SfC Home > Physical Science > Physics >
Explanation of the Universal Gravitation Equation - Succeed in Understanding Physics. Also refer to physical science, Isaac Newton, matter, attraction, force, mass, radius, gravitational constant, Henry Cavendish, Ron Kurtus, School for Champions. Copyright © Restrictions
Universal Gravitation Equation
by Ron Kurtus (revised 25 September 2009)
Isaac Newton's Law of Universal Gravitation states that quantities of matter attract other matter to it. The force of attraction between objects is defined in the Universal Gravitation Equation, which states that the gravitational force is proportional to the masses of the two objects and inversely proportional to the square of distance between them.
This equation is not exact but provides a close approximate to the actual force. The value of the gravitational constant was originally established and verified experimentally by Henry Cavendish in 1798.
Questions you may have include:
- What is the Universal Gravitation Equation?
- How is the equation an approximation?
- What are some newer theories?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Universal Equation
Isaac Newton formulated the Universal Gravitation Equation, which defines the gravitational force between two objects. The equation is:
F = GMm/R2
where
- F is the force of attraction between two objects in newtons (N)
- G is the universal gravitational constant
- M and m are the masses of the two objects in kilograms (kg)
- R is the distance in meters (m) between the objects, as measured from their centers of mass
Universal gravitational constant
The universal gravitational constant, G, has been determined experimentally to be:
G = 6.67*10−11 N-m2/kg2
Note: The number 10−11 is 1/1011 or 0.000000000001 with 11 zeros after the decimal point.
A newton can also be stated in terms of kg-m/s2, so you may also see G defined as: G = 6.67*10−11 m3/kg-s2. Since the unit of force is in newtons (N), the units for G used in the Universal Gravitation Equation should be N-m2/kg2.
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:
F N = (G N-m2/kg2)*(M kg)*(m kg)/(R m)2
Just considering the units:
N = (N-m2/kg2)*(kg)*(kg)/(m)2
N = (N)*(m2)*(kg)*(kg)/(m2)*(kg2)
N = N
Thus, the units used are correct.
Equation an approximation
The Universal Gravitation Equation for the force between two objects makes some assumptions that result in the equation representing an approximation of the real force, especially concerning large objects and small distances.
Center of mass
The gravitation equation defines R as the distance between the objects, as measured from their centers of mass. 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.
In deriving the equation, it was assumed that the gravitational force can be measured as if the mass of each object was concentrated at its center of mass. That point is also called the center of gravity for the object in some situations.
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 distances 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.
Problem with short distances
When the distance between small objects approaches zero, the gravitational force becomes very large, approaching infinity. This means that there must be some small distance at which the Universal Gravitation Equation breaks down, perhaps at quantum distances.
Measuring gravitational constant
In an effort to measure the density of the Earth in 1798, Henry Cavendish also was able to measure the value of the gravitational constant, G.
He used a device with two objects on a rod that was hung on a wire. These masses were attracted to two larger objects, twisting the rod slightly. The measurement of the masses, distance between them and the torque on the wire allowed G to be determined.
Cavendish experiment using torque to measure gravitation
The most recent measurement used a device called an atomic interferometer to measure G.
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.
See the Side Menu for more Gravitation and Gravity topics
Excel in what you do
Resources
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Gravitation and Gravity Resources
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
What do you think?
Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. We will try to get back to you as soon as possible.
Share link
Feel free to establish a link from your website to pages in this site.
Or use our form to send this link to yourself or a friend.
Students and researchers
The Web address of this page is:
www.school-for-champions.com/
science/gravitation_universal_equation.htm
Please include it as a reference in your report, document, or thesis.
Where are you now?
Universal Gravitation Equation