One of the important concepts of introductory physics is the notion of vectors and their addition. Combining forces in the laboratory provides a concrete experience for students to encounter these notions. Over the years we have struggled to make the process as facile as possible. Our method in the past was much like that described recently in The Physics Teacher by Bracikowski, et. al.¹ In our experiments the forces were generated by using spring balances secured with strings to heavy weights.

As the simplest example we want students to observe experimentally that two arbitrary forces can be balanced by a third force of equal magnitude and opposite direction to the vector sum of these two forces. Setting up three spring balances to do this with force magnitudes that are neither too large, sending the balances off scale, nor too small, producing unacceptably large errors, is daunting to students who may have only a tenuous idea of what the experiment is about. Faced with results that are only accurate to ten percent the student are often frustrated and confused.

Introduction of low-friction pulleys combined with the traditional force table improves the accuracy of the experiment, but this approach has limitations. With a judicious choice of weights it is possible to balance the forces by simultaneously adjusting the angles of the two pulleys. But this defeats our objective of adding arbitrarily directed forces. Alternately these forces can be fixed and the student can attempt to find the required combinations of weights to obtain the equilibrant. This approach is tedious and limited by the discrete character of the weight sets.

The popular strain gauge force sensors provide an excellent solution to these problems. One force sensor is mounted on a force table and is easily adjusted to provide the required equilibrant for two or more weights. This restores simplicity to teaching the idea of adding several forces with an experimental error of one percent or less.

The setup is shown in Fig. 1. We used the Pasco Force Table², the Dual Range Force Sensor³ and bracket⁴ made by AU Physics Enterprises (distributed by Vernier Software), and the Pasco Science Workshop 700 Interface⁵. Three strings with lengths about three times the table diameter are tied together at their center yielding six strands. Up to five of these support weights on pulleys and the sixth is connected to the force sensor. Several pulleys are placed at arbitrary angles around the perimeter of the table with hooked weights attached. The only steps necessary to balance the forces are to position the sensor at the appropriate angle and to make a slight adjustment of the radial position of the force sensor using a bracket screw so that the knot is at the center of the table. The force acting on the sensor can then be read.

Finding the equilibrant of five forces is not substantially more difficult than doing it with two.. Table I, done with a spread sheet, summarizes the results of six runs, each adding five forces. These tables are generated with a spreadsheet with nested if statements to select the proper quadrants for the inverse tangent. The difference between the magnitudes of the calculated sum of forces and the equilibrant are consistently less than 1% . The noise level in the sensor we tested was about 10 mN on the 10 N range. Therefore, for equilibrants less than 1 N the error in measurement would be greater than one percent. To avoid damage to the pulleys Pasco recommends limiting weights to masses of less than 1 kg.

In the summer and fall of 1998 our students first used the sensor in conjunction with the force table. In our experiment the students hung two masses at specified directions. Next they adjusted the location of the force sensor so that its force was toward the table center. Then they turned the bracket screw, moving the force sensor radially, until the string knot was at the table center. They recorded in Science Workshop the force for a few seconds and entered in their report the average value and direction of the force sensor force. The students used graphical and component vector addition techniques to add the forces produced by the two masses and compared this sum with the measured equilibrant force measured from the force sensor. The average experimental error obtained by these students was about one percent. The students repeated the experiment finding the sum of forces produced by three hanging masses with similarly accurate results.

The bracket-mounted force sensor is a major improvement over the spring balance method of teaching force vector addition in the lab. When used in conjunction with the force table, students can measure equilibrants with errors less than one percent, and do so very quickly and easily. With the precision of this approach the teacher can easily see if the student is doing careful laboratory work. Its ease of performance frees the student from time consuming tedius manipulation giving them more opportunity to concentrate on understanding the physics.

References

1. C. Bracikowski, P. J. Garcia, and D. J. Harper, "Getting the Feel for Vector Addition of Forces," The Phys. Teach. 36, 114 (1998).

2. Super Pulley Force Table (ME-9447), available from Pasco Scientific, Roseville, CA; 800-772-8700.

3. Dual-Range Force Sensor (DS-DIN), made by AU Physics Enterprises and distributed by Vernier Software, Portland, OR; 503-297-5317.

4. Dual-Range Force Sensor bracket (new, designed by Dr. Bruce Lee, Professor Emeritus, Andrews University), made by AU Physics Enterprises and distributed by Vernier Software, Portland, OR; 503-297-5317.

5. Science Workshop 700 Interface (CI-6565A), available from Pasco Scientific, Roseville, CA; 800-772-8700.

3. and 4. combined into new 3.

3. Dual-Range Force Sensor (DFS-DIN) and Bracket (FTA-DFS, $25.00.) distributed by Vernier Software, Portland, OR; 503-297-5317. The Force Sensor and new bracket were designed by Dr. Bruce Lee, Professor Emeritus, Andrews University and are made by AU Physics Enterprises.
 

Table I. A Comparison of Predicted and Measured Equilibrants
	Run #1		Run #2		Run #3		Run #4		Run #5		Run #6
	|F|	qº	|F|	qº	|F|	qº	|F|	qº	|F|	qº	|F|	qº
F1	6.27	74	4.51	52	2.84	54	4.41	75	1.76	51	1.67	66
F2	5.29	139	1.47	120	3.53	116	3.23	132	1.57	128	1.37	129
F3	7.35	197	6.76	183	4.41	167	2.74	177	1.76	192	1.76	168
F4	2.55	224	4.41	221	1.76	222	4.80	243	1.86	225	1.18	239
F5	5.00	313	5.19	308	1.86	288	4.02	300	1.86	283	1.76	306
Sum	7.96	166.0	5.46	207.4	6.03	144.4	4.05	193.6	2.63	199.8	1.57	160.6
Epred	7.96	346.0	5.46	27.4	6.03	324.4	4.05	13.6	2.63	19.8	1.57	340.6
Emeas	8.03	346.1	5.49	27.5	6.09	324.5	4.08	13.4	2.64	19.5	1.58	340.1
Error	0.96%	0.1	0.64%	0.1	0.90%	0.1	0.70%	0.2	0.31%	0.3	0.78%	0.5