13.1 Newton's Law of Universal Gravitation

The History of Gravitation

The earliest philosophers wondered why objects naturally tend to fall toward the ground. Aristotle (384–322 BCE) believed that it was the nature of rocks to seek Earth and the nature of fire to seek the Heavens. Brahmagupta (598~665 CE) postulated that Earth was a sphere and that objects possessed a natural affinity for it, falling toward the center from wherever they were located.

The motions of the Sun, our Moon, and the planets have been studied for thousands of years as well. These motions were described with amazing accuracy by Ptolemy (90–168 CE), whose method of epicycles described the paths of the planets as circles within circles. However, there is little evidence that anyone connected the motion of astronomical bodies with the motion of objects falling to Earth—until the seventeenth century.

The idea of heliocentrism, in which the Sun is at the center of the solar system, dates at least as far back as Aristarchus of Samos in the third century BCE, but attracted little attention, and Ptolemy’s geocentric (Earth-centered) model held sway in Europe until the sixteenth century. Nicolaus Copernicus (1473–1543) is generally credited as being the first to produce a mathematical model of a heliocentric system and to challenge Ptolemy’s geocentric system. This idea was supported by the incredibly precise naked-eye measurements of planetary motions by Tycho Brahe and their analysis by Johannes Kepler and Galileo Galilei. Kepler showed that the motion of each planet is an ellipse (the first of his three laws, discussed in Kepler’s Laws of Planetary Motion), and Robert Hooke (the same Hooke who formulated Hooke’s law for springs) intuitively suggested that these motions are due to the planets being attracted to the Sun. However, it was Isaac Newton who connected the acceleration of objects near Earth’s surface with the centripetal acceleration of the Moon in its orbit about Earth.

Other prominent scientists and mathematicians of the time, particularly those outside of England, were reluctant to accept Newton's principles. It took the work of another prominent philosopher, writer, and scientist, Émilie du Châtelet, to establish the Newtonian gravitation as the accurate and overarching law. Du Châtelet, who had earlier laid the foundation for the understanding of conservation of energy as well as the principle that light had no mass, translated and augmented Newton's key work. She also utilized calculus to explain gravity, which helped lead to its acceptance.

Finally, in Einstein’s Theory of Gravity, we look at the theory of general relativity proposed by Albert Einstein in 1916. His theory comes from a vastly different perspective, in which gravity is a manifestation of mass warping space and time. The consequences of his theory gave rise to many remarkable predictions, essentially all of which have been confirmed over the many decades following the publication of the theory (including the 2015 measurement of gravitational waves from the merger of two black holes).

Newton’s Law of Universal Gravitation

Newton noted that objects at Earth’s surface (hence at a distance of $R_{\text{E}}$ from the center of Earth) have an acceleration of g, but the Moon, at a distance of about $60R_{\text{E}}$, has a centripetal acceleration about ${(60)}^2$ times smaller than g. He could explain this by postulating that a force exists between any two objects, whose magnitude is given by the product of the two masses divided by the square of the distance between them. We now know that this inverse square law is ubiquitous in nature, a function of geometry for point sources. The strength of any source at a distance r is spread over the surface of a sphere centered about the mass. The surface area of that sphere is proportional to $r^2$. In later chapters, we see this same form in the electromagnetic force.

As shown in Figure 13.2, the ${\overrightarrow{F}}_{12}$ vector points from object 1 toward object 2, and hence represents an attractive force between the objects. The equal but opposite force ${\overrightarrow{F}}_{21}$ is the force on object 2 exerted by object 1.

These equal but opposite forces reflect Newton’s third law, which we discussed earlier. Note that strictly speaking, Equation 13.1 applies to point masses—all the mass is located at one point. But it applies equally to any spherically symmetric objects, where r is the distance between the centers of mass of those objects. In many cases, it works reasonably well for nonsymmetrical objects, if their separation is large compared to their size, and we take r to be the distance between the center of mass of each body.

The Cavendish Experiment

A century after Newton published his law of universal gravitation, Henry Cavendish determined the proportionality constant G by performing a painstaking experiment. He constructed a device similar to that shown in Figure 13.3, in which small masses are suspended from a wire. Once in equilibrium, two fixed, larger masses are placed symmetrically near the smaller ones. The gravitational attraction creates a torsion (twisting) in the supporting wire that can be measured.

The constant G is called the universal gravitational constant and Cavendish determined it to be $G=6.67\times {10}^{\mathrm{-11}}\text{N}·{\text{m}}^2{\text{/kg}}^2$. The word ‘universal’ indicates that scientists think that this constant applies to masses of any composition and that it is the same throughout the Universe. The value of G is an incredibly small number, showing that the force of gravity is very weak. The attraction between masses as small as our bodies, or even objects the size of skyscrapers, is incredibly small. For example, two 1.0-kg masses located 1.0 meter apart exert a force of $6.7\times {10}^{\mathrm{-11}}\text{N}$ on each other. This is the weight of a typical grain of pollen.

Although gravity is the weakest of the four fundamental forces of nature, its attractive nature is what holds us to Earth, causes the planets to orbit the Sun and the Sun to orbit our galaxy, and binds galaxies into clusters, ranging from a few to millions. Gravity is the force that forms the Universe.

The effect of gravity between two objects with masses on the order of these space vehicles is indeed small. Yet, the effect of gravity on you from Earth is significant enough that a fall into Earth of only a few feet can be dangerous. We examine the force of gravity near Earth’s surface in the next section.

Summary

• All masses attract one another with a gravitational force proportional to their masses and inversely proportional to the square of the distance between them.
• Spherically symmetrical masses can be treated as if all their mass were located at the center.
• Nonsymmetrical objects can be treated as if their mass were concentrated at their center of mass, provided their distance from other masses is large compared to their size.

Conceptual Questions

Problems

The International Space Station has a mass of approximately 370,000 kg. (a) What is the force on a 150-kg suited astronaut if she is 20 m from the center of mass of the station? (b) How accurate do you think your answer would be?