Experiment 2

Vector Addition

Objectives:

To (1) learn the graphical methods (parallelogram and polygon methods) of vector addition, and (2) compare graphical results with analytical solutions to get an idea of how accurate graphical methods are.

Equipment:

A protractor, a Metric ruler, and a few sheets of graphing paper

Theory:

The resultant of two or more vectors is a vector that is equivalent in its physical effects to the action of the original vectors.  For example, if three force vectors were acting on an object, these three forces could be replaced by their resultant, and the object would experience the same net effect.

Note:  In the following sections, "gf" means gram-force.  1gf is the force of gravity on the mass of one gram.

I) Finding the resultant of three forces by the parallelogram method:

Example:

        Given:  A = 200.gf at 0.0°, B = 150.gf at 35.0°, and C = 250. gf at 130.°      

As shown under solution above, first construct a parallelogram on vectors A and B.   To increase precision, use a larger scale.  For instance, use 5.0 cm for 100gf.   The  diagonal that is drawn from the tails of vectors A or B is the resultant R_AB.  Construct another parallelogram using the last vector C and the resultant R_AB.  The diagonal of this second parallelogram is the resultant of the three original vectors, A, B, and C.   It is labeled R_ABC.  Measure the magnitude (length) of R_ABC .   Convert it back to the units of grams using the chosen scale.  Also, find the angle of R_ABC with respect to the positive x-axis.  These are considered the measured values. 

II) Finding the resultant of the same three forces by the polygon method:

     Same Example: Given:  A = 200.gf at 0.0°,  B = 150.gf at 35.0°, and C = 250.gf at 130.°

As shown in the example above, draw a polygon with the three given vectors A, B, C by placing the vectors, one after another, on a tail-to-tip basis.  First draw A.  Then from the tip of A, draw B.  Next, from the tip of B, draw C.  Finally connect the tail of A ( the first one) to the tip of C (the last one) to obtain the resultant.  The resultant is formed by a vector drawn from the tail of the first vector to the tip of the last vector.  Find the gram-force equivalent of the length of the vector.  That is the magnitude of the resultant.  The angle it makes with the positive x-axis is its direction.  Measure it with the protractor.  Again, choose a scale large enough to make the drawing cover almost the whole sheet of the graphing  paper.  Make sure not to change the original directions of each vector as you complete the polygon step-by-step.

III) Finding the resultant of the same three forces by the analytical method:

      Same Example:

      Given:  A = 200. gf at 0.00⁰, B = 150. gf at 35.0⁰, C = 250. gf at 130.0⁰

         The x- and y-components of the vectors are:

            A_x = 200. cos 0.00˚    =   200. gf         Ay = 200. sin 0.00˚  =   0.00  gf

B_x = 150. cos 35.˚   =  123.0  gf          B_y = 150. sin 35.0˚  = 86.0  gf

C_x= 250. cos 130˚  = -161.0  gf          C_y = 250. sin 130.0˚  = 192.   gf

R_x = A_x + B_x + C_x = 200.+ 123. – 161. = 162 gf    

R_y = A_y + B_y + C_y =      0 +  86.0 + 192 = 278 gf

R = R_ABC = 320. gf, 59.8˚

Step 1: Calculate the horizontal and vertical components of each force A, B, and C.

            Step 2:   Sum the components in the x-direction to obtain R_x.

Step 3:  Sum the components in the y-direction to obtain R_y.

Step 4:  Compute the magnitude and direction of the resultant using 

Step 5:  Draw a sketch of R_x and R_y, and calculate θ by using the tan ^–1 function

Procedure:

1.      The data for this experiment are the three vectors (A, B, and C), as given below.

 2.      Find the following resultants R₁, R₂, and R₃ ( one at a time) using a graphical method (preferably polygon).

 R₁ = A + B (polygon)                       R₂ = B + C (polygon)

R₃ = A + B - C (polygon)               

Be sure to use a scale that is as large as possible so the most accurate graphical results are obtained.  The scale used should be shown on each drawing.  Use a separate page for each graph.

Solve for the resultants (R₁, R₂,  and R₃) using the analytical method (calculation).  Be sure to indicate magnitude and direction of the resultant in each case.

Make a table of results showing the magnitude and direction of each resultant as computed by the graphical versus analytical methods.  Calculate percent errors in magnitude and direction using the following:

 Data:

Calculation(s):

         Provide the necessary calculations.

 Comparison of the results: 

         Provide the percent error formula used as well as the calculation of the percent errors.

 Conclusion: 

         State your conclusions of the experiment.

 Discussion: 

        Provide a discussion if necessary.

Questions:

         Which method is the most precise, graphical or analytical method?

         Why is the polygon method generally considered to be the most reasonable graphical technique?