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PHY 2053 L. Lab Vectors, using the simulator. Name: Date: Open the simulator: https://phet.colorado.edu/sims/html/vector-addition/latest/vector-addition_en.html Objective The objective of this lab is adding vectors using both the tail-to-head method and the component method and to verify the results using a simulator. Theory A scalar quantity is a number that has only a magnitude. When scalar quantities are added together the result is a sum. Vectors are quantities that have both magnitude and direction; specific methods of addition are required. When vector quantities are added, the result is a resultant. For example, if you walk 100 m north, then 100m east, you will walk a total distance of 200 m (distance is a scalar quantity). However the displacement 𝛥𝑟 is a vector, involves both distance and direction. So, the same 200 m walk results, in a displacement of approximating 141 m northeast of where you began (141 m, northeast of your starting position). A negative vector has the same length as the corresponding positive vector, but with the opposite direction. Making a vector negative can be accomplished either by changing the sign of the magnitude or by simply adjusting the direction by 180°. Suppose you have a vector with magnitude 5 m in the direction of 100° (related to the positive X-axis). You can describe it specifying the magnitude and direction such as ° . ⃗ , which will be Also, you can identify the opposite vector to 𝑉 ° or – ° Experiment: Vector Addition. Tail-to-Head Method. Geometric method. Part A. Vectors can be added together graphically 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 does not change the resultant. • Draw each vector on a coordinate system; begin each from the origin. • Choose any vector drawn to be the first vector. • Choose a second vector and redraw it, beginning from the end of the first. • Repeat, adding as many vectors as are desired to the end of the “train” of vectors. • The resultant is a vector that begins at the originand ends at the tip of the last vector drawn. It is the shortest distance between the beginning and the end of the path created. PHY 2053 L. Lab Vectors, using the simulator. ~ +F ~ =F ~ +F ~ =R ~ F 1 2 2 1 Figure 1: Adding 2 Vectors, Tail-to-Head The tail-to-head method is often useful when working problems. A quick sketch, rather than measurements, can help verify your solutions. VECTOR ADDITION. Component Method. Part B To add vectors by components, calculate how far each vector extends in each dimension. The lengths of the xand y-components of a vector depend on the length of the vector and the sine or cosine of its direction, ✓: Use algebra to solve for each component, F1x and F1y, from these equations. (1) (2) (3) Figure 2 PHY 2053 L. Lab Vectors, using the simulator. When each vector is broken into components, add the x-components of each vector: 𝑛 ∑𝑖=1 𝐹𝑖𝑦 = 𝑅𝑦 , (4) Then add all of the y-components: 𝑛 ∑𝑖=1 𝐹𝑖𝑦 = 𝑅𝑦 (5) The sums are the x- and y-components of the resultant vector, R The components of R can be converted back into original form determining the magnitude and direction of the vector polar form (R, ) using the Pythagorean theorem (Eq. 6) and the tangent function (3): |𝑅⃗ | = 𝑅 = √𝑅𝑥2 + 𝑅𝑦2 (6) 𝑅𝑦 𝜃𝑅 = 𝑎𝑟𝑐𝑡𝑎𝑛 ( ) (7) 𝑅𝑥 Note: Verify the quadrant! A calculator will return only one of two possible angles (7). To verify the quadrant, determine if Rx, Ry are positive or negative. If your calculation puts the resultant in quadrant I, but Rx and Ry are both negative, it must be in quadrant III; simply add 180 to the angle. PHY 2053 L. Lab Vectors, using the simulator. Lab Report. Write the goal of the experiment. Part 1. Using the simulator “Explore 2D option”. Activities. 1.1) Using the simulator, Explore 2D, and given the value for vectors a, b and c, represent all vectors in the simulator and complete the data table 1, show all formulas and calculations. 1.2) Attach the diagram to the lab report with three vectors and their components. 1.3) In which quadrant are the vectors a, b and c. Describe the signs of its components. Could you establish some rule about their signs? Explain. Table 1.EXPLORE 2D Vector a Component x Component y Vector b Vector c Magnitude Direction Magnitude Direction Magnitude Direction 23.3 59° 25 143.1° 13.9 -159 PHY 2053 L. Lab Vectors, using the simulator. Part 2. Using the simulator , “Lab Option” Table 2. Lab in simulator Trial No Vector a Magn. Direction of a degree Vector b in Magn. Direction of b degree trial 1 26.7 13.0° 17.8 128.2° trial 2 18 13° 10 -90 trial 3 15 180° 25.5 -11.3 trial 4 7.1 45° 22.4 -26.6 Compon. a Compon. b in ax ay bx by Resultant vector Magn. Direct. of R of R PHY 2053 L. Lab Vectors, using the simulator. Activities. 2.1) Using the simulator, Lab option, and given the value for vectors a and b on the table 2, represent all vectors in the simulator and complete the data table 2, show all formulas and calculations. 2.2) Using the simulator represent all trials in your computer and check that your answers using the theoretical equations in the point 2.1 are the same with the result of the simulator. Show your formulas and calculations 2.3) Attach the diagram only for trial 4 to the lab report with all vectors a, b and R, where it is shown all simulator calculations for R. Part 3. Using simulator “Equation option”. Table 3.”Equation option” Vector a Vector c components Vector b Cases ax ay bx by c=a+b 0 5 5 5 c=a-b 0 5 5 5 a+b+c=0 0 5 5 5 cx Vector c magn and direct. cy c Ɵ PHY 2053 L. Lab Vectors, using the simulator. ACTIVITIES . about your experiment 3.1) Using the simulator, Equation option, and given the value of the corresponding components of vectors a and b on the table 3, represent all vectors in the simulator and complete the data table 3, show all formulas and calculations. 3.2) Using the simulator represent all three cases in your computer and check that your answers using the theoretical equations in the point 3.1 are the same with the result of the simulator. 3.3) Attach the diagram only for case 3 to the lab report with all vectors a, b and c, where it is shown all simulator calculations for R. 3.4) Calculate the percent error in case 1 fro the magnitude of vector c, where the experimental value using a ruler in the diagram was c=11.3. Note : remember the percent error for any experiment will be: 0⁄ 𝑒𝑟𝑟𝑜𝑟 = |𝐴−𝐸| 100% , A is the acceptable value (in our case will be the calculation) and E ; will be the value 0 𝐴 obtained from experiment with some instrument of measurement. -Show all formulas and calculations. -Write the conclusions for experiment.
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