Experiment 2

Vector Addition (Graphical Method)

Objectives:

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

Equipment:

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

Theory:

The resultant of two or more vectors is a single 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.

Example:

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

find R = A + B + C(1) by graphical method, and (2) by analytical method.

Solution:

1) Graphical Method (Here the polygon method is used)

As shown above, a polygon is drawn with the given vectors A, B, C by placing the vectors one after another, on a tail-to-tip basis.  First A is drawn.  Then from the tip of A,  B is drawn.  Next, from the tip of B, C drawn.  Finally,  tail of A ( the first one) is connected to the tip of C (the last one) to obtain the resultant.   Now, the gram-force equivalent (or the length of the resultant) must be measured.  This length gives us the magnitude of the resultant.  The angle that the resultant makes with the positive x-axis gives us its direction.  It should be measured with a protractor. 

2) Analytical Method (Here the Rectangular Components method is used).

a) Calculate the x and y component of each of A, B, and C.

b) Sum the components in the x-direction to obtain Rx .

c) Sum the components in the y-direction to obtain Ry .

d) Compute the magnitude and direction of the resultant using 

        

e) Draw a sketch of Rx and Ry, and find θ by using the tan-1.

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 = 162gf.     and     Ry = Ay + By + Cy = 278gf.

R = (Rx+ Ry)1/2   &  θ = tan-1(Ry /Rx).    We get R = (320 gf, 59.8o).

 

Procedure:

Three vectors A, B, and C are given in Table 1 under the "Data" section.

The purpose is use a ruler and a protractor and apply the polygon method to find Resultants R1, R2, and R3 (one at a time) as shown below:

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

For each of R1,  R2,  and R3 take the following steps:             

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

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

3) Solve for the same resultant that 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 Table 2.

4) Calculate a % error on magnitude and a % error on direction and record them in the space provided in Table 2.  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 the % errors.  It is not necessary to show the repeated substitution of numbers in the % error formula.

 Data:

Table 1

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

Table 2

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            

 

Calculations:

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 the most precise, graphical or analytical method?

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