Newton's Law of Universal Gravitation Calculator
Newton's law of universal gravitation states that every pair of masses attracts each other with a force proportional to the product of the masses and inversely proportional to the square of the distance. The force is always attractive — there is no negative mass to repel. This tool accepts inputs from grams to solar masses, and from millimetres to astronomical units, covering both everyday and astronomical scales.
Enter valid non-negative masses; distance must be greater than 0.
Force magnitude F
—
Gravity is always attractive
Other units
- Millinewtons (mN)
- —
- Micronewtons (μN)
- —
- Kilonewtons (kN)
- —
Formula
F = G · m₁ · m₂ / r², G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻²
Uses the CODATA 2018 recommended value G = 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻². Mass must be non-negative.
Formula
F = G · m₁ · m₂ / r², G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻²
- · Uses the CODATA 2018 recommended value G = 6.67430 × 10⁻¹¹ m³·kg⁻¹·s⁻² (relative uncertainty ~2.2 × 10⁻⁵ — one of the least precisely known fundamental constants).
- · Mass must be non-negative; distance must be greater than 0 (r = 0 is a singularity).
- · Result is in newtons (N), with mN / μN / kN shown alongside so you can size it up against everyday forces.
- · Distance is measured between the two centres of mass. For a uniform sphere viewed from outside, mass behaves as if concentrated at the centre (shell theorem).
- · Reference conversions: 1 lb = 0.45359237 kg; 1 au = 1.495978707 × 10¹¹ m (exact since IAU 2012 Resolution B2); M⊕ ≈ 5.9722 × 10²⁴ kg; M☉ ≈ 1.98892 × 10³⁰ kg.
- · Like Coulomb's law, gravity obeys an inverse-square law — but at the atomic scale, the electrostatic force between an electron and a proton is ~10³⁶ times stronger than their gravitational attraction.
- · Sources: Halliday-Resnick-Walker Fundamentals of Physics §13, CODATA 2018, IAU 2012 Resolution B2, NASA Earth/Sun fact sheets.
Frequently asked
Why is the result always attractive (no sign for direction)?
Mass has no sign — there is no negative mass — so gravity is always attractive, unlike Coulomb's law where like charges repel. For directional problems (e.g. components along x / y axes), treat each pairwise force as a vector and add them using your chosen coordinate axes.
Can I use this tool to compute the Earth-surface gravitational acceleration g?
Yes. Set m₁ = 1 M⊕, m₂ = 1 kg, r = 1 R⊕. The result is ~9.82 N, which is the weight of a 1-kg mass at the surface and equals the local g. Actual g varies slightly with latitude, altitude and crustal density (typically 9.78–9.83 m/s²).
How much gravity acts between two people standing next to each other?
Tiny. Two 70 kg people 1 m apart experience F = 6.6743 × 10⁻¹¹ · 70 · 70 / 1² ≈ 3.3 × 10⁻⁷ N — about the weight of a single grain of sand. In daily life we only feel Earth's gravity.
Why does the Moon orbit the Earth?
Gravity supplies the centripetal force that bends the Moon's tangential motion into a (slightly elliptical) orbit. With M⊕ = 5.972 × 10²⁴ kg, M_Moon = 7.342 × 10²² kg and an average separation of 3.844 × 10⁸ m, the gravitational pull is ~1.98 × 10²⁰ N — exactly what is needed to keep the Moon moving at its orbital speed of ~1.022 km/s.
Is this formula accurate near a black hole or for highly relativistic objects?
Newton's law is the weak-field, low-velocity limit of general relativity. Near a black hole or neutron star, in strong-lensing regimes, or for speeds approaching c, you need Einstein's field equations (GR). For everyday physics — Earth, Solar-System dynamics — Newton's formula is accurate to better than 0.1 %.
Related tools
Ohm's Law Calculator (V / I / R / P)
Enter any two of voltage, current, resistance, or power — the calculator solves for the other two using V = IR and P = VI.
Speed, Distance & Time Calculator
Enter any two of distance, time and speed to get the third — with km/h, mph, m/s, km, miles, hours and minutes supported.
Density Calculator (mass / volume)
Compute density from mass and volume (ρ = m / V), or solve for the missing variable. Built-in reference table for 19 common substances.
Projectile Motion Calculator
Enter launch speed, angle and height to compute projectile range, peak height and flight time (no air resistance). Pick from Earth, Moon, Mars and more.
