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 built-in 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 F1, F2, and F3 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 F1, F2, and F3 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 F1 and F2 is neutralized by F3, or the resultant of F2 and F3 is neutralized by F1.  Whatever the magnitude of the resultant R of F1 and F2 is, the magnitude of F3 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 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.   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 F1, F2, and F3.  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 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 purpose is to find the resultant R of F1 and F2 in this experiment.  Now you have found F3, the opposite of the resultant.

4) Use the x- and y-components of F3 to calculate the magnitude and direction (angle) of F3.  The F3 magnitude  that you calculate is to be used as the measured value of F3 or the measured value of R.  Add 180.° to or subtract 180.° from the angle of F3 to find the direction of R, the resultant of F1 and F2.   Now, you have the measured values for the magnitude and direction of R.

5) Also, find the resultant R of F1 and F2 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.

Rx = Ax + Bx = 200.cos(35.0) + 300.cos(115.0) = ......  .

Ry = Ay + By = 200.sin(35.0) + 300.sin(115.0) = ......  .

R = SQRT( Rx2 + Ry2) =........  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 ( ° ) Magnitude (units of force) Angle ( ° ) Magnitude (units of force) Angle ( ° ) 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 ? ?

### Provide the necessary calculations.

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:

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