Gravitational Force Calculator finds the pull between two masses from Newton’s law, or solves for a mass or the distance, and shows every step, the local gravity and orbital speed.
Type big numbers as 5.9722e24 or 5.9722*10^24. Distance (r) is measured center to center.
Result Unit, Height and G
Step-by-Step Working
- Known values: m₁ = 5.9722 × 10²⁴ kg, m₂ = 60 kg, r = 6,371 km
- Convert to SI units: r = 6.371 × 10⁶ m
- Gravitational constant: G = 6.6743 × 10⁻¹¹ N·m²/kg²
- Formula: F = G × m₁ × m₂ ÷ r²
- Substitute: 6.6743 × 10⁻¹¹ × 5.9722 × 10²⁴ × 60 ÷ (6.371 × 10⁶)²
- Result: F = 589.218 N
Force at Other Distances
| Distance | Center gap (km) | Force (N) | Potential energy (J) |
|---|---|---|---|
| ¼ r | 1,593 | 9,427 | −1.502 × 10¹⁰ |
| ½ r | 3,186 | 2,357 | −7.508 × 10⁹ |
| r (yours) | 6,371 | 589.2 | −3.754 × 10⁹ |
| 2 r | 12,740 | 147.3 | −1.877 × 10⁹ |
| 3 r | 19,110 | 65.47 | −1.251 × 10⁹ |
| 5 r | 31,860 | 23.57 | −7.508 × 10⁸ |
| 10 r | 63,710 | 5.892 | −3.754 × 10⁸ |
How the Gravitational Force Calculator Works
The Gravitational Force Calculator applies Newton’s law of universal gravitation to any two masses. Set Solve For to the force, to either mass or to the distance, and the three remaining values become the inputs.
It opens on Earth and a 60 kg person standing at the surface. Earth’s mass is 5.9722 × 10²⁴ kg and its mean radius is 6,371 km, so the pull comes out at 589.2 N on each of the two bodies.
The Example Pair list also loads Earth and the Moon, the Sun and Earth, and two 70 kg people standing 1 m apart. Large values can be typed as 5.9722e24 or 5.9722*10^24. Each field has its own unit menu, with Earth masses, solar masses and astronomical units next to the everyday units.
It sits with the other Dynamics and Statics calculators, because gravity is one more force that obeys Newton’s laws of motion. The four result cards then turn that single force into local gravity, other force units, orbital figures and the change with distance.
Gravitational Force Formula
Two masses attract each other with a force that grows with the product of the masses and falls with the square of the distance between their centers. OpenStax University Physics, section 13.1, states Newton’s law in this form.
$$F = G \frac{m_1 m_2}{r^2}$$
F is the gravitational force in newtons, m₁ and m₂ are the two masses in kilograms, and r is the center-to-center distance in meters. G is the gravitational constant, 6.6743 × 10⁻¹¹ N·m²/kg², taken from the 2022 CODATA values published by NIST.
Many people type the law as f = (g * m1 * m2) / r2, which is the same equation written on one line. The g in that version is the constant G and not the 9.8 m/s² of free-fall problems. Swapping one for the other makes the answer about 10¹¹ times too large.
The Gravitational Force Calculator handles the other three unknowns by rearranging the same law. Solving for one mass, or for the distance, gives the two forms below, and the Mass 2 mode simply swaps m₁ and m₂.
$$m_1 = \frac{F r^2}{G m_2}$$
$$r = \sqrt{\frac{G m_1 m_2}{F}}$$
The distance is always measured between centers, never between surfaces. For a person on the ground that means Earth’s full radius, even though the gap between feet and soil is zero. The law is exact for point masses and for spheres, which act as if all their mass sat at the center.
Gravitational force unit
The SI unit of gravitational force is the newton, the force that gives 1 kg an acceleration of 1 m/s². The Result Unit menu switches the answer to pounds-force, kilograms-force or dynes. For a push or pull that comes from mass and acceleration instead of two attracting bodies, the Force Calculator uses F = ma.
Finding the Force from Mass and Distance
To find gravitational force with mass and distance, put both masses in kilograms and the distance in meters. Multiply the two masses by G, then divide by the distance squared. The Step-by-Step Working panel writes out these lines for whatever has been entered, which makes it a gravitational force calculator with steps for checking physics homework.
For the opening example of Earth and a 60 kg person, the working runs in the order below. The figures match the lines the panel prints, including the conversion of 6,371 km to meters.
- Known values are m₁ = 5.9722 × 10²⁴ kg, m₂ = 60 kg and r = 6,371 km.
- Convert the distance to SI units, which gives r = 6.371 × 10⁶ m.
- Multiply G × m₁ × m₂ = 6.6743 × 10⁻¹¹ × 5.9722 × 10²⁴ × 60 = 2.3916 × 10¹⁶.
- Square the distance, (6.371 × 10⁶)² = 4.0590 × 10¹³ m².
- Divide the first number by the second to get F = 589.2 N.
