Vector Addition of Forces

Purpose/Introduction: The purpose of this lab was to study vector addition using graphs and also using components. We also used a circular table to confirm our results. To begin, we were given three different vector magnitudes and their angles. The vectors that were given to us are represented in this table.

Vector	Magnitude (grams)	Angle (Degrees)
A	200	0
B	100	55
C	200	135

We decided that instead of drawing a vector with a magnitude of 200 cm, we would scale the magnitudes to 1 cm = 50 g. Therefore, the magnitudes to draw were 4 cm (A), 2 cm (B), and 4 cm (C). The next procedure was to draw a vector diagram of the three vectors and find the resultant vector using a protractor and straight edge.

Vector diagram of vectors A, B, and C. Vector D is A + B + C.

The resultant force from the vector diagram was D. The magnitude of D was approximately 4.8 centimeters or about 240 grams. This means that in order to balance the circular force table, you would need to put approximately 240 grams on the fourth pulley.

The second part of the lab was to find the resultant vector (D) by using components. We graphed the vectors according to the scale and the angle given. We used sine and cosine to determine the opposite and adjacent sides. 

Data:
î represents the x coordinate

ĵ represents the y coordinate

cosθ= î /hypotenuse

sinθ= ĵ / hypotenuse

Vector	î substituted	î	ĵ substituted	ĵ
A	4cos0 = î	4	4sin0 = ĵ	0
B	2cos55 = î	1.147	2sin55 = ĵ	1.638
C	4cos45 = î	-2.828	4sin45 = ĵ	2.828

We calculated Vector D to have a magnitude of 251 grams at 62.5°. The way we found that out is that we added the three vectors and received two components, î and ĵ. We took the arctangent of (ĵ/ î) and received the angle of Vector D. In order to find the magnitude of D, we added î² and ĵ² and took the square root of the sum. We received an answer of 251, which was consequently our magnitude.

The next step was to confirm our results by setting up a force table and arranging the pulleys so that they represent our given vectors at the given angles. After placing the three pulleys on the table, the ring in the center was NOT at equilibrium. We placed a fourth pulley at 242.5° (180° + 62.5°) with 250 grams and the ring equaled out.

Circular force table that shows Vectors A, B, C, and D (resultant vector)

The next and final step was to confirm our results using a university website. We confirmed that the sum of our vectors was approximately 252 grams and about 62.5°.

Vector program from http://phet.colorado.edu/en/simulation/vector-addition

Conclusion: This lab showed that vectors can be calculated in many ways. Today’s lab demonstrated two ways to calculate the sum of more than one vector. The first way is by a graph. Although this method is less accurate than components, it is still effective. The reason why some calculations are off is because the measuring devices (rulers, protractors, ect.) are not always accurate and one is basing the measurement of the resultant on the measuring device. This is one source of error of the graphing method. The other method has a more accurate measurement. The component method uses trigonometry to calculate the angle and magnitude of the resultant vector. This is far more accurate than the graphical method because one is using angles and magnitudes rather than a ruler to measure a vector. There is hardly any error with this method. With the website that we were supposed to check our resultant vector, the reason that was off is because the program only allowed certain magnitudes and angles; consequently, one is not able to put in the correct magnitudes and angles or the given vectors. Using the component method will ultimately give one the most accurate resultant vector.