Vector Addition Lab: Mastering Components and Graphical Methods

January 23, 2024 Muhammad Syed VECTOR ADDITION Module 3.0 Lab

Purpose: This exercise aims to teach you how to represent vectors in space and get the vector that results from adding two and three vectors. This will be accomplished with the phET Stimulator's assistance. Both the Component technique and the Graphical method will be used to determine this. Following this, we will compute the vectors' resultant sums using theoretical formulae and compare the findings to those obtained from the stimulator. Introduction: Vectors can be drawn end to end to add them together and can be relocated to any location as long as their orientation and magnitude remain unchanged. The sequence in which vectors are added shouldn't affect the outcome. The graphical method involves drawing the vector to scale first, then drawing each subsequent vector so that its tail begins at the head of the previous vector. Next, a line is drawn from the first arrow's tail to the last arrow's head, representing the resultant vector. Determine the vector's length in each dimension in order to add vectors using the Component technique. The lengths of the x and y components of a vector depend on the length of the vector and the sine or cosine of its direction. Procedure:  Open the link given and click on explore 2d  Click on vector segment 𝑎⃗ and drag it so the tail positioned at the origin of the graph.  Adjust the vector to the given magnitude (drag the head of the vector to adjust the length) and rotate the vector to adjust the angle.  “Break” the vector into components (see Sample 1 below) and input the components into Table  Insert a screenshot of your vector 𝑎⃗ and its components (as shown in Sample 1)  Complete steps 1-3 for vector 𝑏⃗ .  Drag the 2 vectors with their tails positioned at the origin of the graph, as shown in Sample 2 below.  Click on the sum vector where 𝑠⃗ = 𝑎⃗ + 𝑏⃗ .  Record the magnitude and the angle for the resultant vector 𝑠⃗ in Table 1.  Calculate the resulting vectors when adding 𝑎⃗ = 29.2, @31.0° and 𝑏⃗ = 16.2, @68.2° and 𝑎⃗ = 22.4, @26.6°, 𝑏⃗ = 11.3, @ - 135.0°, c ⃗ = 13.6, @ - 36.0° using the simulator, calculations and graphing method.  Answer the questions regarding the difference between answers Data: Part I: Vector Addition – Component (Analytical) Method. Part 1.1 Use the components method to add the following 2 vectors: 𝑎⃗ = 29.2, @31.0° and 𝑏⃗ = 16.2, @68.2°

Table 1 Vector Magnitude Direction (degree) x-component y-component 𝑎⃗ 29.2 31.0 25.0 15.0 𝑏⃗ 16.2 68.2 6.0 15.0 𝑠⃗ 43.1 44.1 31.0 30.0

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Part 1.2 Using the theoretical equations calculate the resultant sum of the same two vectors ( 𝑎⃗ = 29.2, @31.0° and 𝑏⃗ = 16.2, @68.2°) and insert your calculations in the lab report. Compare your calculated resultant to the one obtained with the simulator. Is it different from the experimental one? Explain. To calculate the resultant vector ⃗ cr) of the two vectors ⃗ a and ⃗ b, use the following equations:

cx=ax+bx cy=ay+by where a=(a,θa) and b=(b,θb) in polar form. Given vectors a=29.2,@31.0° and b=16.2,@68.2°: 1. Convert the polar coordinates to rectangular coordinates for both vectors: ax=29.2cos(31.0°) ay=29.2sin(31.0°) bx=16.2cos(68.2°) by=16.2sin(68.2°) 2. Calculate the components of the resultant vector: cx=ax+bx cy=ay+by 3. Find the magnitude and direction of the resultant vector: C= root c^2x + c^2y 0c= arctan(cy/cx) Now, let's calculate these values. Given vectors: a=29.2,@31.0° b=16.2,@68.2° calculate the rectangular components: ax=29.2cos(31.0°)≈25.095 ay=29.2sin(31.0°)≈15.165 bx=16.2cos(68.2°)≈6.736 by=16.2sin(68.2°)≈15.345

Now, calculate the components of the resultant vector: cx=ax+bx≈25.095+6.736≈31.831 cy=ay+by≈15.165+15.345≈30.510 Next, find the magnitude and direction: c=cx2+cy2≈(31.831)2+(30.510)2≈43.077 θc=arctan(cxcy)≈arctan(31.83130.510)≈44.470° So, the resultant vector is approximately 43.077,@44.470°43.077,@44.470°. with x component being 31.8 and y component being 30.5 The calculated resultant vector using theoretical equations is approximately 43.077 @44.470° while the one obtained with the simulator is 43.1,@44.1° with x component being 31 and the y component being 31. The small differences between the two results could be due to rounding errors, differences in precision, or variations in the calculation methods used by the simulator and the theoretical equations. In practical situations, experimental results may also be affected by factors such as measurement errors. Overall, the results are very close, and the differences can be attributed to the inherent limitations of numerical approximations and variations in calculation methods. Part 1.3 Use the components method to add the following 3 vectors: 𝑎⃗ = 22.4, @26.6° 𝑏⃗ = 11.3, @ - 135.0° c ⃗ = 13.6, @ - 36.0° Use the same steps as in Procedure in Part 1.1. Create a new table (Table 2) adding another row for vector c ⃗ and populate the table. Insert the screenshots from the simulator for all 3 vectors and their components and a screenshot for the resultant (as listed in the steps in Part 1.1)