Both bodies feel that same 589.2 N, pointing toward each other. This is Newton’s third law, and it is why the main result reads “on Each Mass”. The lighter body moves far more because the same force acts on far less mass.
Earth’s Gravity and the 9.8 Value
The gravitational force formula for Earth is the general law with Earth’s mass as m₁ and the distance from Earth’s center as r. Dividing the force by the second mass leaves the acceleration that every object shares at that distance.
$$g = \frac{G M}{R^2}$$
With M = 5.9722 × 10²⁴ kg and R = 6.371 × 10⁶ m, g comes to 9.820 m/s². That is how the familiar 9.8 is calculated, and it agrees with the mean surface gravity on NASA’s Earth Fact Sheet. The first result card shows it as Gravity at m₂.
The value shifts a little from place to place. NASA lists 9.780 m/s² at the equator and 9.832 m/s² at the poles, while the standard value used to define units is exactly 9.80665 m/s². This is why the 60 kg person weighs 589.2 N here but 588.4 N when multiplied by standard gravity.
The same card lists Gravity at m₁, the acceleration of Earth toward the person, at 9.866 × 10⁻²³ m/s². The force on each is equal, and a = F ÷ m does the rest. The Acceleration Calculator covers that step for any force and mass.
Gravity also fades with height. Open the accordion, switch Distance Entered As to Height above surface and enter 400 km over Earth’s 6,371 km radius. The field drops to 8.694 m/s² and the pull on the person to 521.7 N, about 89% of the surface figure.
Weight only halves at 2,639 km above the ground, where the distance from Earth’s center has grown by a factor of √2. Near the surface the change is small enough that the Free Fall Calculator can treat g as constant during a drop.
Gravitational Force of a Planet or Moon
The Gravitational Force Calculator also gives weight on other worlds. To calculate the gravitational force of a planet on something at its surface, enter the planet’s mass as Mass 1 and the object as Mass 2. Use the planet’s mean radius as the distance. The force is the object’s weight on that world, and the first card gives the surface gravity.
For the Moon, NASA’s Moon Fact Sheet gives a mass of 7.346 × 10²² kg and a mean radius of 1,737.4 km. A 60 kg person there is pulled with 97.46 N, or 9.94 kgf, under a surface gravity of 1.624 m/s². That is about one sixth of the Earth figure.
The built-in pairs give the gravitational force value for the best-known cases. The table lists each one with the orbital speed and period from the third result card, rounded the way the tool displays them.
| Pair | Mass 1 | Mass 2 | Distance | Force | Orbital speed | Period |
|---|---|---|---|---|---|---|
| Earth and a 60 kg person | 5.9722 × 10²⁴ kg | 60 kg | 6,371 km | 589.2 N | 7.91 km/s | 84.35 min |
| Earth and the Moon | 5.9722 × 10²⁴ kg | 7.346 × 10²² kg | 384,400 km | 1.982 × 10²⁰ N | 1.025 km/s | 27.28 days |
| Sun and the Earth | 1.9884 × 10³⁰ kg | 5.9722 × 10²⁴ kg | 1.49598 × 10⁸ km | 3.542 × 10²² N | 29.78 km/s | 365.3 days |
| Two 70 kg people | 70 kg | 70 kg | 1 m | 3.27 × 10⁻⁷ N | 9.666 × 10⁻⁵ m/s | 18.06 h |
The masses and distances are the ones on NASA’s Earth, Moon and Sun fact sheets. NASA lists the Moon’s mean orbital velocity as 1.022 km/s over 27.32 days, and Earth’s as 29.78 km/s over 365.256 days. The circular-orbit results land within half a percent of both.
Newtons, kgf and lbf
The second card of the Gravitational Force Calculator restates the pull in kilograms-force, pounds-force and US tons-force. One kilogram-force is the weight of 1 kg under standard gravity, exactly 9.80665 N. One pound-force is 4.448 N, and a US ton-force is 2,000 lbf.
These units answer questions such as how much force 60 kg is. Under standard gravity 60 kg weighs 60 kgf, which is 588.4 N or 132.3 lbf. The card shows 60.08 kgf for the opening example because gravity at Earth’s mean radius is slightly above the standard value.
| Mass | Weight in newtons | Weight in kgf | Weight in lbf |
|---|---|---|---|
| 1 kg | 9.807 N | 1 kgf | 2.205 lbf |
| 5.2 kg | 51.0 N | 5.2 kgf | 11.46 lbf |
| 60 kg | 588.4 N | 60 kgf | 132.3 lbf |
| 68.04 kg (150 lb) | 667.2 N | 68.04 kgf | 150 lbf |
Every row uses standard gravity, so the kgf number equals the mass in kilograms. Away from Earth that shortcut fails. The 60 kg person on the Moon still has 60 kg of mass but a weight of only 9.94 kgf.
Inverse-Square Law and Distance
The fourth card shows what the square in r² does. At twice the distance the 589.2 N pull falls to 147.3 N, one quarter. At half the distance it rises to 2,357 N, four times as much. Tripling the distance would leave one ninth.