Wind Chill Calculator
Compute the wind chill (feels-like temperature) from air temperature and wind speed using the 2001 Environment Canada / US NWS formula, with frostbite risk levels.
Dew Point Calculator
Compute dew point from air temperature and relative humidity using the Magnus formula — handy for HVAC, photography and weather analysis.
Kinetic Energy Calculator (KE = ½ m v²)
Compute kinetic energy KE = ½ m v² with mixed units (kg / g / lb and m/s / km/h / mph) and see the result in joules, kilojoules, food calories, foot-pounds and watt-hours.
Half-Life & Exponential Decay Calculator
Enter any three of initial amount, remaining amount, elapsed time and half-life to solve for the fourth — useful for radioactive decay, drug pharmacokinetics and radiometric dating.
Resistor Color Code Calculator (4 / 5 band)
Pick the colour bands and instantly read the resistance and tolerance — 4-band and 5-band notations supported, with Ω / kΩ / MΩ formatting and a closest E12 / E24 preferred-value check.
GPS Distance Calculator (Haversine)
Enter two latitude/longitude pairs to compute the great-circle distance using the haversine formula (km, miles, nautical miles), with bearing and midpoint.
Solution Dilution Calculator (C₁V₁ = C₂V₂)
Solve any one of C₁, V₁, C₂, V₂ from the dilution equation C₁V₁ = C₂V₂ — a daily lab essential for chemistry, biology and pharmacy work.
Decibel (dB) Sum Calculator
Two 80 dB sound sources do not equal 160 dB. Enter multiple dB values to compute the combined SPL, and subtract background noise to recover the signal alone.
Resistor Parallel / Series Calculator
Enter up to 8 resistor values to see the series total (R₁ + R₂ + …) and the parallel total (1 / Σ(1/Rᵢ)) at the same time.
Wavelength ↔ Frequency Calculator
Convert between electromagnetic wavelength and frequency via c = λf, with the matching spectrum band (radio / microwave / visible / X-ray / γ) and photon energy.
Tank Volume Calculator
Compute the capacity of vertical or horizontal cylindrical, rectangular and spherical tanks, including partial-fill volumes at a given liquid level.
Pendulum Period Calculator (T = 2π√L/g)
Enter the pendulum length and local gravity to get the period, frequency and angular frequency, with Earth/Moon/Mars/Jupiter presets — and reverse-solve for the length needed to hit a target period.
Heat Index Calculator
Enter air temperature and relative humidity to get the apparent temperature (NOAA Rothfusz heat index) and the corresponding heat-stress risk band.
Vehicle Stopping Distance Calculator
Enter speed, reaction time and road friction to estimate reaction, braking and total stopping distance.
Snell's Law Refraction Calculator
Enter the refractive indices of two media and an angle of incidence — get the refraction angle and critical angle from Snell's law (n₁ sin θ₁ = n₂ sin θ₂).
Capacitor Energy Calculator
Enter capacitance (F, mF, µF, nF, pF) and voltage to compute the stored energy (E = ½CV²) and charge (Q = CV) on a capacitor.
Boiling Point at Altitude Calculator
Enter altitude to compute the boiling point of water (°C / °F) and local air pressure using the ICAO standard atmosphere and the Antoine equation — useful for hiking, cooking and high-altitude baking.
Specific Heat (Q = mcΔT) Calculator
Solve Q = m × c × ΔT for any one of heat energy, mass, specific heat capacity or temperature change — with presets for water, aluminium, iron, copper, glass, air and more.
pH and Hydrogen Ion Concentration Calculator
Convert between pH, pOH, hydrogen-ion concentration [H⁺] and hydroxide concentration [OH⁻] — with acid / neutral / alkaline classification.
Ideal Gas Law (PV = nRT) Calculator
Pick the unknown (P, V, n or T), enter the other three and PV = nRT is solved instantly — works in Pa / kPa / atm / bar / mmHg / psi, m³ / L / mL, mol / mmol / kmol and K / °C / °F.
Coulomb's Law Calculator
Enter two point charges and the separation distance to compute the electrostatic force between them via F = kₑ·q₁·q₂/r² (attractive or repulsive).
Distance to Horizon Calculator
Enter the observer eye height above the surface to compute the distance to the geometric and refraction-corrected horizon.
Hooke's Law (Spring Force) Calculator
Given any two of spring constant k, displacement x or restoring force F, solve for the third and the elastic potential energy U = ½kx².