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 the graphical methods are.

Equipment:  A protractor, a metric ruler, a few sheets of graph paper, a computer with the Internet connection, a calculator (The built-in calculator of the computer may be used.), regular paper, and pencil

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 gram-force 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:  Click on the following link (1):  http://www.walter-fendt.de/ph14e/resultant.htm . This applet allows you add 2 or more vectors by the "polygon method."  You can position each vector the way you wish by grabbing its tip with the mouse and move it around giving it the magnitude and direction you want.  Try this for several examples of your own.

Now, click on this link (2):    http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=68 .   An x-y coordinates system appears.  Anywhere you click on this applet, a blue vector (A) appears whose tip is connected to the mouse.  You can place the tip anywhere you want by making another click.  The coordinates of its tip can be read from the data box at the top left corner.   Once you fix the blue vector A, the tip of the mouse starts dragging a green vector B.  Again, you can choose where this 2nd vector should end.  As soon as you click this 2nd vector, the applet starts adding A and B and showing how the resultant R or C is found.   Practice this 2nd applet for a few times as well each time trying a different example of your own.   As soon as you feel comfortable with the idea of vector addition, you may start the experiment as follows:

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

2.      The experiment has 3 separate parts.   Find the following resultants R₁, R₂, and R₃ ( one at a time) using a graphical method (preferably polygon).

 Part 1 :   R₁ = A + B                      

 Part 2 :   R₂ =A + B + C

Part 3 :   R₃ = A + B - C

Be sure to use an appropriate scale each time such that your drawing becomes as large as possible within the boundaries of the graphing paper.  That should result in more precise measurements by your ruler and protractor.  The scale used each time should be shown on each drawing.  It is better to use a separate graphing paper for each part.

3.   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.  All calculations must be shown under "calculations" or on the drawing.

4.   Make a table of results showing the magnitude and direction of each resultant as measured by the graphical method as well as the calculated values by the analytical methods.  Calculate percent errors both for magnitude and direction in each case using the percent error formula.

 Data:

Calculation(s):

         Provide the necessary calculations.  The calculation for each part my be shown on the corresponding drawing.

 Comparison of the results: 

         Use the following percent error formula to calculate 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?