Physics in the Mathematics Curriculum Summary
Monday, July 8, 2002
Addition of Force Vectors Through a Geometrical Construction
Mathematics Involved: Properties of parallelograms, measuring angles, sum of vectors, trig identities, complex numbers
- Have two students hold their arms out palm out and push against their hands so that the two hands meeting do not move (show photo here). What is the angle between the two student's arms?
- Add a third student pushing against the other two. In order for the point at which the three hands meet, how hard does each student need to push? (clean this up)
- Model this situation with three strings knotted together to keep fixed lengths with each string attached to a spring scale. (Show photo here.) Have students read off values on the three scales, note that as the angle between the strings changes, the composing forces also changes. Can you get any combination of scale readings? Are there any patterns to the triples you can achieve?
- Give students three scale measurements and have them create a construction which diagrams this situation. Show construction diagram here. The construction is created on graph paper by using a compass to draw an arc of radius length matching one of the three scale values. Then, opening the compass to one the other three scale readings, draw another arc. Return to the starting point, and open the compass to the third scale reading drawing a third arc. The intersection of this third and second arc gives us a second point on the third vector.
- Have students draw lines on their graph paper representing the three strings.
- Use strings and scales with one student pulling on each string at the specified force. Slide the construction on the graph paper under the three strings and line up the knot in the string with the intersection of the three force vectors.
- Give students several examples to construct.
- Give students examples which are impossible, why can't they make the construction work?
Getting a car out of the ditch with a chain held with tension.
- Use law of sines or cosines to calculate the angles given the triples.
- Any tie in to our complex numbers?
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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.