Universal Gravitation Calculator
Calculate gravitational force, mass or distance using Newton's universal law of gravitation, with clear worked equations.
Universal Gravitation Diagram and Values
Step-by-Step Solution
Universal Gravitation Calculator
This Universal Gravitation Calculator helps you calculate the gravitational force between any two objects. It uses Newton's law of universal gravitation, the standard physics equation used to describe the attraction between planets, moons, satellites, stars, and everyday objects. Enter two masses and the centre-to-centre distance to find force in newtons. You can also choose to solve for a missing second mass or separation distance when force is known.
The calculator is designed for students, teachers, engineers, and anyone checking a gravitation problem. Instead of showing only a final answer, it displays the formula, substitutes your values in scientific notation, and presents the calculation in clear mathematical steps. For the Earth-Moon example, it also shows the related gravitational acceleration and orbital velocity calculations. This makes it useful for homework, exam revision, astronomy lessons, and quick physics checks.
Newton's Universal Law of Gravitation Formula
Newton's law of universal gravitation is written as F = Gm1m2/r squared. In this formula, F is the gravitational force in newtons, G is the universal gravitational constant, m1 and m2 are the masses in kilograms, and r is the distance between the centres of the two masses in metres. The value of G is approximately 6.67430 x 10 to the power of negative 11 N m squared per kg squared.
Every object with mass attracts every other object with mass. The force acts along the line connecting their centres. Although gravity between ordinary small objects is extremely weak, it becomes powerful when one or both objects have planetary or stellar mass. Newton's equation accurately explains much of classical orbital motion, including the motion of moons around planets and planets around the Sun.
How to Use the Gravitational Force Calculator
First, select the quantity you want to calculate: gravitational force, second mass, or distance. Next, enter the known values. Use kilograms for each mass and metres for distance. Scientific notation is accepted, so an Earth mass can be entered as 5.972e24 and a Moon mass as 7.342e22. Select force if both masses and their separation are known. The answer will be given in newtons.
To find a missing mass, select Second mass and enter the force, first mass, and distance. To find separation, select Distance and enter both masses and the force. Check that every value is positive and that distance means the separation from centre to centre, not simply the gap between visible surfaces. The live worked solution updates immediately whenever an input changes.
Step-by-Step Earth and Moon Example
A classic universal gravitation example uses Earth and the Moon. Earth's mass is about 5.972 x 10 to the power of 24 kg, the Moon's mass is about 7.342 x 10 to the power of 22 kg, and their average centre-to-centre distance is 384,400,000 m. Substituting these values into F = Gm1m2/r squared gives a gravitational force of approximately 1.98 x 10 to the power of 20 N.
This equal and opposite force pulls both bodies toward their common centre of mass. Earth moves only slightly because it is much more massive, while the Moon has a much larger acceleration. The calculator also evaluates the gravitational acceleration at the Moon's distance from Earth and the approximate orbital speed. These related values connect Newton's gravity equation to the motion observed in an orbit.
Why Distance Is Squared in the Gravity Formula
Distance has a major effect on gravity because it appears as r squared in the denominator. This is called the inverse-square law. If the distance between two objects doubles, gravitational force becomes one quarter as large. If the distance triples, force becomes one ninth as large. Reducing the distance has the opposite effect: halving it makes the force four times stronger.
The inverse-square relationship occurs because the effect of a source spreads through three-dimensional space. It is essential when comparing gravity at a planet's surface with gravity far away from that planet. Always square the complete distance after converting it to metres. A common error is to square only part of a scientific-notation expression or to use kilometres without converting them first.
How Mass Affects Gravitational Force
Unlike distance, each mass has a direct linear relationship with gravitational force. Doubling the first mass doubles the force. Doubling the second mass also doubles the force. Doubling both masses makes the force four times larger. This simple relationship helps you estimate whether a calculation is reasonable before relying on a calculator result.
Mass is not the same as weight. Mass is the amount of matter in an object and is measured in kilograms. Weight is a force caused by gravity and is measured in newtons. Near Earth's surface, weight can be found from W = mg, where g is approximately 9.81 m/s squared. Universal gravitation explains where the local value of g comes from: g = GM/r squared.
