James
P. Hoare
Research Laboratories General Motors Corooration Warren, Michigan
I I
I
A Simple Method for Finding Slopes
It is the purpose of this paper to describe a simple method for determining quite reproducibly the tangent a t any point on a given curve of any shape. One proceeds as follows: A solid glass rod, G, in. in diameter and 6 in. long, is placed on a plot of a given curve, XY, as shown in Figure 1. The rod is rotated until an unbroken image of the curve is reflected through the glass rod as demonstrated in Figure 2. A straight line, AB, is drawn along and parallel to the axis of the glass rod. Next, as in Figure 3, the glass rod is placed across line AB so that again the line appears unbroken as it passes under the rod. As before, a line, CD, is drawn along and parallel to the axis of the glass rod. This line, CD, is the tangent to the curve a t point P and is the slope of X Y a t point P. This procedure may be repeated for any point along XY to find the required slope a t that point. When a ray of light passes from one medium into another, its direction is changed and the magnitude
FIG. 3
570
/
Journd o f Chemical Education
FIG.
of this change is a function of the angle of incidence.' If a plane of light strikes a rectangular prism normal to the surface of the prism, it will not be bent as it passes through and emerges from the other side because the angle of incidence is zero. In the present case, instead of a rectangular prism, a glass rod is placed on a plot of the given curve. A plane of light perpendicular to both the plane of the paper and the axis of the glass rod will not experience any bending when passing through the rod. If the segment of X Y covered by the rod is contained within such a plane, the light reflected from this line segment will not be bent and the curve will appear unbroken as it passes under the rod. A line, AB, drawn parallel to the axis of the rod, is perpendicular to this plane containing the line segment of XY and is therefore perpendicular to XY at P.. In like manner, the line, CD, is perpendicular to AB and is, therefore, the tangent to X Y a t P. Since the intersection of a plane normal to a cylinder describes a circle, the reflection of the light from the segment of X Y will produce a magnifying or lens e f f e ~ t . ~ This is why the broken images in Figure 2 have a sigmoid appearance which would not be observed if a rectangular prism were used. The attractive feature of this method is the use of an instnunent (glass rod) readily available to the experimenter in any laboratory. A somewhat more accurate method of determining CD is obtained by using two glass rods about '/la in. in diameter and held parallel to one another as shown in Figure 4. This is true because the line segment covered by the rod is much less than that covered by the '/,in. rod and, therefore, more closely approximates a straight line segment. Two rods are used because this procedure brings out to a greater extent small deviations from the perpendicular position.
' HAUSMANN, E., AND SLACK,E.P., "Physics," D. Van Nostrand Co., Inc., Princeton, N. J., 4th ed., 1957, p. 572. Ibid., p. 607.
