Experiment 2

Vector Addition

Objectives:

The objective is to (1) practice the polygon method of vector addition, and (2) compare the graphical results with calculation (analytical solution) to get an idea of how accurate the graphical method used is.

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 effect 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 a single vector (the 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.

 

Finding the resultant of three forces by the polygon method:

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

 

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, drawC.  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 that it makes with the positive x-axis is its direction.  Measure it with the protractor.   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.

 

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

      Same Example:

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

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

            Ax = 200 cos 0.00˚ = 200 gf      Ay = 200 sin 0.00˚ = 0.00 gf

Bx = 150 cos 35.0˚  =123 gf       By = 150 sin 35.0˚ = 86.0 gf

Cx= 250 cos 130˚ = -161 gf       Cy = 250 sin 130˚  = 192 gf

Rx = Ax + Bx + Cx = 200 + 123 - 161 = 162 gf    

Ry = Ay+ By + Cy =   0 + 86.0 + 192 = 278 gf

R = RABC = (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 Rx.

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

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

        

      Step 5: Draw a sketch of Rx and Ry, and calculate θ by using the tan-1 function.

Procedure:

The data for this experiment are the three vectors (A, B, and C), as "Given" the Table 2 below.

The purpose is to find the following resultants: R1, R2, and R3 (one at a time) using the polygon method as shown in Table 1.

Table 1

R1 = A + B R2 = A + B + C R3 = A + B - C

To find each of the resultants R1,  R2,  and R3 take the following steps             

1) Choose a reasonable scale that gives you a drawing that is big enough for a good precision and at the same time it does not go out of page.  Indicate the selected scale somewhere on the drawing.  For these vectors, if your x-axis is 1 inch above the lower edge of the paper and your y-axis is 1 inch from the left edge of the paper, none of R1,  R2,  and R3 will go out of page provided that you choose your scale as 1cm = 2N.

 2) Add the vectors by the polygon method to find the resultant.  Record the magnitude and direction of the resultant (that you measure by your ruler-protractor set) in the Table 3 shown below.  These are your measured values.

3) Solve for the same resultant as you found in Step 2, but this time by using the analytical method (by calculation and use of trigonometry).  Calculate the magnitude and direction of the resultant and record it under calculated (accepted) values in the Table 3.

4) Calculate a %error on the magnitude and a %error on the direction and record them in the space provided in the Table 3.  The following %error formula is the one we will be using in this lab throughout the semester Note: In your report, the percent error formula must be shown under " The Comparison of the Results" section. You simply show this formula and only the calculated values of  %errors.  It is not necessary to show the repeated substitution of numbers in the %error formula.

 

 Data:

 

Table 2

Vector Magnitude Direction
A 25.0N 35.0o
B 10.0N 120.0o
C 15.0N 155.0o
     

 

Table 3

Resultant

Measured

Calculated (Accepted)

%error on Magnitude %error on Direction
Magnitude (N) Angle (o) Magnitude (N) Angle ( o )
 R1=A+ B            
 R2 =A+ B+C            
 R3 =A+B-C            

 

Calculation(s):

Show sample calculation, for example, the complete calculation for R1.

 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 more precise, graphical or analytical method?

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