The Per meter farther row gives the slope of the force at the entered distance, in newtons per meter. It comes from differentiating Newton’s law with respect to r, which leaves the short ratio below.
$$\frac{dF}{dr} = -\frac{2F}{r}$$
For the person at Earth’s surface the slope is −1.85 × 10⁻⁴ N per meter. Climbing a 100 m tower therefore trims the weight by about 0.0185 N, far less than a bathroom scale can show.
The Force at Other Distances panel repeats the calculation from one quarter of r out to ten times r. For the opening example it runs from 9,427 N down to 5.892 N. Its last column is the gravitational potential energy of the pair, set to zero at infinite separation.
$$U = -\frac{G m_1 m_2}{r}$$
The person at the surface sits at −3.754 × 10⁹ J. The value is negative because energy has to be added to pull the two bodies apart, and it rises toward zero as the distance grows.
Orbital Speed, Period and Escape Speed
The third card treats the entered distance as the radius of a circular orbit. In orbit, gravity supplies the whole centripetal force. A turning car gets that inward force from the road instead, as in the Angle of Banking Calculator.
$$v = \sqrt{\frac{G (m_1 + m_2)}{r}}$$
$$T = 2 \pi \sqrt{\frac{r^3}{G (m_1 + m_2)}}$$
Escape speed is √2 times the orbital speed at the same distance. At Earth’s surface radius the card gives 7.91 km/s for a grazing orbit, a period of 84.35 minutes and an escape speed of 11.19 km/s. NASA’s Earth sheet lists 11.186 km/s.
Both masses appear in the formulas because the two bodies circle their shared center of mass. For a satellite the second mass changes nothing visible. For equal bodies it matters, and two 70 kg people 1 m apart in empty space would circle each other once every 18.06 hours.
The message under the cards changes with the result. It reads Too Small to Feel below one millionth of a newton and Planet-Scale Force from 10¹⁵ N upward. When escape speed passes 10% of the speed of light it warns that Newton’s law no longer holds and general relativity is needed.
Gravitational Force Calculator FAQ
How much is 51 N of gravitational force in kg?
About 5.2 kg. Set Solve For to Mass 2, keep Earth’s mass and the 6,371 km radius, and enter 51 N as the force. The Gravitational Force Calculator returns 5.193 kg. Dividing 51 N by standard gravity, 9.80665 m/s², gives 5.200 kg, and the small gap is the same difference in g described above.
Is 1 kg equal to 9.8 newtons?
Not exactly, because a kilogram measures mass and a newton measures force. What is true is that the force of gravity on 1 kg at Earth’s surface is about 9.8 N. Under standard gravity it is 9.80665 N, which is the definition of 1 kgf and equals 2.205 lbf.
Newton’s law with Earth’s mean radius gives 9.820 N for the same kilogram. On the Moon that kilogram would weigh only 1.624 N, although its mass has not changed. Scales that read in kilograms assume Earth’s gravity, which is why the difference rarely shows up in daily life.
Is g 9.8 or 10, and is gravity 9.8 kg?
Gravity at Earth’s surface is an acceleration, so its unit is m/s² and never kg. One g, or “1 gravity”, is defined as exactly 9.80665 m/s². Writing 9.8 keeps two significant figures and 9.81 keeps three, so match the figure to the precision of the other numbers in the problem.
Using 10 m/s² is a rounding for mental estimates and school problems that ask for it. It overstates every weight by about 2%, which is fine for a rough check and too coarse for lab work.
How many kg is a 5 g force?
A g-force multiplies weight, so the answer depends on the mass being accelerated. A 60 kg person at 5 g feels 5 × 60 × 9.80665 = 2,942 N, the same as 300 kgf. Loads of that size last only an instant in a collision, and the Car Crash Calculator works them out from speed and stopping distance.
Can you feel 1 newton of force?
Yes. One newton is the weight of a 102 g object under standard gravity, close to a small apple resting on the palm. It equals 0.102 kgf or 0.2248 lbf. By comparison, the pull between two 70 kg people 1 m apart is 3.27 × 10⁻⁷ N, about three million times weaker than that apple’s weight.
How many kg is a 150 pound woman?
A 150 lb person has a mass of 68.04 kg, since one pound is 0.45359237 kg. Her weight under standard gravity is 667.2 N. Entered in the Gravitational Force Calculator as 150 lb against Earth at 6,371 km, the result is 668.2 N, or 150.2 lbf. The extra 0.2 lbf appears because g at the mean radius is 9.820 m/s².
Is gravitation a weak force?
Between ordinary objects, yes. Two 70 kg people standing 1 m apart attract each other with 3.27 × 10⁻⁷ N. Each of them weighs about 686 N, so Earth’s pull on one person is roughly two billion times stronger than the pull of the neighbor.
Gravity still rules at large scales because it only attracts and it adds up with mass. With planet-sized bodies the numbers become enormous. The Sun holds Earth with 3.542 × 10²² N, and Earth holds the Moon with 1.982 × 10²⁰ N.