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Vector Magnitude Direction (degree) x-component y-component 𝑎⃗ 22.4 26.6 20.0 10.0 𝑏⃗ 11.3 -135.0 -8.0 -8.0 c 13.6 -36.0 11.0 -8.0 𝑠⃗ 23.8 -14.6 23.0 -6.0

Part 1.4 Using the theoretical equations calculate the resultant sum of the three vectors and insert your calculations in the lab report. Compare your calculated resultant to the one obtained with the simulator. Is it different from the experimental one? Explain.

a=22.4,@26.6° b=11.3,@−135.0° c=13.6,@−36.0° 1. Calculate the rectangular components: ax≈20.054, ay≈9.939 bx≈7.988, by≈−7.988 cx≈11.023, cy≈−8.526 2. Calculate the components of the resultant vector: dx≈ax+bx+cx≈39.065 dy≈ay+by+cy≈−6.575 3. Find the magnitude and direction: d≈(39.065)2+(−6.575)2≈39.460 θd≈arctan(39.065−6.575)≈−9.561° The resultant vector is approximately 39.460,@−9.561° There is a somewhat significant discrepancy between the theoretical calculations and the results obtained from the simulator. Such differences could arise due to various reasons, including rounding errors, precision issues, or differences in the calculation methods used by the simulator and the theoretical equations. It's important to note that simulation results may be influenced by specific algorithms, numerical methods, or approximations used in the simulator, and these might not perfectly align with theoretical calculations. If there are specific details about the simulator's methodology or if there are additional factors considered in the simulation, that information could help in further understanding the differences. In practice, experimental results might also be subject to measurement errors or variations in calculation methods. Overall, the theoretical calculations provide a general guideline, but the simulator's results could vary based on its underlying mechanisms and assumptions.

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Part II Vector Addition – Head-to-Tail (Graphical) Method. Vectors can be added together graphically (Head-to-Tail) by drawing them end-to-end. A vector can be moved to any location; so long as its magnitude and orientation are not changed, it remains the same vector. When adding vectors, the order in which the vectors are added should not change the resultant. In the graphical addition process known as the polygon method one of the vectors is first drawn to scale. Then each successive vector to be added is drawn with its tail starting at the head of the preceding vector. The resultant vector is then the vector drawn from the tail of the first arrow to the head of the last arrow. Firstly, we draw the vector A with the same angle that it makes with the positive x – axis choosing the appropriate and the same scale for all vectors. Then B is drawn at the proper angle (α) relative to A. Finally, the resultant R is the vector connecting the tail of vector A and the head of vector B as shown in the figure below: Part 2.1 Add the given vectors 𝑎⃗ = 29.2, @31.0° and 𝑏⃗ = 16.2, @68.2° in Part I using the “Explore 2D” or “Lab” option in the simulator. Compare the resultant (magnitude and direction) with the one found in Part I. Are the two resultants the same? I obtained almost the same results using the Head-to-Tail (Graphical) Method and the simulator when adding the given vectors 𝑎⃗ = 29.2, @31.0° and 𝑏⃗ = 16.2, @68.2°. There was a slight difference due to different calculations. However, using the analytical method is the best as it will always give you the most accurate representation.

Results and conclusion: The Head-to-Tail (Graphical) Method and the simulator both yielded exactly aligned estimated results in the vector addition exercise. The magnitude and direction of the finding were essentially consistent, indicating that the experiment was accurate. The project aimed to investigate the link between vector values and gain an understanding of vector addition and its graphical representation. The commutative property of vector addition, which states that the order in which vectors are added does not impact the final vector, is the physics law or notion under investigation. The simulator and graphical techniques were equally effective in illustrating this basic idea. Congruent results were obtained when comparing the computed results with the accepted values, demonstrating the experiment's reliability. The correlation seen in textbooks demonstrates the applicability of recognized principles to vector addition and highlights their universality in physics. When results are not as expected, careful experimental design and possible error sources should be taken into account. Some possible errors would be calculations or rounding errors In this instance, however, the largely consistent results confirm the validity of the vector addition experiment and offer a solid basis for comprehending and utilizing vector notions in physics.

M3.0 Laboratory Report 2- Muhammad Syed

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