Section II, Chapter Eight: Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation is an example of an inverse square law—the magnitude of the force due to gravity falls with the square of the distance between the two objects in question.
Note—this is a draft of Chapter Eleven of a book I’m writing. The book is going to cover humanity’s deepest ideas from philosophy, physics, epistemology, and economics. Each chapter is meant to be short and digestible. Most of the chapters will explain just one or two ideas.
‘Chapter []’ indicates a future chapter that I’ve not yet written.
As I wrote in Chapter Eight, Newton’s Law of Universal Gravitation quantifies the direction and magnitude of the gravitational force between any two massive objects. Algebraically, it is written as F = -G*m*M / r2, where F is the attractive force between two objects, G is a fundamental constant of Nature, m and M are the masses of each object, and r is the distance between the objects.
Strictly speaking, the right hand side of the equation should be multiplied by a unit vector that points from mass M to mass m (see Chapter Nine for a brief explanation of vectors). Furthermore, it is conventional to assume that the object with mass M is far more massive than the object with mass m (they may correspond, for example, to the Sun and the Earth, respectively). When M is much greater than m, we can approximate the object with mass M as stationary and so choose its point in space as an origin in our reference frame. The minus sign therefore indicates that the gravitational force on m due to M goes in the opposite direction of the unit vector on the right hand side—that is to say, gravity is always an attractive force between massive objects.
But we could have just as well chosen the (moving) position of m as the origin of our frame of reference. For instance, we could have chosen Earth as our frame of reference, rather than the Sun. The magnitude of the gravitational force would be the same, but its direction would be flipped. That is to say: the gravitational force levied on object m by object M pulls object m towards object M with magnitude A, while the same gravitational force levied on object m by object m pulls object M towards object m with magnitude A. In this way, Newton’s Law of Universal Gravitation reflects his Third Law: that every force engenders an equal and opposite force in kind.
For example, the Sun pulls the Earth towards it with a gravitational force of about F = -G*m*M / r2 = (6.674*10-11 m3 / k*s2)*(5.972*1024 kg)*(1.989x1030 kg) / (1.496*10^11 m2)2 = 3.54x1022 Newtons. But, as per Newton’s Third Law, the Earth pulls the Sun towards it with a gravitational force of the exact same magnitude. And yet the Sun is very nearly still, while the Earth accelerates around the Sun at a rate of about 5.93x10-3 m/s2 (the Earth does not fall straight into the Sun because the component of the Earth’s velocity that is perpendicular to the Sun’s gravitational pull is high enough).
Newton’s Second Law, F = m*a, explains the apparent discrepancy. The force each body exerts on the other might be equal and opposite, but their masses differ by orders of magnitude—and, therefore, so do their subsequent accelerations. We can rearrange the terms in Newton’s Second Law so that a = F / m. Therefore, the acceleration of the Earth due to the Sun’s gravitational force is a = (3.54x1022 N) / (5.972*1024 kg) = 5.93x10-3 m/s2. On the other hand, the acceleration of the Sun due to the Earth’s gravitational force is a = (3.54x1022 N) / (1.989x1030 kg) = 1.78x10-8 m/s2. So, while the two bodies exert the same gravitational pull on each other, the Sun causes the Earth to accelerate at a rate roughly five orders of magnitude more than the Earth imposes on the Sun.
There are many situations in which we wish to understand the dynamics of more than just two massive bodies. In this case, we may determine the net gravitational force acting on an object by summing over the gravitational force acting on it by each other body in the system. For instance, consider a system comprised of three stars all orbiting each other, M1, M2, and M3. The net gravitational force on M1 due to M2 and M3 is F = -G*M1*M2 / r12 + -G*M1*M3 / r22, where r1 is the distance between M1 and M2 and r2 is the distance between M1 and M3.
Why an Inverse Square Law?
Newton’s Law of Universal Gravitation is an example of an inverse square law—the magnitude of the force due to gravity falls with the square of the distance between the two objects in question. That is to say, F is inversely proportional to r2. But why should that exponent be two, rather than three, or 1.5, or any other number?
You may think of gravity as a field that propagates from a source in all directions. If we imagine a point of mass m, then its gravitational field at a distance r is a constant. Geometrically, this corresponds to a spherical shell of radius r. No matter where you are along the surface of this shell, the magnitude of the gravitational force due to the point mass m is the same—the point mass’s gravitational field can be thought to be evenly distributed across this shell’s surface area. Now, imagine the same point mass’s gravitational field to be twice as far out as before. Here, the gravitational force is evenly distributed across a spherical shell of radius 2*r. Because surface area increases with the square of the radius, the gravitational field now must ‘cover’ not twice but four times as much surface area as it did at distance r. In short, as an object’s gravitational field propagates from its source, it dilutes with the square of the distance from its source because the surface area that it covers increases with the square of the distance from its source.
The Principle of Equivalence
Earlier, we’d shown that Newton’s Second Law may be rearranged so that m = F / a. In this light, an object’s mass can be thought of as a measure of how much force is required to cause it to accelerate. The greater an object’s mass, the more force is needed to change its motion. In this role as resistor to acceleration, mass is described as inertial mass, or mi.
But as we’ve seen in this chapter, mass also plays a role in gravitational phenomena—namely as the source of gravitational attraction. In this role, mass is described as (passive) gravitational mass, mg.
The equivalence principle in Newtonian mechanics tells us that these masses are always equal: mi = mg. For instance, we might calculate an object’s inertial mass if we know the force acting on it and its resultant acceleration: mi = F / a. Now, imagine we wish to determine the gravitational force between the same object and some other object M using Newton’s Law of Universal Gravitation: F = -G*mg*M / r2. Following the equivalence principle, once we know mi, we don’t have to separately determine mg. Rather, we may simply substitute mi for mg and proceed with our calculations.
There is no reason a priori why this equivalence principle should hold, and pondering why it does would lead Einstein to develop his own ideas about mass and gravity roughly two hundred years after Newton’s death (see Section Nine).
Thanks to Dennis Hackethal for early feedback.