(This article originally appeared in the Q1 1993 STAR news letter)
The Basics of Gravitational Force and Motion
by Jerry Watson
In the latter 1660's when Isaac Newton was about 25 years old, much of his innovative work concerning the relationship of force and motion had already been formulated. Of particular interest here are his three laws of motion and the law of universal gravitation. These insights into the physical workings of the natural world, and the new mathematics needed to quantify them (calculus), were not formally introduced to the world until 1687 in Newton's book Principia.
This subject matter is usually found in the early chapters of any basic textbook on astronomy. I can recommend Introductory Astronomy and Astrophysics by Michael Zeilick and Elske Smith, 2nd edition (1987), CBS College Publishing, pp. 503. Chapter 1, entitled "Celestial Mechanics and the Solar System" is written at the undergraduate college level. The discussion to follow is a considerably condensed summary of this material with the purpose of getting at the basics. This level of understanding will be adequate to explore the amazing acrobatics of celestial objects moving under the influence of their gravitational attraction.
1. The Force of Gravity
We shall consider the simplest of all physical systems: just two objects alone in the universe. Much can be learned from this so-called 'two-body problem', even about the behavior of planets, satellites, and other solar system objects under the control of the sun. For such a system Newton proposed that the force (F) of object 1 on object 2, and vice versa, is given by:
On the right hand side m1 and m2 represent the respective masses of the two objects. Sometimes in our everyday conversation "mass" and "weight" are used synonymously; they are related, but they are not the same. Newton meant by "mass" the quantity of matter (atoms and molecules) of which something is composed. The Apollo astronauts took their mass with them when they went to the Moon. However, on the Moon their mass weighed only about one-sixth the value at the Earth's surface. As a matter of fact your "weight" is actually a measure of the 'force of gravity' exerted on you by the mass of the other object (e.g., the Earth or Moon).
A basic unit of mass, the gram, was defined as the mass of 1 cubic centimeter of pure water. The usual unit of mass for calculation purposes is the kilogram (kg), equal to 1000 grams. At the Earth's surface (strictly, at mean sea level) 1 kg weighs 2.2 pounds. Hence, a 220 pound object has a mass of 100 kg. On the Moon, that mass would weigh about 220/6 ≈ 37 pounds.
The law of gravitation in (1) above tells us that the force is proportional to the product of the two masses, and is inversely proportional to the square of the distance (r) between the two objects. Newton proved that even when dealing with large spherical objects (e.g., Sun or Earth), the objects gravitationally interact with one another as though all of their mass were concentrated at their respective centers. Thus, r is the distance between the centers of the two objects. The fact that the force depends inversely on the 'square' of the separation means that the force decreases rapidly with increasing distance (and vice versa). If the distance between two masses is doubled, the attractive force decreases to one-fourth; if the separation becomes 10 times some original value, then the force is only 1/100th of the original. Relationships of the form of equation (1) are referred to as 'inverse square' laws.
Finally, the right hand sides of equation (1) contain the universal constant of gravitation (G). The determination of the value of this constant has been a challenge for experimentalists since the first attempt by Henry Cavendish in England in 1798. A summary of the history of measurement apparatuses, and also of the reasons why an accurate value of G is so important to several areas of astronomy, are the subjects of a recent (April 1993) Sky and Telescope article: "Getting The Measure Of Gravity", p. 28. The current 'best-value' is G = 6.6726 x 10-11 (SI units).
Now that we have examined the individual physical factors contributing to the law of gravitation, let us look at the equations as a whole. If you haven't already noticed, the right sides of both equations are the same. That is, F1 = F2. The force exerted by object 1 on object 2 is exactly the same, but in the opposite direction, to that of object 2 on object 1. This relationship is a manifestation of Newton's 3rd law of motion. To paraphrase: for every action there is an equal and opposite reaction!
Based on the experiments of the Italian scientist Galileo Galilei some 50 years earlier, and upon his own intuition, Isaac Newton proposed his other two laws of motion. The 1st law of motion defines what is meant by a 'force'. This law states that an object at rest, or an object moving in a straight line at constant speed, will forever retain this motion, unless acted upon by a 'force'. In other words, to change either the speed or direction of motion of an object requires action by a force. Such a change in the state of motion is called an 'acceleration'.
The quantitative relationship between force and acceleration is contained in the 2nd law of motion. This law states that the product of the mass (m) times acceleration (a) is equal to the sum of all of the forces (F) acting on an object. Symbolically, F = ma. In the case of the gravitational force between two objects, the respective relationships are:
We have already noted that F1 = F2, and therefore equation (2) shows that:
The ratio of accelerations is inversely proportional to the ratio of masses. In other words, the less massive object will be accelerated to higher speeds than the more massive object. That is certainly consistent with our own experience. If you put all of your strength (force) into pushing a stopped automobile, you might be able to accelerate it to a few miles per hour. However, the same force applied to a much more massive dump-truck may not produce any motion at all!
Now let us combine equation (2) with the law of gravitation (1). If we divide through by the mass common to both expressions for the force on the same object, we get expressions for the accelerations. That is,
We note that the acceleration of an object depends upon the mass of the other object, but not upon its own mass. Equation (3) is a quantitative expression of Galileo's observation that objects of different mass (or weight) dropped from some height (the Leaning Tower of Pisa?) will accelerate toward the ground at the same rate. That is, in a vacuum so that air resistance will not affect an object's fall speed, a feather and a 10 ton boulder will reach the ground at exactly the same time!
3. Some Conclusions
The law of gravitation (1) has revealed that a mass of (say) 100 kg is attracted 'downward' by the Earth's mass with a force (weight) of 220 pounds. The Earth is also attracted 'upward' with the same force. However, the Earth is so much more massive (about 6 x 1024 kg), the planet's acceleration in response to such a tiny force is correspondingly minuscule. Equations (2) and (3), show that the accelerating effect of the Earth's mass on the 100 kg object (or any object) is, however, quite substantial. As we all know, a jump from even a height of 10 feet can produce bone-jarring speeds at impact with the ground.
The acceleration of gravity (g) for objects at the Earth's surface is:
Here, m is the Earth's mass and r the Earth's radius, and the numerical values are given in the english and metric system of units. Using the more familiar english value of g, a falling object (neglecting air resistance) would be travelling at a speed of 32 feet/second after 1 second, at 64 feet/second after 2 seconds, ... , 320 feet/second (28 miles/hour) after 10 seconds, and faster and faster!
To conclude our discussion, we note that Newton's basic laws of motion and gravitation have performed so many useful tasks on behalf of astronomy and other sciences over the past 3 centuries. Newton himself was able to prove Kepler's laws of motion which were based entirely on observations of the planets, and to show that the same law that governs a falling 'apple' at the Earth's surface also controls the motion of the Moon about the Earth. In future articles and talks at Club meetings, the present writer will use the concepts discussed here as a starting point for exploring the motion of objects subjected to mutual gravitational interaction.