Experiment 3

Vector Addition: Force Table

 Objective:

The objective is to experimentally verify the parallelogram law of vector addition by using a force table.

Equipment:

A computer with Internet connection, a calculator (The built-in calculator of the computer may be used.), paper, and pencil

Theory:

Concurrent forces are forces that pass through the same point.  Resultant is a single force that can replace the effect of a number of  forces.  "Equilibrantis a force that is exactly opposite to a resultant.  Equilibrant and resultant have equal magnitudes but opposite directions.  Review the introduction section of Experiment 2 for additional information on the graphical method as well as the analytical method of finding a resultant, if necessary.

Procedure:

The force table in a regular lab setting has a schematic diagram as shown below.  It has 3 or 4 double pulley systems that allow 3 or 4 forces to be applied onto a central tiny ring.   The forces are shown as F1, F2, and F3 . 

 Figure 1: The schematic diagram of a force table

Here for this online experiment, you will use a suitable physics applet. 

Click on the following link:    http://lectureonline.cl.msu.edu/~mmp/kap4/cd082.htm

You will see F1 in red, F2 in green, and F3 in blue.  With the mouse you can move any of these three vectors by holding their tips.   For just practice and getting good at it, try to set each vector at certain x and y components.  As you move any of the vectors around you will see that the values corresponding to its components change on the applet.  The black vector shows the resultant of the three vectors F1, F2, and F3.  If you make the black vector to have a zero magnitude, then the three red, green, and blue vectors are in equilibrium and their sum is equal to zero as shown:

F1 + F2 + F3 = 0    or,    F1 + F2 =  - F3       or,    R = - F3 .

This means that when the sum of the 3 vectors is zero,  "the resultant of any two has a magnitude equal to the magnitude of the third one, and a direction opposite to direction of that third one."

Refer to Table 1 under the Data Section.  There are four cases (experiments) to be done.

1) In Line 1 of Table 1, first calculate the x- and y-components of F1 and F2.  Round the numbers to the nearest integer.  Make sure your calculator is in "Degrees" mode.

2) Place the mouse on the tip of F1 and move it around until its x- and y-components match your calculated values for F1.    Repeat this procedure for F2.

3) Move the tip of F3 around until the black vector shrinks to zero.  Record the x- and y-component of F3 F3 is the equilibrant.   It is the opposite of the resultant R of F1 and F2.  The goal is to find the resultant R of F1 and F2 .  You have found F3, the opposite of R.

4) Use the x- and y-components of F3 that you read from the applet to calculate the magnitude and direction (angle) of F3 using the following formulas:

F3 = [(F3x)2 + (F3y)2 ]1/2     and     θ3 = tan-1( /F3x)

The F3 magnitude that you calculate is to be used as the measured value of F3 or the measured magnitude of R.   Adding 180.o to or subtracting 180.o from the angle of F3  gives you the measured direction of R.   

Record the magnitude and direction (so found) for R as the measured values for R in Table 1.

5) Also, completely aside from this applet, find the resultant R of F1 and F2 by performing calculations on paper.   The magnitude and direction of R that you calculate using the following formulas, give you the accepted values for the magnitude and direction of R.

Rx = F1x + F2x = 50.0Ncos(35.0o) + 75.0Ncos(115.0o) = ...... N.

Ry = F1y + F2y = 50.0Nsin(35.0o) + 75.0Nsin(115.0o) = ...... N.

R = (Rx2 + Ry2)1/2  .......N    and    tan-1 (Ry / Rx) = .......o. 

Record the above R and θ under the accepted values in Table 1.  

6) Use the % error formula to calculate a % error on R and a %error on θ.  At this point Line 1 of the Table is finished.

7) Apply the above method to Lines 2, 3, and 4 of the Table to complete the experiment.

 Data:  

Given:  The magnitude and direction of each set of vectors to be added are given in Table 1.  

Measured:   Record your measured values in Table 1 in the space provided.

 Table 1

Case Given Measured

F3

Measured

R     

Calculated

R

%error
  F1 F2
  Magn.(N) Angle Magn.(N) Angle Magn.(N) Angle Magn.(N) Angle Magn.(N) Angle on R on θ
1 50.0 35.0 75.0 115                
2 38.0 130.0 80.0 210                
3 50.0 0.0 50.0 120                
4 75.0 40.0 75.0 -40.0                

 

 

Calculations:

 

Provide a full calculation for one case here, but only show the results for other cases in Table 1.

Comparison of the results: 

         Provide the percent error formula used as well as the calculated values of percent errors in this section as well as Table 1.

Conclusion: 

         State your conclusions of the experiment.

Discussion: 

         Provide a discussion if necessary.

 Questions:

Include the following questions and their answers in your report:

1) Two forces, one 500gf and the other 800gf, act on a body.  What are the maximum and minimum possible magnitudes of the resultant force? 

2) Could four forces be placed in the same quadrant or in two adjacent quadrants and still be in equilibrium?  Draw a sketch and explain your answer.

3) What is the relationship between the equilibrant vector and the resultant?