Warped Passages - Lisa Randall [43]
Newton’s gravitational force law also says how the gravitational force depends on the distance between the two objects. As discussed in Chapter 2, the law says that the force between two objects is proportional to the inverse square of their separation. This inverse square law was where the famous apple entered in.* Newton could deduce the acceleration due to the Earth’s gravitational pull on an apple located near the Earth’s surface and compare it with the acceleration induced on the Moon, which is located sixty times further away than the Earth’s surface is from its center. The acceleration of the Moon due to the earth’s gravity is 3,600 times smaller (3,600 is the square of 60) than the acceleration of the apple. This is in accordance with the gravitational force decreasing as the square of the distance from the Earth’s center.7
However, even when we know the dependence of the gravitational attraction on mass and distance, we still need another piece of information before we can determine the overall strength of gravitational attraction. The missing piece is a number, called Newton’s gravitational constant, that factors into the calculation of any classical gravitational force. Gravity is very weak, and this is reflected in the tiny size of Newton’s constant, to which all gravitational effects are proportional.
The Earth’s gravitational pull or the gravitational attraction between the Sun and the planets might seem pretty big. But that’s only because the Earth, the Sun, and the planets are so massive. Newton’s constant is very small, and the gravitational attraction between elementary particles is an extremely weak force. This feebleness of gravity is itself a big puzzle that we will return to later on.
Although his theory was correct, Newton delayed its publication for twenty years, until 1687, while he tried to justify a critical assumption of his theory: that the Earth’s gravitational pull was exerted as if its mass were all concentrated at the center. While Newton was hard at work developing calculus to solve this problem, Edmund Halley, Christopher Wren, Robert Hooke, and Newton himself made tremendous progress in determining the gravitational force law by analyzing the motion of the planets, whose orbits Johannes Kepler had measured and found to be elliptical.
These men all made major contributions to the problem of planetary motion, but it is Newton who gets credited with the inverse square law. That is because Newton ultimately showed that elliptical orbits would arise as a result of a central force (that of the Sun) only if the inverse square law was true, and he showed with calculus that the mass of a spherical body did in fact act as if it were concentrated at the center. Newton did, however, acknowledge the significance of others’ contributions in his words, “If I have seen further, it is because I have stood upon the shoulders of giants.”* (However, rumor has it that he said this only because of his intense dislike for Hooke, who was very short.)
In high school physics, we learned Newton’s laws and calculated the behavior of interesting (if somewhat contrived) systems. I remember my outrage when our teacher, Mr Baumel, informed us that the gravitational theory we had just learned was wrong. Why teach us a theory that we know to be incorrect? In my high school view of the world, the whole merit of science was that it could be true and reliable, and could make accurate and factual predictions.
But Mr Baumel was simplifying, perhaps for dramatic effect. Newton’s theory was not wrong: it was merely an approximation, one that works incredibly well in most circumstances. For a large range of parameters (speed, distance, mass, and so on), it predicts gravitational forces quite accurately. The more precise underlying theory is relativity, and you