PROBLEM 1. Newton's law of gravitation, one of the most important ideas in space science, states that the force of gravitational attraction between two bodies of masses M sub 1 and M sub 2, is proportional to the product of the two masses and inversely proportional to the square of the distance R separating the two masses. If G is the constant of proportionality, called the "universal gravitational constant", this law can be stated in symbols in (1) below. What must be the unit for G in the mks system?
Solution: Using dimensional analysis, we equate the known units in accordance with the relationship (1) without worrying about the numbers, then solve algebraically to get the unknown unit.
Space science is based on a mathematical description of the universe. This mathematical description is in turn based on defining physical quantities clearly and precisely so that all observers can agree on any measurement of these quantities. Every measurement has two parts: a number and a unit. In mathematics, we tend to focus on the numbers and assume that the units are taken care of; but in scientific work, units receive careful attention through a procedure known as dimensional analysis, which is illustrated in the first problem.
Among the physical quantities used to describe the universe, some are considered fundamental quantities whereas others are derived quantities, comparable to the designation of definitions and undefined terms in a mathematical system. Although it does not really matter which particular quantities are the ones designated as fundamental, the most common are length, mass, and time. In scientific work the two major systems of units for these quantities are the mks (meter-kilogram-second) and the cgs (centimeter-gram-second). Every measurement is a comparison with the standards that are universally accepted as definitions of these fundamental units. In astronomy and space science, where large distances are common, the meter and even the kilometer are too small to be convenient; in Problems 5, 9, and 10 of this chapter, we show how more suitable units for length are defined.
Dimensional analysis (manipulation of units according to the rules of algebra) is the procedure used to ensure consistency in the definition and use of units. For example, since force is, by definition, the product of mass and acceleration, measured respectively in kg and m/s in the mks system, the unit of force in this system must be equivalent to kg-m/s. A new term, the newton, was created to describe the unit of force: 1 newton = 1 kg-m/s.