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. "Equilibrant" is 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 F_{1}, F_{2}, and F_{3 }.
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 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. 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 F_{1}, F_{2}, and F_{3}. 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:
F_{1} + F_{2} + F_{3} = 0 or, F_{1} + F_{2} = - F_{3} or, R = - F_{3 }.
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 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 y-components 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 y-component of F_{3}. F_{3} is the equilibrant. It is the opposite of the resultant R of F_{1} and F_{2}. The goal is to find the resultant R of F_{1} and F_{2} . You have found F_{3}, the opposite of R.
4) Use the x- and y-components of F_{3} that you read from the applet to calculate the magnitude and direction (angle) of F_{3 } using the following formulas:
F_{3} = [(F_{3}_{x})^{2} + (F_{3}_{y})^{2} ]^{1/2} and θ_{3} = tan^{-1}( /F_{3}_{x})
The F_{3} magnitude that you calculate is to be used as the measured value of F_{3} or the measured magnitude of R. Adding 180.^{o} to or subtracting 180.^{o} from the angle of F_{3} 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 F_{1} and F_{2} 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.
R_{x} = F_{1x} + F_{2x} = 50.0Ncos(35.0^{o}) + 75.0Ncos(115.0^{o}) = ...... N.
R_{y} = F_{1y} + F_{2y} = 50.0Nsin(35.0^{o}) + 75.0Nsin(115.0^{o}) = ...... N.
R = (R_{x}^{2} + R_{y}^{2})^{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 F_{3} |
Measured R |
Calculated R |
%error | |||||||
F_{1} | F_{2} | |||||||||||
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 |
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?