Gravitational Force Calculator

This tool to calculate the gravitational force allows you to determine the gravitational force between any two objects and vice versa. By entering the masses of both objects as well as the distance between their centers of mass, the gravitational force can be calculated.

If any field is left blank, the tool will figure out that value for you. The units can be converted at any time from grams or kilograms to Earth Masses or Solar Masses. In addition, it also displays the potential energy of gravity, the acceleration each object experiences, and the escape velocity of the heavier object.

What is gravity?

Gravity is the force of attraction that all objects with mass exert on other objects with mass. The falling of an apple, the movement of the moon around Earth and the maintenance of planetary orbits by the sun are examples of gravity in action.

Isaac Newton formalized this concept in 1687 as the Law of Universal Gravitation. His groundbreaking contribution was not the discovery that objects fall, but rather the insight that the same force that makes an apple fall also keeps the moon in orbit at a great distance. The strength of gravitational attraction increases proportionally to the product of the masses of both objects and decreases inversely with the square of the distance between their centers of mass.

The formula for gravity is:

The gravitational force between two masses is equal to the product of their masses multiplied by the gravitational constant divided by the square of the distance between them.

where F is the force of gravity, m1 and m2 are the two masses, r is the distance between their centers of mass, and G is the gravitational constant. No matter how different the masses of two objects are, the magnitude of the gravitational force they exert on each other is always the same, with opposite directions. You pull down on Earth with exactly the same amount of force that Earth pulls up on you, but in the opposite direction.

The gravitational constant G

The constant G determines the strength of gravity throughout the universe.

This is a very small number, which is why the force of gravity is much weaker than the four fundamental forces. You can only feel gravity strongly if at least one of the objects involved has an extremely large mass, such as a planet. Henry Cavendish first measured the gravitational constant G in 1798 using a very precise method that relied solely on rotational motion. This experiment was so accurate that it actually made it possible to determine the mass of the Earth.

How to use this calculator:

Pick three known values and enter them in any units you like, as needed. Leave the fourth field blank.

If all three values are entered, the tool will fill in the blank fields and display the resulting force as the main result. At the same time, it also displays the potential energy of the attraction, the acceleration of each mass, and a simple summary. To calculate another value, you just need to leave one field blank. There is no need to select a mode, since leaving a field blank tells the tool which value to calculate.

Determine mass or distance:

As this law contains an equation with four variables, it can be converted into a form that allows any one variable to be determined. The form used by the tool is as follows:

So if the force of attraction is measured and the masses of two objects are known, then the distance between them can be calculated. If the force of attraction, the distance, and one of the masses are known, then the other mass can be worked out. This is how astronomers work out the mass of planets or stars that can never be weighed.

Reformulate Newton's laws.

This table shows the same equation expressed in four different forms, each of which corresponds to a possible missing value.

Your weight.

Your weight is the force of attraction between you and Earth. Imagine a person with a mass of 70 kilograms standing on the surface of the earth.

The mass of the Earth is about 5.97 times 10 to the power of 24, expressed in kilograms, while the radius of the Earth, which is the distance from the ground to the center of the Earth, is about 6.371 million meters. So...

This force of 687 Newtons is the weight of this person. If you divide that by a mass of 70 kilograms, then you get an acceleration of about 9.8 meters per second squared. This is a known value for the acceleration due to gravity on Earth's surface. The tool shows each object's acceleration separately so that the strength of the gravitational force can be seen at a glance.

The inverse square law.

The square of the parameter r in the denominator is a characteristic feature of the inverse-square law. When the distance between two masses doubles, the gravitational force does not decrease by half but to one fourth. If the distance triples, then the gravitational force decreases to one ninth. The curves for gravity and distance on the calculators show this steep decline corresponding to the mass value entered.

This is why astronauts in low Earth orbit continue to experience almost the full force of gravity on Earth even though they are only a few hundred miles above the surface. In contrast, spacecraft farther away from Earth experience very little gravity. Because distance is measured from the center of the Earth, small changes in altitude have little effect on the strength of gravity.

