EASILY CONSTRUCTED TANGENT METER* ROBERT M. HOFFMAN University of Wisconsin, Madison, Wisconsin
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N PROBLEMS of kinetics and thermodynamics it is often necessary to determine the slope or tangent at various points along a plotted curve. The familiar mirror method of Latishaw1involves turning the mirror until the line and its reflection appear contimuous. Excellent tangent-findimg devices are now manufacturedf which, by means of a prism, cause an actual break to appear in the line until the true tangent position is found. Obviously, it is much easier for the eye to detect a disjointed break in the line than to estimate whether the line is merely bent sharply in one direction or the other.
two pieces about twenty centimeters long. These are pressed side by side on a flat surface and fastened together with a small amount of sealing wax applied on top of the rods near each end. When the double rod is placed over a line a t any angle other than ninety degrees, the line appears broken; i. e., the ends break away from each other in opposite directions. The rod is turned until the two ends of the plot meet. The double rod is now a t right angles to the tangent of the curve at that point. From this position, the derivative of the curve a t that point may be found by any of several obvious methods. For example, a right triangle may be placed against the rod and the tangent slope obtained graphically by determining the ratio of an increment along the y-axis to the mrrespondiig increment along the x-axis. Another convenient method is to place a protractor scale against the rod and determine the angle a between the tangent and any line parallel to the abscissa. Then
2
=,,a
The numerical value of tan a can be found in tables or read from a good slide-rule. This value, of course, should then be multiplied by the constant ratio of the values assigned to unit distances along the ordinate and abscissa. This gives the derivative the same dimensions as those used in the graph; for example, cubic centimeters per minute, etc. The average student, to whom the more expensive To aid in placing the double rod exactly over any tangimeter may not be readily available, can easily certain point, it is well to inscribe a quarter-inch circle construct a device which will give a similar effect withabout the point. out requiring a prism. This simple device will prove especially useful on A straight piece of clear glass stirring rod about six curves that are rapidly changing their slope. The millimeters in diameter is obtained. From it are cut accuracy may be easily tested over a finely ruled straight line. In twenty trials the greatest deviation from a zero slope was one-half degree, and the average was two-tenths of a degree.
