For centuries, in the Middle Ages and early Renaissance, triangulation was a specialized skill, almost synonymous with mathematics.† This skill was what mathematicians had to offer princes, as it was especially useful in military campaigns, for surveying the battlefield, constructing fortifications, directing bombardment, etc.† At least this was how mathematicians presented it when they looked for employment.
Through this practical art every mathematically minded person in those days learned an easy familiarity with such notions as angular size and the application of geometry to real situations.† Physics inherited all this.† Triangulation is still a good exercise, and, as we will see, it is a challenge to do it well.
Figure 1: Triangle with legs H and D and angle q.
The basic relationship, shown in Fig. 1 above, for a measurement of a height H, is
This looks like a formula for H in terms of two things one can measure easily, the distance D from the foot, and the angle .†† We will choose to consider it as a relationship between D and †by rewriting it as
This says that D is proportional to 1/tan.† If we graph D vs 1/tan, for several measured values (always trying to measure the same H, but from different distances away), then we expect a straight line with slope H.† This is better than just using the formula (1.1) blindly, because it tests the relationship, and also gives some idea of how accurate the measurement is (by the scatter of the measured points around the trend line).
A variant of this problem is to measure the height without actually being able to get to the foot of H.† (Perhaps it is a tower with hostile archers in it, who will not let you approach!)†
Figure 2: A triangle such that the length of one leg cannot be measured directly, requiring that the leg be treated as two portions, D and X.
As the geometry of Fig. 2 shows,
Here you measure the distance D from some point that is itself a distance X from the foot of H.† Divide through by tan, and think how you could plot measured data to lie on a straight line, such that you can find an unknown height H at an unknown distance X.
For each height you measure this way, one by the method of Fig. 1 and one by the method of Fig. 2, try estimating it ahead of time.† You donít have to report your estimate with your actual measurement of H, but it might be fun for you to see how well you do.
For both these measurements, your raw data will be distances D and angles .† Make a neat data table in Excel with these values in columns.† Make Excel find tan() for you in a new column.† If your first angular value is in cell B2, then in C2 you could type ď=tan(B2*3.14159/180)Ē.† Notice that Excel expects angles to be in radians, the usual default unit for angles, so you have to convert the angle to radians before taking the tangent of it.† Proceed to get a chart (XY scatter), a trendline, a best fit slope, etc., as described in the Excel handout.
Sketch the actual situation of your triangulation, labeling what you will measure:
Sketch of graph, with axes labeled, and with equation of best fit line:
Write a paragraph explaining what you did, and what the slope and intercept mean in your graph.† Comment quantitatively on the uncertainty in your measurement, and give not just a height H but also an indication of its uncertainty.† What is the main source of uncertainty in this measurement?† How could the measurement be improved?†
OPTIONAL:†† If you wish, please say what you liked or didnít like about this lab and how it might be improved.