More Precisely 2.2 - The Moon Is Falling!

Posted by Andri Fadillah Martin on Tuesday, February 14, 2012

The Moon Is Falling!

The story of Isaac Newton seeing an apple fall to the ground and "discovering" gravity is well known, in one form or another, to most high-school students. However, the real importance of Newton’s observation was his realization that, by observing falling bodies on Earth and elsewhere, he could quantify the properties of the gravitational force and deduce the mathematical form of his law of gravitation.

Galileo Galilei had demonstrated some years earlier, by the simple experiment of dropping different objects from a great height (the top of the Tower of Pisa, according to lore) and noting that they hit the ground at the same time (at least, to the extent that air resistance was unimportant), that gravity causes the same acceleration in all bodies, regardless of mass. Since acceleration is proportional to force divided by mass (Newton’s second law), this meant that the gravitational force on one body due to another had to be directly proportional to the first body’s mass. Applying the same reasoning to the other body and using Newton’s third law, it follows that the force must also be proportional to the mass of the second body, hence the two "mass" terms in the law of gravity. (The experimental finding that the gravitational force is precisely proportional to mass is now known as the equivalence principle. It forms an essential part of the modern theory of gravity; see More Precisely 22-1).

What about the "inverse-square" part of Newton’s law? At Earth’s surface, the acceleration due to gravity is approximately 9.80 m/s2. It is denoted by the letter g. Where else other than on Earth could a falling body be seen? As illustrated in the accompanying figures, Newton realized that he could tell how gravity varies with distance by studying another object influenced by our planet’s gravity—the Moon. Here’s how he did it.

Let’s assume for the sake of simplicity that the Moon’s orbit around Earth is circular. As shown in the second figure, even though the Moon’s orbital speed is constant, its velocity (red arrows) is not—the direction of the Moon’s motion is steadily changing. In other words, the Moon is accelerating, constantly falling toward Earth. In fact, the acceleration of any body moving with speed v in a circular orbit of radius r may be shown to be

always directed toward the center of the circle (toward Earth, in the case of the Moon). This acceleration is sometimes called centripetal ("center-seeking") acceleration. You probably already have an intuitive experience for this equation—just think of the acceleration you feel as you take a tight corner (small r) at high speed (large v) in your car.

Knowing the Moon’s distance r = 384,000 km (measured by triangulation), and the Moon’s sidereal orbit period P = 27.3 days, Newton computed the Moon’s orbital speed v = 2πr/P = 1.02 km/s, and hence determined its acceleration to be a = 0.00272 m/s2, or 0.000278 g. (More Precisely 1-4) Thus, Newton found, the Moon, lying 60 times farther from Earth’s center than the apple falling from the tree in his garden (taking Earth’s radius to be 6400 km), experiences an acceleration 3600, or 602, times smaller. In other words, the acceleration due to gravity is inversely proportional to the square of the distance. Isaac Newton’s application of simple geometric reasoning and some very basic laws of motion resulted in a breakthrough that would revolutionize astronomers’ view of the solar system, and pave the way for humanity’s exploration of the universe.

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