The Gravitational Constant in Newton’s Gravity Equation

FROM THE LECTURE SERIES: THE JOY OF SCIENCE

By Robert M. Hazen, Ph.D., George Mason University

Several aspects of Isaac Newton’s law of gravity have incredible intuitive powers. Gravity, in some sort of deep and mysterious way, is an attribute of mass. More mass means more gravitational force, and it is in direct proportion. Read on to learn more about the gravitational constant.

The universal gravity equation with illustrations of the Moon and the Earth under the equation.
Newton determined the capital G, or gravitational constant, in the gravity equation. (Image: Designua/Shutterstock)

The Inverse-Square Relationship

The bottom half of Newton’s equation is the 1 over d2 part of it. As the distance of two objects becomes greater, the gravitational force between them drops off. And it doesn’t happen just in proportion to distance, but also in proportion to the square of the distance. That’s called an inverse-square relationship and is actually a very common relationship.

In other words, the difference between the intensity at one distance, and a distance twice as far away would be the ratio of one to one-quarter, or 1 over d2.

To think about gravity in a geometrical way, a gravitational field and an imaginary array of lines that radiates straight out from any object can be imagined. In this suggestion, the gravitational force is the number of those lines that intersect between two masses. As the distance is doubled, the number of intersections of lines lessens.

This is a transcript from the video series The Joy of Science. Watch it now, on The Great Courses Plus.

The Capital G in the Gravity Force Equation

Concept of the gravity force equation and Isaac Newton with an apple in his hand.
Newton’s gravity equation did not contain the numerical value of the gravitational constant. (Image: Lia Koltyrina/Shutterstock)

Even after Newton proposed his gravity equation, there was one big gap, and that was the numerical value of the gravitational constant shown with capital G in the equation. This constant is tremendously important, and it helped estimate the magnitude of the gravitational force between any two objects. 

Determining G is extremely difficult because the experiment has to be completely independent of the Earth’s huge gravitational force that swamps out most other measurements. The experiment also has to eliminate any other contributions by stray electrical or magnetic fields, and that’s not very easy.

After all, a slight static electric charge on a comb can pick up a piece of paper. Also, a magnet, which is a very tiny device compared to the entire Earth, can pick up objects against the Earth’s entire gravitational force. So to think about gravity, it’s a very weak force compared to some of the other forces that are around all the time.

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Henry Cavendish and His Experiment to Determine G

The most famous attempt to determine G was undertaken two centuries ago by the English chemist Henry Cavendish. He was born in 1731 in Nice into one of the wealthiest families in all of Britain. He was educated at Cambridge University and spent most of his life, more than 50 years, in London performing scientific experiments. He then died a recluse in a London home in 1810.

One of Henry’s important experiments was devising a very ingenious method for determining the gravitational constant G. In 1798, he suspended a dumbbell with lead spheres from a wire, such that the two suspended spheres were in proximity to two much larger lead spheres that were fixed. The slight force of gravity between the large spheres and the smaller lead spheres caused the suspended dumbbell to rotate, to torque slightly.

Well, Cavendish knew how much torque it took to twist his wire. He measured that twist, and therefore he was able to measure the force of gravity because he knew how much the two lead masses weighed, and the distance between the weights, and so he could solve G. 

Substituting the measured value for G, the gravitational acceleration of the Earth’s surface, and the Earth’s radius all into a modified form of Newton’s equation, yielded the Earth’s mass.

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Calculating the Earth’s Mass

An image of an asteroid being pulled into the Earth’s gravity.
The Earth is more massive than many objects, so it attracts them with its force of gravity. (Image: Muratart/Shutterstock)

There are two equivalent descriptions of force—one using Newton’s equation for gravity, and the other using Newton’s second law: force equals mass times acceleration. Because these two equations describe the same force, they can be set equal to each. So the acceleration due to gravity at the Earth’s surface or (g) is 9.8 m/s2 , and this is equal to capital G. 

Indeed, the mass of the Earth is equal to little g times r squared over big G [(g)(r2/G)]. Plugging in the numbers, the answer ends up with the mass of the Earth six times 1024 kilograms. The Earth weighs six trillion, trillion kilograms. That number comes from Cavendish’s discovery of the value of the big G, the gravitational constant.

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Other Ways to Determine G

Cavendish presented this calculation for the first time in his now-classic paper, Experiments to Determine the Density of the Earth, published in 1798. That paper stands as a great landmark in geophysical science.

But there were a couple of new ways of determining G as well. There was a group in Zurich that weighed a kilogram mass on an exquisitely sensitive balance. 

In one case, they did it with a 1,000 kilogram mass just below sort of a donut-shaped mass, and then they raised that donut-shaped mass and put it just above the kilogram mass, and they looked at the difference in the weight of the object when that 1,000 kilogram was just below, or just above the suspended mass on the balance. A very difficult measurement, but they were able to do it and come up with a value for G.

Another American group tried a similar approach to solve the gravitational constant or G. They measured the time that it took an object to fall. Times also can be measured to millionths or billionths of a second. So if the time it takes an object to fall is measured with a heavy mass below, as opposed to a heavy mass above, a slightly different time would be shown.

All of these techniques now yield similar values for G. For the record, the best estimate of G is now about 6.674 times 10-11 m3/kg/sec2. So it’s a number that people are still converging on, but that’s pretty much the accepted value now.

Common Questions about the Gravitational Constant in Newton’s Gravity Equation

Q: What does G refer to in the gravity force equation?

In the gravity force equation, the capital G refers to the gravitational constant.

Q: What is the gravity force?

Gravity is the energy between any two objects in the universe that attract each other with a certain amount of force. The more massive object attracts the less massive one.

Q: How can the Earth’s mass be calculated?

There are two ways to calculate the Earth’s mass. It can be either by Newton’s second law or by Newton’s gravity equation. In both, the answer is exactly the same.

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