Gravitational Force Due to a Sphere: A Noncalculus Calculation

In teaching noncalculus physics, an instructor is tempted to introduce an occasional formula with the explanation, “This formula is easily proved through the use of calculus.” This rabbit-from-the-hat approach is regrettable from the standpoint that physics should be presented as a logical development of experiment and theory. Yet, from the experience of this author, it is very rare that a student will object to the explanation quoted above. Nevertheless, an instructor should be prepared with a noncalculus justification if so challenged, and in most instances such is available. For example, the areas, volumes, and moments of inertia of simple geometric figures can be calculated by means of The Method of Exhaustion developed by Archimedes.¹ The purpose of this paper is to treat two theorems that are important in introductory physics, and that cannot be established by such a method. Instead, we shall provide an alternative justification. However, we anticipate that students will be satisfied to hear only a description of the methods used in the calculations and of the final results.