Experiment 2: Vector AdditionLaboratory Report

Kate Auditor, Bethanee Baes, Keana Balverde, Lina Lou Berdijo

Department of Occupational TherapyCollege of Rehabilitation Sciences, University of Santo TomasEspaa Street, Manila Philippines

Abstract

1. IntroductionIn order to explain the various phenomena, man utilizes many physical quantities. Furthermore, each of these quantities can be described as scalar or vector quantity. Scalar quantity is a quantity that is described by a magnitude. Examples of these would include quantity of mass, time, distance, and temperature. On the other hand, vector quantity is a quantity that is completely described by both magnitude and direction. Velocity and force are instances of this.

A distinguishable characteristic of a scalar quantity is the ability of the quantities to be used (added, subtracted, etc.) like ordinary numbers. Contrast to this, a vector quantity is more complex because it is also described by direction, aside from just magnitude.

Vectors are manipulated by many mathematical operations, generally called vector algebra. However, in this experiment, the focus would be on addition of vectors. Vector addition is a process of combining two or more vectors. The sum of two or more vectors would then be called a resultant vector (R). Different ways can be used to achieve the resultant. Hence for this experiment, the group aims to: (1) determine the resultant displacement by the component method, parallelogram method, and polygon method; (2) show that vector addition is commutative and associative.

2. Theory

Different ways can be utilized to find the resultant vector. One way to determine the resultant vector is through the use of graphical methods. An example of this is the polygon method. Graphing the vector quantities in a head-to-tail manner, resultant vector is calculated by lining up the head of the last vector to the tail of the first vector. This method is used to find the resultant of three of more vectors and states that theresultantof two or more vectors is a vector that is equivalent in its physical effects to the action of the original vectors. This entails that no order of addition must be followed, as long as the direction and length of each vector is not changed. Another method is the parallelogram method. This involves drawing the vector to scale in the indicated direction, sketching the same length parallel to the original vector, all which creates a parallelogram. A diagonal from the origin drawn inside is the resultant vector. Because it follows a parallelogram, only two vectors can be accommodated, indicating that a formula must be used to accommodate the other vectors. Since this is the case for the experiment, below are the formulas used:(A + B) + C = resultantA + (B + C) = resultantNote that the letters represent the given vectors. The resultant vectors from the parentheses will be derived from the parallelogram.Analytical methods were also used to determine the resultant. Most commonly used the Pythagorean Theorem; however this is only limited to vectors that are perpendicular to each other. If vectors involved are oriented to each other at angles other than 90 degrees, the analytical method called component method can be used. This includes drawing each vector and finding each x-component and y-component of each vector using the formula:

After which, the sum of the x-components and y-components are calculated:

Their sum was then used to find the resultant. The Pythagorean Theorem is utilized here: Unlike the graphical method that uses a protractor to measure the direction of the resultant, the angle is found from the definition of tangent function:

3. Methodology

The meter stick, chalk, graphing paper and protractor were used to determine the resultant displacement for this experiment.

For the first activity, the Polygon Method, the initial position of the group member was marked. This member underwent the following displacement, successively: 1m E, 2.5m N, and 3m 30 N of W. As this member's final position was marked, an arrow from the initial to the final position was drawn; representing the resultant displacement. Using the ruler and protractor, the magnitude and direction of the resultant displacement was determined. This member was then asked to walk in the following order: 2.5m N, 3m 30 N of W and 1m E. Then the resultant displacement was noted. It was repeated, following this order: 3m 30 N of W, 2.5m N and 1m E.

Image 1: Polygon Method

In the second activity, the following displacements were used: = 1m E, = 2.5m N, and = 3m 30 N of W. A suitable scale was utilized to determine the resultant displacement with the use of the Parallogram Method. A and B were added for the first determination. Its resultant was then added to C. Afterwards, B and C were added for the second determination. The result was then added to A.

Image 2: Determining the Resultant Displacement for the Parallelogram and Component Method

The Component Method was used to determine the resultant displacement in the third activity. Using this resultant as the accepted value, the % error of magnitude and direction of the resultant displacement obtained in activities 1 and 2, were computed.

