How does a vector equilibrant differ from a vector resultant?

How can the equilibrant of a system of vectors be determined graphically?

How can the equilibrant of a system of vectors be determined mathematically?

How can we better understand the importance and fundamental significance of the equilibrant

in solving problems involving multiple concurrent vectors?

Laboratory Exercise

Objectives. We will determine the equilibrant of two force vectors acting on a point by applying a third force vector at that point which will act to equalize the effect of both vectors combined. That is, we will determine a third force which, when added to the other two, create a vector sum of zero for the entire system of forces. This third force is referred to as the "equilibrant" of the other two and is equal in magnitude yet antiparallel to the resultant of the two forces if they were acting by themselves alone. Since the equilibrant is equal in magnitude but opposite in direction to the resultant of the two forces, we can compare our results obtained from direct measurement to those obtained from both a graphical solution as well as a mathematical solution to such a problem. This exercise will allow you to better understand the way in which concurrent vectors combine.

Equipment

Force table

Triple spring scale-ring-chain assembly

Protractor

Metric ruler

Legal size unlined paper

Pencil with eraser

Scientific calculator

Procedure

1. Arrange the triple spring scale-ring-chain assembly on the force table by placing the connecting ring in the center of the force table and orienting the three spring scales and attached chains radially outwards to the edge of the circular table.

2. Attach the triple spring scale-ring-chain assembly to the force table by slipping a selected chain link of one of the spring scale and chain arms into one of the serrations cut into the margin of the force table. Repeat this process for the other two arms while applying a measure of tensile force while doing so. Be mindful not to overstretch the spring, that ism do not apply a force equal to or in exceedance of the yield force for our spring scales. (Remember the lessons of Hooke's Law and the elastic limit!) Try to ensure that the connecting ring of your assembled apparatus is situate near the center of the force table (within a few cm). Make sure the ring and chains are straight and not twisted and that you are able to read the force values on the spring scales.

3. Slip a sheet of legal size paper between the force table and the triple spring scale-ring-chain assembly. Mark a dot indicating the location of the center of the ring on the paper with a pencil. Then draft a straight line extending radially outward from this point beneath the mean center of one of the spring scale arms. Repeat this process for the other two spring scale arms. Label each of the three drafted lines with the force value indicated on the adjacent spring scale. Remove the paper from the force table. Measure the angle between the two radial lines drawn that possess the two smallest force values and label this angle A. This diagram is an "acting" vector diagram of your system of concurrent forces. Record all data into the data table.

4. Repeat this process for a total of five trials, in which the angles between each of the arms of the triple spring scale-ring-chain assembly are different from the previous. Provide an acting force diagram for each of the five trials for your lab report.

Data

Trial

1

2

3

4

5

Force 1

______

______

______

______

______

Force 3

______

______

______

______

______

Force 2

______

______

______

______

______

Angle A

______

______

______

______

______

180 - A

______

______

______

______

______

Data Analysis

After collecting all your data from the force table measurements and entering them in the data table above, you are to then check your results graphically. Create a scale drawing of each of the force vectors as they are combining with one another. That is, make a scaled adding diagram of your system of forces for each of the five trials. Compare your graphic results with those obtained by direct measurement. The procedure for adding vectors to from an equilibrant is detailed below. Assuming that two of the measured forces are correct, perform an error analysis between the measured and calculated values of your equilibrant.

Appendix: Adding Vectors to Form and Equilibrant

1. On a large sheet of paper, establish a magnitude-length scale and direction-angular indices that will provide you sufficient paper surface area to construct your vectors. Your metric ruler is the tool used to perform this function. The grid spacing of graph paper, for instance (and if so provided), may be alternately used to establish a magnitude-length scale.

2. Draw your first vector from a point chosen so that all the drawn vectors will fit within the confines of your paper. The first vector should appear as a line whose length and direction have been drawn with reference to your established scales. The starting point is located at the "tail" of the vector. The end of the vector should be adorned with an arrow tip. This is the "head" of the second vector. The graphical method of adding vectors is accomplished by drawing them in a sequential "head-to-tail" chain. Additional vectors may be added to each previously drawn vector.

3. Starting from the head of the first vector, construct the second vector following the same procedure used to construct the first vector. The staring point (tail) for this vector will begin at the arrow-tip (head) of the first vector. The end of the second vector should also be adorned with an arrow-tip, This is the head of the second vector. The graphical method of adding vectors is accomplished by drawing them in a sequential "head-to-tail" chain, preserving both length (magnitude) and orientation (direction). Additional vectors may be added to each previously drawn vector in turn.

4. The order in which vectors are placed (added) is irrelevant. As long as all of the vectors are drawn to the appropriate length and in their proper orientation, the sum (resultant) will be the same.

5. Once all of the concurrent vectors have been drawn, it is time to construct the equilibrant vector (or vector "equalizer"). This is accomplished by drawing a line from the head of the last vector to the tail of the first vector. An arrow-tip is placed at the end of this line, that is, at the tail of the first vector drawn in the chain. This is the equilibrant (anti-sum) of all the vectors you have drawn in the chain. It is the vector which must be added to the entire system of vectors you have previously drawn to render the sum of the vector system equal to zero. Label this Line E. Measure the length of this line with your ruler and reference it to your magnitude scale. This is a measure of its magnitude, that is, the amount of the vector anti-sum. Finally, measure the orientation of the line with your protractor with reference to your directional indices scale, in degrees. This is the direction of your equilibrant.

Questions

1. Do your results indicate that forces combine vectorially (algebraically/trigonometrically)? Briefly summarize.

2. How do your measured equilibrants compare with your graphically determined ones? Explain any errors, inconsistencies, or unexpected results.

3. Is it possible for all three concurrent forces to possess equal magnitudes? Explain why or why not.

4. Is it possible for one of the forces to be larger than the sum of the other two in this situation? Explain why or why not.