Gravitational potential energy

In addition to gravity, the tool also calculates the gravitational potential energy stored in this mass configuration. It uses a linear power instead of the square of the distance and there is a minus sign.

The negative energy means that the two masses are bound together. The minus sign indicates that it takes energy to pull them apart. The smaller and more negative the value is, the tighter they are bound. As you push them infinitely far apart, the energy goes to zero. This physical quantity determines whether a comet will be captured by the Sun or flung into space.

Weight, mass and gravity

Mass and weight are not the same thing. Mass is the amount of matter contained in an object, and it does not change. Weight is the force of gravity acting on that mass, and it depends on the object exerting the gravity. For example, an astronaut weighs about one-sixth as much on the moon as he or she does on Earth, while his or her mass remains the same.

According to Newton's third law the forces exerted by two masses are equal in magnitude but the accelerations do not have to be the same. Acceleration is the result of dividing force by mass, so a lighter object will move much faster. When you drop an apple it actually makes the earth rise up to meet it. However because the earth has such a large mass its acceleration would be incredibly small. The tool calculates the accelerations of both masses separately and shows this difference clearly.

Escape velocity and orbital speed

When the Escape Velocity and Orbital Velocity levels are enabled, the tool considers the heavier of the two objects to be the central body and calculates three quantities for an object orbiting at a distance r from that body.

Escape velocity is the speed that an object needs to achieve in order to permanently break free from the gravitational pull of a central body without requiring additional propulsion.

For example, the escape velocity from Earth is about 11.2 km/s. Orbital velocity is the speed required to maintain a stable orbit and it is equal to the escape velocity divided by the square root of two. The orbital period, or time taken for one revolution, follows Kepler's third law and is given by T = 2 pi r3/2 / sqrt(GM). These relationships allow satellites to be placed in orbit and probes sent to other planets.

Comparison of gravitational and electrostatic forces

Newton's law of gravitation and Coulomb's law, which deals with electric charges, look almost the same. Both decrease with the square of the distance. The difference is in strength and direction. Gravitational force only acts attractively, while electrical force can act both attractively and repulsively.

Gravity is also an astonishingly weak force. The electrical force between a proton and electron is about 10 to the power of 39 times stronger than gravity. Gravity only becomes dominant on planetary or galactic scales because large objects are usually electrically neutral. So even though it's weak, gravity is the only force for which there is no known comparable counter-force.

Where does gravity come in?

Gravity shapes the universe on all scales. It holds your feet to the ground and pulls rain back down to Earth. It keeps the moon in orbit around us and causes tides in the ocean. The tides are strongest when the sun and moon line up and work together.

On the largest scales gravity holds stars together into galaxies, galaxies into clusters of galaxies and drives the formation of black holes. In our immediate vicinity satellite navigation systems such as GPS have to take account of gravity very precisely because even Einstein's corrections to Newton's law change timekeeping in a noticeable way.

Assumptions and limitations

In this simple form of Newton's law, it is assumed that objects are considered to be point masses or perfect spheres with their entire mass concentrated at the center. Thus, the distance between centers is measured. For most everyday problems and astronomical questions, this makes a very good approximation.

When more than two masses are involved, the resulting force acting on any given object is the vector addition of the forces exerted by all other objects. This principle is known as the superposition principle. In extreme gravitational environments, i.e., very close to massive objects or at speeds approaching that of light, Newton's law is superseded by Einstein's general theory of relativity. For normal applications, however, you can use this formula for calculations with many decimal places of accuracy.

Tips to improve accuracy:

Make sure to measure the distance from center to center and not surface to surface as the formula uses the distance between centers of mass. Also be aware of the magnitudes involved, a planet's mass can range in values up to 10 times 24 so use units of Earth masses or solar masses or scientific notation and avoid long strings of zeros. Leave unit conversions to the tool and do not mix units manually. Also remember that gravity is always an attractive force and never a repulsive one.