Weighing the Earth

The Idea

USSR, President Dwight David Eisenhower, asked his generals to tell him, based on its orbit, how massive the satellite was. Unfortunately, they were unable to provide the U.S. president with the information he requested. However, they would have been able, instead, to tell him the mass of the Earth (which Eisenhower wasn't concerned about). In this project, you use a different satellite—the moon—to determine the mass of the Earth. You also explore how the scientist Cavendish performed some painstaking calculations of gravitational attraction and was able to accomplish the same thing.

What You Need

• moon
• calendar

Method

where G is the constant of Universal Gravitation = 6.67 ×10–¹¹m³/kgs²

1. Determine how long it takes for the moon to circle the Earth.
2. A calendar can give a reasonable result. A more accurate value is the sidereal period, which indicates only the time it takes for the moon to circle the Earth, without consideration for how long it takes to return to a particular phase. This can be obtained from a sidereal table or by subtracting 2.2 days from the value obtained by observing the number of days from one full moon to another.
3. Calculate the velocity of the moon in its orbit based on its average radius, r, of 384,400 kilometers (3.844 ×10⁸m). You can do this using the equation:
4. Calculate the mass of the Earth using the equation

Expected Results

Using the following values:

T = 27.322 days = 2,360,621 seconds

v = 2 πr/T = 1,023 meters/second

r = 3.844 ×10⁸m

= 6.03 ×1024 kilograms

This is within 1 percent of the accepted value for the mass of the Earth of 5.97 ×10²⁴ kilograms.

Why It Works

Newton's law of universal gravitation states there is an attractive force between any two masses in the universe. The attractive force is related to how massive the objects are and how far apart they are from each other. The gravitational force is linked to the mass and distance by a constant, "big G," called the universal gravitation constant. Since the gravitational force is the force that provides the centripetal force that keeps a satellite in orbit, we can solve for the mass of the Earth if we know the other variables in the equation. Similarly, knowing that the gravitational force equals the weight of an object, we can solve for the mass of the Earth.

Other Things to Try

Cavendish's famous experiment is one of our "wish-list" experiments that can be used to determine the big G and, as a result, the mass of the Earth. Gravitational force between two masses can be measured using an apparatus shown in Figures 118-1 and 118-2. The relatively small force is detected by measuring the torsion it produces in a thin filament between the masses.

The Point

The mass of a body that a satellite rotates around can be determined by the orbital period of that satellite. A key component of the force is the universal gravitational constant, G. By knowing G, it is possible to determine the mass of the Earth, using either the weight of objects on the Earth's surface or the orbital period of satellites circling around the Earth.

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