Units for Universal Gravitation Problems
Correct SI units are essential for a correct answer. Use kilograms for m1 and m2, metres for r, and newtons for F. Convert tonnes, grams, kilometres, miles, and astronomical units before using the formula. For example, 384,400 km must become 384,400,000 m. Because distance is squared, a missed kilometre-to-metre conversion produces an error of one million in the final force.
Scientific notation keeps very large and very small values readable. The calculator accepts e notation, such as 6.6743e-11. It displays significant values in a form that is easier to follow in the worked solution. Include units with every final answer, particularly in written physics work. A number without its unit does not fully communicate the physical quantity calculated.
Rearranging the Universal Gravitation Equation
To calculate an unknown second mass, rearrange the formula to m2 = Fr squared divided by Gm1. Multiply the force by the square of the distance, then divide by the product of G and the known mass. This form is useful when determining the mass of an object from a measured force, or when studying a simplified two-body system.
To calculate distance, rearrange the formula to r = square root of Gm1m2 divided by F. The square root is necessary because r was squared in the original formula. A distance answer must be positive. These rearranged forms use the same unit requirements as the original equation and are automatically selected by this universal gravitation calculator when you change the calculation mode.
Gravity, Orbital Velocity, and Satellites
Gravity provides the inward acceleration needed to keep a satellite in orbit. For a circular orbit, the approximate orbital speed is v = square root of GM/r, where M is the mass of the central body and r is the orbital radius measured from its centre. A lower orbit requires a higher speed, while a higher orbit has a lower speed. This relationship follows directly from universal gravitation.
Satellites do not escape gravity when they orbit Earth. They continually fall toward Earth, but their sideways motion carries them around the planet. The same principle applies to the Moon, artificial satellites, and planets around the Sun. Real orbital planning requires additional effects such as atmospheric drag, non-uniform gravity, and other celestial bodies, but Newton's law is the foundation for first estimates.
Applications of Newton's Law of Gravitation
The universal gravitation formula has many practical and scientific applications. It is used to estimate forces between planets and moons, calculate satellite paths, study tides, determine escape velocity, and infer the mass of distant astronomical objects. Astronomers can estimate a star's mass by observing the orbit of a companion object. Space scientists use gravitational models when planning probes, flybys, and planetary missions.
In education, the formula teaches force, vectors, inverse-square relationships, units, and scientific notation in one problem. In engineering and research, more detailed models may be required for high-precision or safety-critical work. Relativity, irregular body shapes, atmospheric effects, and multiple gravitational sources can matter. This calculator provides the standard Newtonian two-body approximation for learning and straightforward calculations.
Common Universal Gravitation Calculation Mistakes
The most common mistake is using surface-to-surface distance when the formula requires centre-to-centre distance. For large spherical objects, add the radii if the given separation is a surface gap. Another frequent mistake is forgetting to square r, entering kilometres instead of metres, or mixing mass and weight. Make sure force is in newtons, not kilograms or kilograms-force.
Also remember that the forces on both objects are equal in magnitude and opposite in direction. Their accelerations are not usually equal because acceleration depends on mass. Avoid rounding too early when a calculation uses large exponents. Keep several significant figures through the intermediate steps, then round the final answer appropriately. The equation display on this page makes these steps easier to check.
Universal Gravitation Calculator FAQ
Does gravity act only on planets? No. All objects with mass attract one another, although the force between small everyday objects is usually too weak to notice. Is the gravitational constant the same everywhere? Yes. G is a universal constant; the local gravitational acceleration g changes with mass and distance. Can this calculator be used for the Sun and planets? Yes, provided you enter consistent SI units and use centre-to-centre distance.
For the best result, use accurate input values, convert all measurements to SI units, and check whether the Newtonian two-body model fits your problem. The Universal Gravitation Calculator gives a fast numerical result along with transparent steps, helping you understand both the answer and the physics behind it.
Frequently Asked Questions
What is Newton's law of universal gravitation?
It states that every mass attracts every other mass with a force F = Gm1m2/r squared.
What value should I use for the gravitational constant G?
Use 6.67430 x 10 to the power of negative 11 N m squared per kg squared.
Should the distance be measured centre to centre?
Yes. For spherical bodies, use the distance between their centres of mass.
Why does gravity get weaker with distance?
The force follows an inverse-square law, so it is divided by the square of the separation distance.
Are the gravitational forces on both objects equal?
Yes. They are equal in magnitude and opposite in direction, as required by Newton's third law.