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# Universal Gravitation Equation

by Ron Kurtus (revised 14 March 2011)

The * Universal Gravitation Equation* states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of separation between them. This equation is a result of Isaac Newton's

*Law of Universal Gravitation*, which states that quantities of matter attract other matter to it.

The proportionality constant in the equation is called the *Universal Gravitational Constant*. The value of that constant was determined experimentally by Henry Cavendish in 1798.

This Universal Gravitation Equation originally applied to point masses but was extended to masses of finite size with the assumption that their mass was concentrated at their center of mass.

Questions you may have include:

- What is the Universal Gravitation Equation?
- What is the Universal Gravitational Constant?
- How is the equation an approximation?

This lesson will answer those questions. Useful tool: Units Conversion

## Universal Equation

In 1687, Isaac Newton formulated the *Universal Gravitation Equation*, which defines the gravitational force between two objects. The equation is:

F = GMm/R^{2}

where

**F**is the force of attraction between two objects in newtons (N)**G**is the Universal Gravitational Constant in N-m^{2}/kg^{2}**M**and**m**are the masses of the two objects in kilograms (kg)**R**is the separation in meters (m) between the objects, as measured from their centers of mass

This equation has proven highly effective in explaining the forces between objects, as well as leading into the effects of gravity.

## Universal Gravitational Constant

When Newton originally stated the equation, he simply said that **F** was proportional to **Mm/R ^{2}**. The value of the proportionality constant or

*Universal Gravitational Constant*,

**G**, was not even considered for many years and not officially calculated until 1873, 186 years after Newton defined the equation.

The *Cavendish Experiment* has since been used to determine Universal Gravitational Constant as:

G= 6.674*10^{−11}N-m^{2}/kg^{2}

Note: The number 10^{−11}is 1/10^{11 }or 0.000000000001 with 11 zeros after the decimal point.(

See Cavendish Experiment to Measure Gravitational Constant for more information.)

### Check on units

It is important to make sure you are using the correct units for each item in your equation. Check by adding units to the gravitation equation and then seeing that the result is correct:

FN = (GN-m^{2}/kg^{2})*(Mkg)*(mkg)/(Rm)^{2}

Just considering the units:

N = (N-m

^{2}/kg^{2})*(kg)*(kg)/(m)^{2}N = (N)*(m

^{2})*(kg)*(kg)/(m^{2})*(kg^{2})N = N

Thus, the units used are correct.

### Other units

**G** can also be stated in other terms, depending on its usage. Since a newton (N) equals kg-m/s^{2}, you may also see **G** defined as:

G= 6.674*10^{−11}m^{3}/kg-s^{2}

Also, in applications where greater separations are studied, it is more convenient to use kilometers^{} instead of meters. Since 1 m = 10^{−3 }km, the value of **G** is:

G= 6.674*10^{−20}km^{3}/kg-s^{2}

When comparing a force in newtons with gravitational force with km, the value is the strange combination of units:

G= 6.674*10^{−17}N-km^{2}/kg^{2}

You can use whichever set of units that fulfill your requirements.

## Equation an approximation

Newton originally stated the Universal Gravitation Equation as the force between two point masses, separated by **R**.

### Problem with point masses

A problem exists when considering point masses, and that is the situation when the separation between point masses approaches zero. In such a case, the gravitational force becomes approaches infinity. Since that is impossible, there must be some small separation at which the Universal Gravitation Equation breaks down, perhaps at quantum distances.

### Objects of finite size

The concept was later extended to objects of finite size, using the assumption that the mass was concentrated at the center of mass of each object.

In a system of particles, the center of mass is the average of the particle positions, weighted by their masses. The center of mass of a sphere that has its mass evenly distributed is the center of the sphere.

(

See Gravitation and Center of Mass for more information.)

Thus, the separation **R** in the Universal Gravitation Equation is the separation between the objects, as measured from their centers of mass.

### Summation of forces

The true gravitational force between two objects is a summation of the forces from each point on both objects.

Various points on object attract points on other object

Calculus is used to integrate over all the points on the surfaces and within each object. Unfortunately, the mathematics for the exact equation is highly complex, and it is easier to make some assumptions to simplify the math.

By considering the mass of the objects concentrated is their center of mass, we get an equation that is close enough for practical purposes in most cases.

### Distribution of matter in spheres

Most of the objects where the Universal Gravitation Equation applies are large spheres, such as planets, moons and stars. Often the distribution of mass in those objects is not even, and the objects are often not exact spheres.

For example, the density of matter in the Earth is unevenly distributed, plus the Earth is not an exact sphere but is flattened near its poles.

Since the separation between astronomical objects—such as the Earth and the Moon or Sun—are so large, assuming the center of mass as the center of the object is an acceptable approximation.

### Consider atoms as points

Atoms, molecules and even subatomic particles are considered so small and separated by great distances relative to their size that they can be considered point sources of gravitation, and the Universal Gravitation Equation applies to these small particles.

Atoms considered points separated by distance **R**

However, since molecules and atoms are normally in rapid motion, you would seldom calculate the gravitational force between them, except perhaps as an average.

## Summary

Isaac Newton formulated the *Law of Universal Gravitation*, stating that all matter attracts other matter to it. This force of attraction is defined in the theory's *Universal Gravitation Equation*. This equation is actually a close approximation, to simplify the mathematics. The measurement of the gravitational constant was first made by Henry Cavendish.

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