Abstract
Newton in Principia gives us a mathematical method of finding the center of force for a body moving on an ellipse in Proposition V, Problem I. The same thinking can be applied also to the case of a hyperbola and also a parabola, only that in the last case the center of force is at infinite distance. For the first two cases there are 3 cases of possible forces: a) An force proportional to the distance from the center. For ellipse an attractive force for hyperbola a repulsive, b) a force proportional to the inverse of the square of the distance from the left focus, for ellipse an attractive and for hyperbola a repulsive force, c) an attractive force inversely proportional to the square of the distance, inversely proportional to the square of the distance for both the ellipse and the hyperbola. This method when applied to the case of circular orbits for which we can find the center of force with the same method: Newton studied a semicircular orbit with center of force at infinite distance, and the case of a central force whose center is located on the circular orbit or inside the circle studied the case of a spiral orbit. In each the law of the force was derived by using the law of areas.
Highlights
In Newton’s words: There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common center: to find that center (Newton, n.d.). His proof is simple and needs some explanation: Newton uses the constancy of angular momentum which he explained in by PROPOSITION I, Theorem
If from the center K we draw the radii KA, KB, KC, the areas KAB, KBC, are equal, since the triangles KAB, KMN have equal bases and equal heights, and the triangles KBC and KBM have the same base (KB) and heights DC=NM As we can see in Figure 1, the constancy of the areas is valid for central force which for Newton’s model is given by impulses along the lines which join the end of the velocity vector to the center of force
Newton in principia gives us a mathematical method of finding the center of force for a body moving on an ellipse in Proposition V, Problem I
Summary
In Newton’s words: There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common center: to find that center (Newton, n.d.). The areas, which bodies made to move describe by radii drawn to an immovable center of force lie unmoving planes, and are proportional to the times in which they are described (Newton, 1999) This proposition is very useful for teaching purposes. We draw the three lines PT, TQZ, ZR which touch the curve at the points, P, Q, R, and meet in T and Z (see Figure 3) On the tangents erect the perpendiculars PA, QB, RC, reciprocally proportional to the velocities of the body in the points P, Q, R, from which the perpendiculars were raised.
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