4. Results and Discussion

Data gathered using the Polygon method, Parallelogram method and Component method are all summarized in Tables 1, 2, and 3 respectively. A total of three trials were done using the polygon method, while a total of two trials were done using the parallelogram method.

Table 1. Polygon MethodTrial 1Trial 2Trial 3

Magnitude of R4.5m4.5m4.5m

% Error of Magnitude4.41%4.41%4.41%

Direction of R67 N of W69 N of W68 N of W

% Error of Direction1.76%1.17%0.29%

All the trials were made using the polygon method. However, each trial differed in the sequence of successive displacements. All three trials showed a 4.5m magnitude of R. Using the resultant from the component method as the accepted value, a 4.41% error of magnitude was derived for all the three trials. For the direction of R, varied results were measured: 67 N of W from trial 1, 69 N of W from trial 2, and 68 N of W from trial 3. Using the direction from the component method as the accepted value, a 1.76%, 1.17% and 0.29% error were computed for trials 1, 2 and 3 respectively.

Table 2. Parallelogram MethodScale: ___1 cm 1m______Trial 1Trial 2

Length of arrow respecting R4.4 cm4.3 cm

Magnitude of R4.4 m4.3 m

% Error of Magnitude2.09%0.23%

Direction of R65N of W68 N of W

% Error of Direction4.69%0.29%

Trials 1 and 2 were executed using the Parallelogram method. However, each of the trial had a different set of determination (see Methodology). With a scale of 1 cm: 1m, Trial 1 showed a length of 4.4 cm, hence a magnitude of 4.4 m, while trial 2 showed a length of 4.3 cm was measured, hence a magnitude of 4.3 m. Computing for the % error of magnitude using the results from the component method as the accepted value, a 2.09% error of magnitude for trial 1 and a 0.23% error of magnitude for trial 2 were gotten. A direction of 65 N of W, which yields a 4.69% error of direction and 68 N of W, which yields a 0.29% error were measured and calculated from trials 1 and 2 respectively.

Table 3. Component MethodDisplacementx-componenty-component

A10

B02.5

C-2.61.5

Magnitude of R = 4.3m, Direction of R = 68.2 N of W

Using given values of A = 1m, B = 2.5m N and C = 3m 30 N of W (see Methodology), the x-component of each displacement were computed using cosine: 1 for A, 0 for B and -2.6 (negative because of its direction) C, yielding a summation of -1.6. Using sine, the y-component for each displacement was calculated: 0 for A, 2.5 for B and 1.5 for C, yielding a summation of 4. With the sum of the x-component and the y-component, a magnitude of 4.3 m and a direction of 68.2 N of W were derived.

5. ConclusionVector quantities cannot be manipulated like scalar quantities. Instead, graphical and analytical methods must be used to determine the resultant. Three of those methods are the Polygon method, which uses the head-to-tail graphical fashion, Parallelogram method, which uses the concept of parallelograms, and Component method, which accommodates vectors with angles other than 90 degrees. From the experiment, results from the polygon method showed a 4.5 m magnitude of R on all trials done. This demonstrates that vector addition is commutative, noting that the successive displacements were varied in each trial. From the Parallelogram method, a resultant of 4.4m in trial 1 and 4.3m in trial 2 were drawn. This deviates from the theory of vector addition being associative and is most likely perhaps by error that this property was not shown. Lastly, a resultant of 4.31m was derived using the component method.

6. Applications

1) You are given only the magnitudes of two vectors: 3 units and 4 units. What is the range of magnitude of resultant? What must be the angle between these vectors to get A) maximum resultant B) minimum resultant C) a resultant of magnitude 5 units D) a resultant of 6 units?

2) Differentiate distance from displacement. Is it possible for you to have no displacement even though you have travelled a great distance? Explain by giving examples.

3) To go to a grocery, a student has to walk 8.25m S, 4.0 m E, and then 2.5m SE from his dormitory. Specify the distance and bearing of the grocery relative to the students dormitory.

7. References

Pedrosa, Ciriaco O.P., College Physics: A Laboratory Giode and Notebook, Manila: UST Cooperative, 1981.

Siddons, Collins, Experiments in Physics. Oxford: Basil Blackwell, 1988.