant Frequency

f = \frac{1}{2\pi\sqrt{LC}}

Electricity & MagnetismWaves & OscillationsPhysicsThe natural oscillation frequency of an inductor-capacitor pair.