Physics Calculators

Density

ρ = m / V

Density from mass and volume, or its inverse for any material.

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Force Equation

F = m · a

Newton's second law — force, mass, and acceleration.

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Static Friction

F_s ≤ μ_s · N

Maximum static friction force from coefficient and normal force.

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Kinetic Friction

F_k = μ_k · N

Kinetic friction force from kinetic coefficient and normal force.

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Circular Motion

v = ω · r · a_c = v² / r

Tangential velocity, centripetal acceleration, and centripetal force in circular motion.

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Tangential Velocity

v = ω · r = 2πr / T

Tangential (linear) speed of a particle moving along a circular path.

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Centripetal Acceleration

a_c = v² / r = ω² · r

Center-directed acceleration of an object moving in a circular path.

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Centripetal Force

F_c = m · v² / r

Inward force required to keep a body in circular motion.

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Orbital Period

T² = (4π² / GM) · a³

Orbital period from semi-major axis via Kepler's third law.

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Constant Acceleration Motion

v = v₀ + a · t · x = v₀·t + ½at²

Kinematic equations for one-dimensional motion under constant acceleration.

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Coulomb's Law

F = k · |q₁ · q₂| / r²

Electrostatic force between two point charges.

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Doppler Effect Equations

f' = f · (c ± v_o) / (c ∓ v_s)

Frequency shift for a moving source or receiver in the Doppler effect.

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Einstein Equation

E = m · c²

Mass-energy equivalence for any rest mass.

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Escape Velocity

v_e = √(2GM / r)

Minimum velocity to escape the gravitational pull of a planet or star.

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Gravitational Acceleration

g = G · M / r²

Gravitational acceleration at the surface of any planet or moon.

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Gravitational Force

F = G · m₁ · m₂ / r²

Gravitational force between two masses (Newton's law of universal gravitation).

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Gravity Equations

g = G · M / r² · F = G · m₁m₂ / r²

Gravitational acceleration, force, and Kepler's third law equations bundle.

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Hooke's Law

F = −k · x

Spring restoring force and potential energy for a Hookean spring.

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Impulse Momentum Equations

J = Δp = F · Δt

Impulse, momentum, and the impulse-momentum theorem.

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Impulse Force

F = Δp / Δt

Average force from change in momentum and contact time.

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Momentum

p = m · v

Linear momentum from mass and velocity.

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Impulse-Momentum Theorem

F · Δt = Δp = m · Δv

Connect average force × contact time to the change in linear momentum.

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Kepler's Third Law

T² / a³ = 4π² / GM

Orbital period and semi-major axis relationship for any two-body system.

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Kinetic Energy

KE = ½ · m · v²

Translational kinetic energy from mass and velocity.

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Lens & Mirror Equation

1/f = 1/do + 1/di

Thin lens and mirror equation for object distance, image distance, and focal length.

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Moment Equation

M = F · d

Moment of force about a pivot point — solve for force, distance, or moment.

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Newton's Second Law Equations

F_net = m · a

Net force, mass, and acceleration via Newton's second law of motion.

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Pendulum Formulas

T = 2π · √(L / g) (simple)

Simple and physical pendulum period, length, and moment of inertia equations bundle.

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Pendulum Period

T = 2π · √(L / g)

Period of a simple pendulum from length and gravitational acceleration.

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Pendulum Length

L = g · T² / (4π²)

Pendulum length required for a target period.

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Potential Energy

PE = m · g · h

Gravitational potential energy from mass, height, and gravitational acceleration.

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Power

P = W / t = F · v

Power from work over time or from force and velocity.

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Power from Work

P = W / t

Power as work done per unit time.

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Power from Force and Velocity

P = F · v

Instantaneous power from force and the velocity of its application.

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Projectile Motion

Range = v₀² · sin(2θ) / g

Range, max height, time of flight, and horizontal/vertical components of projectile motion.

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Radioactive Material

N = N₀ · e^(−λt)

Number of remaining nuclei, activity, and mass for radioactive decay.

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Half-Life

t_½ = ln(2) / λ

Half-life of a radioactive isotope from its decay constant.

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Radioactive Decay

N(t) = N₀ · e^(−λt)

Remaining quantity of a radioactive isotope after a given decay time.

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Radioactive Activity

A = λ · N = A₀ · e^(−λt)

Decay activity (Becquerels or curies) for a radioactive sample.

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Mean Lifetime

τ = 1 / λ = t_½ / ln(2)

Mean lifetime of a radioactive nucleus from decay constant or half-life.

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Snell's Law

n₁ · sin(θ₁) = n₂ · sin(θ₂)

Refraction of light at the boundary between two transparent media.

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Sound Wave Equations

I = P / A · SPL = 10·log₁₀(I/I₀)

Sound intensity, sound pressure level, and distance attenuation equations bundle.

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Sound Wave Speed

v = f · λ

Sound wave speed from frequency and wavelength.

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Sound Intensity Decibels

β = 10 · log₁₀(I / I₀)

Sound intensity level in decibels from measured and reference intensity.

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Specific Gravity

SG = ρ / ρ_water

Specific gravity from density relative to water.

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Specific Volume

v = V / m = 1 / ρ

Specific volume from total volume and mass, or density inverse.

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Stokes Law

v_t = 2(ρ_p − ρ) · g · r² / (9 · μ)

Terminal velocity of a small sphere settling through a viscous fluid.

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Stress Strain

σ = F / A · ε = ΔL / L · E = σ/ε

Stress, strain, and Young's modulus from force, area, and length change.

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Torque

τ = r × F = I · α

Torque from force times lever arm, or moment of inertia times angular acceleration.

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Wavelength, Frequency & Period

v = f · λ · T = 1 / f

Convert between wavelength, frequency, period, and wave speed.

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Weight Equation

W = m · g

Weight from mass and gravitational acceleration (Earth or other planet).

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Wien's Equations

λ_max · T = 2.898 × 10⁻³ m·K

Peak emission wavelength of a blackbody at a given temperature (Wien's displacement law).

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Wing Lift Equations

L = ½ · ρ · v² · S · C_L

Aerodynamic lift from air density, velocity, wing area, and lift coefficient.

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Work

W = F · d · cos(θ)

Mechanical work done by a force over a distance.

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Physics calculators covering classical mechanics (motion, force, energy, momentum), electromagnetism (Coulomb's law, Ohm's law, capacitors), waves (Doppler effect, sound), optics (lens/mirror equation, Snell's law), and gravitation (orbital mechanics, escape velocity).

When to use these calculators

Use these for physics homework, engineering pre-screening calculations, lab work, and cross-checks against experimental measurements. Each calculator solves for any variable in its equation and shows the substituted arithmetic step-by-step.

Frequently Asked Questions

How does the calculator show its work?
Each calculator displays the substituted arithmetic step-by-step below the solution, so the user can verify the math by hand if needed. The 'Copy result' button captures the full formula → substitution → answer block.
Are the formulas cross-verified?
Every calculator's math has been cross-verified against textbook references and round-trip consistency tests (solve A from B, then solve B from A — the result must match the input). Spot-checks against Wolfram Alpha confirm the precision.
How do I switch between solving for different variables?
Most calculators in this category support multiple solve-for modes. Pick the unknown variable from the pill-toggle (or equation-card group) above the input fields, then enter the known values. The calculator auto-computes as you type.