Experiment 3
Equilibrium of Concurrent Forces
Vector Addition by Using a Force Table
Objective:
To experimentally verify the parallelogram law of vector addition by using a virtual force table
Equipment:
A computer with Internet connection, a calculator (The builtin calculator of the computer may be used.), paper, and pencil
Theory:
Concurrent forces are forces that pass through the same point. A resultant force is a single force whose effect is the same as the sum of a number of forces. The equilibrant of a system of forces is equal in magnitude and opposite in direction to the resultant of those forces.
Procedure:
In Physics Lab, a force table (as shown) is usually used. It is helpful to understand how a force table works even if you are not using one in this experiment.
Description of Force Table: A force table has a disc and three or more pulley mechanisms that mount on it for the application of forces. Three or more weights are hung as shown. The strings and frictionless pulleys transmit the three forces F_{1}, F_{2}, and F_{3} to the ring at the center of the round Plexiglas disc. A vertical short stud is attached to the center of the disc. By adjusting the magnitude and direction of the three forces, the ring can be made to be exactly at the center around the central stud. In such case, the three forces F_{1}, F_{2}, and F_{3} are in equilibrium. This means that the resultant of any two of these forces is neutralized by the third one. For example, the resultant of F_{1 }and F_{2} is neutralized by F_{3}, or the resultant of F_{2} and F_{3 }is neutralized by F_{1}. Whatever the magnitude of the resultant R of F_{1 }and F_{2} is, the magnitude of F_{3} must be equal to R and its direction must be opposite to that of R, for equilibrium. Measuring the third force is like measuring the resultant of the first two forces.

A schematic diagram of a force table 
Click on the following link: http://lectureonline.cl.msu.edu/~mmp/kap4/cd082.htm
You will see F_{1} in red, F_{2} in green, and F_{3} in blue. With the mouse you can move any of these three vectors by holding their tips. Try to set each vector at a certain x and y components. As you move any of the vectors around you will see that the values corresponding to its components change. The black vector shows the resultant of the three vectors F_{1}, F_{2}, and F_{3}. If you make the black vector to have a zero magnitude, then the three vectors are in equilibrium and 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 shown in the Data Section. There are four cases (experiments) to be done.
1) In Line 1 of the Table, calculate the x and ycomponents of F_{1} and F_{2}. Round the numbers to the nearest integer. Make sure your calculator is in "Degrees" mode.
2) Place the mouse on the tip of F_{1} and move it around until its x and ycomponents match your calculated values for F1. Repeat this procedure for F_{2}.
3) Move the tip of F_{3} around until the black vector shrinks to zero. Record the x and ycomponent of F_{3}. F_{3} is the equilibrant. It is the opposite of the resultant R of F_{1} and F_{2}. The purpose is to find the resultant R of F_{1} and F_{2} in this experiment. Now you have found F_{3}, the opposite of the resultant.
4) Use the x and ycomponents of F_{3} to calculate the magnitude and direction (angle) of F_{3}. The F_{3} magnitude that you calculate is to be used as the measured value of F_{3} or the measured value of R. Add 180.^{o} to or subtract 180.^{o} from the angle of F_{3} to find the direction of R, the resultant of F_{1} and F_{2}. Now, you have the measured values for the magnitude and direction of R.
5) Also, find the resultant R of F_{1} and F_{2} by performing calculations on the 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.
R_{x} = A_{x} + B_{x} = 200.cos(35.0) + 300.cos(115.0) = ...... .
R_{y} = A_{y} + B_{y} = 200.sin(35.0) + 300.sin(115.0) = ...... .
R = SQRT( R_{x}^{2} + R_{y}^{2}) =........ and θ = tan^{1} (Ry / Rx) = ......... . SQRT( ) means Square Root of.
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 given values in the following chart.
Measured: specified by the question marks.
Table 1
Trials

F1 (Red)  F2 (Green)  F3 (Blue)  
Magnitude
(units of force) 
Angle ( ^{o} )  Magnitude
(units of force) 
Angle ( ^{o} )  Magnitude
(units of force) 
Angle ( ^{o} )  
1  100.  35.0  150.  115.0  ?  ? 
2  75.0  130.0  140.0  210.0  ?  ? 
3  100.0  0.0  100.  120.0  ?  ? 
4  150.  40.0  150.0   40.0  ?  ? 
Comparison of the results:
Provide the percent error formula used as well as the calculation of percent errors.
Conclusion:
State your conclusions of the experiment.
Discussion:
Provide a discussion if necessary.
Questions:
Two forces, one 500gf and the other 800gf, act upon a body. What are the maximum and minimum possible values of the resultant force?
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.