Section 3: Newton's Law of Universal Gravitation

An underlying theme in science is the idea of unification—the attempt to explain seemingly disparate phenomena under the umbrella of a common theoretical framework. The first major unification in physics was Sir Isaac Newton's realization that the same force that caused an apple to fall at the Earth's surface—gravity—was also responsible for holding the Moon in orbit about the Earth. This universal force would also act between the planets and the Sun, providing a common explanation for both terrestrial and astronomical phenomena.

Figure 5: Newton's law of universal gravitation.

Source: © Blayne Heckel. More info

Newton's law of universal gravitation states that every two particles attract one another with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. The proportionality constant, denoted by G, is called the universal gravitational constant. We can use it to calculate the minute size of the gravitational force inside a hydrogen atom. If we assign m1 the mass of a proton, 1.67 x 10-27 kilograms and m2 the mass of an electron, 9.11 x 10-31 kilograms, and use 5.3 x 10-11 meters as the average separation of the proton and electron in a hydrogen atom, we find the gravitational force to be 3.6 x 10-47 Newtons. This is approximately 39 orders of magnitude smaller than the electromagnetic force that binds the electron to the proton in the hydrogen nucleus. See the math

Local gravitational acceleration

The law of universal gravitation describes the force between point particles. Yet, it also accurately describes the gravitational force between the Earth and Moon if we consider both bodies to be points with all of their masses concentrated at their centers. The fact that the gravitational force from a spherically symmetric object acts as if all of its mass is concentrated at its center is a property of the inverse square dependence of the law of universal gravitation. If the force depended on distance in any other way, the resulting behavior would be much more complicated. A related property of an inverse square law force is that the net force on a particle inside of a spherically symmetric shell vanishes.

Figure 6: GRACE mission gravity map of the Earth.

Source: © Courtesy of The University of Texas Center for Space Research (NASA/DLR Gravity Recovery and Climate Experiment). More info

Just as we define an electric field as the electric force per unit charge, we define a gravitational field as the gravitational force per unit mass. The units of a gravitational field are the same units as acceleration, meters per second squared (m/s2). For a point near the surface of the Earth, we can use Newton's law of universal gravitation to find the local gravitational acceleration, g. If we plug in the mass of the Earth for one of the two masses and the radius of the Earth for the separation between the two masses, we find that g is 9.81 m/s2. This is the rate at which an object dropped near the Earth's surface will accelerate under the influence of gravity. Its velocity will increase by 9.8 meters per second, each second. Unlike big G, the universal gravitational constant, little g is not a constant. As we move up further from the Earth's surface, g decreases (by 3 parts in 105 for each 100 meters of elevation). But it also decreases as we descend down a borehole, because the mass that influences the local gravitational field is no longer that of the entire Earth but rather the total mass within the radius to which we have descended.

Even at constant elevation above sea level, g is not a constant. The Earth's rotation flattens the globe into an oblate spheroid; the radius at the equator is nearly 20 kilometers larger than at the poles, leading to a 0.5 percent larger value for g at the poles than at the equator. Irregular density distributions within the Earth also contribute to variations in g. Scientists can use maps of the gravitational field across the Earth's surface to infer what structures lay below the surface.

Gravitational fields and tides

Every object in the universe creates a gravitational field that pervades the universe. For example, the gravitational acceleration at the surface of the Moon is about one-sixth of that on Earth's surface. The gravitational field of the Sun at the position of the Earth is 5.9 x 10-3 m/s2, while that of the Moon at the position of the Earth is 3.3 x 10-5 m/s2, 180 times weaker than that of the Sun.

Figure 7: Plot of the tidal Water Level (WL) at Port Townsend, Washington.

Source: © NOAA. More info

The tides on Earth result from the gravitational pull of the Moon and Sun. Despite the Sun's far greater gravitational field, the lunar tide exceeds the solar tide. That's because it is not the gravitational field itself that produces the tides but its gradient—the amount the field changes from Earth's near side to its far side. If the Sun's gravitational field were uniform across the Earth, all points on and within the Earth would feel the same force, and there would be no relative motion (or tidal bulge) between them. However, because the gravitational field decreases as the inverse of the distance squared, the side of the Earth facing the Sun or Moon feels a larger field and the side opposite feels a smaller field than the field acting at Earth's center. The result is that water (and the Earth itself to a lesser extent) bulges toward the Moon or Sun on the near side and away on the far side, leading to tides twice a day. Because the Moon is much closer to Earth than the Sun, its gravitational gradient between the near and far sides of the Earth is more than twice as large as that of the Sun.

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