Vectors Remote - Procedure

Last Updated 9/7/2020 1 Physics Lab: Vectors Purpose To understand more about vectors, by using three completely different methods of vector addition. Theory In the first part of this lab you will use the simulation at https://phet.colorado.edu/en/simulation/vector- addition. You will select three vectors from the bucket and adjust their magnitudes and directions to match the vector 𝐴𝐴 ⃗ , 𝐵𝐵 �⃗ , and 𝐶𝐶 ⃗ listed in your template. You will find the sum of these three vectors, 𝑅𝑅 �⃗ = 𝐴𝐴 ⃗ + 𝐵𝐵 �⃗ + 𝐶𝐶 ⃗ , in three different ways. It is difficult to determine exactly what size and direction this resultant force has. However, it is easy to find when the net force is (nearly) zero: the ring will stay put, without needing to rest against the center post. By adding a fourth force (which we will call the “Equilibrant,” 𝐸𝐸 �⃗ ), we will get a net force equal to zero. That is, 𝐴𝐴 ⃗ + 𝐵𝐵 �⃗ + 𝐶𝐶 ⃗ + 𝐸𝐸 �⃗ = 0 �⃗ . But since 𝑅𝑅 �⃗ = 𝐴𝐴 ⃗ + 𝐵𝐵 �⃗ + 𝐶𝐶 ⃗ , that means 𝑅𝑅 �⃗ + 𝐸𝐸 �⃗ = 0 �⃗ , or 𝑅𝑅 �⃗ = −𝐸𝐸 �⃗ . So, if we experimentally determine 𝐸𝐸 �⃗ , we can say that 𝑅𝑅 �⃗ has the same magnitude as 𝐸𝐸 �⃗ , but is in exactly the opposite direction (that is, 180° away). Preparation 1. Go to the https://phet.colorado.edu/en/simulation/vector-addition webpage. 2. Open the simulation by clicking with the left mouse button on the play symbol shown in the following figure. Procedure Part A: Finding the Equilibrant By adding a fourth vector (which we will call the “Equilibrant,” 𝐸𝐸 �⃗ ), we will get a net vector equal to zero. That is, 𝐴𝐴 ⃗ + 𝐵𝐵 �⃗ + 𝐶𝐶 ⃗ + 𝐸𝐸 �⃗ = 0 �⃗ . But since the resultant 𝑅𝑅 �⃗ = 𝐴𝐴 ⃗ + 𝐵𝐵 �⃗ + 𝐶𝐶 ⃗ , that means 𝑅𝑅 �⃗ + 𝐸𝐸 �⃗ = 0 �⃗ , or 𝑅𝑅 �⃗ = −𝐸𝐸 �⃗ . So, if we experimentally determine 𝐸𝐸 �⃗ in our simulation, we can say that 𝑅𝑅 �⃗ has the same magnitude as 𝐸𝐸 �⃗ , but is in exactly the opposite direction (that is, 180° away). In this part of the lab, you will use the simulation to “physically” add the three vectors.

Physics Lab Vectors Penn State Harrisburg 2 1. Double click on the “Lab” icon to start the simulation. Uncheck the box next to the grid in the menu on the right to turn off the background grid. 2. Drag a blue arrow from the box near the 𝑥𝑥 -axis (see diagram below). 3. Move the start (tail) of the arrow to the center of the screen by clicking and holding down the left mouse button over the middle of the arrow and then dragging the arrow. 4. Adjust the length and angle of the arrow by clicking and holding down the left mouse button over the arrow end (tip) while dragging the end using the mouse until the arrow matches the magnitude and direction of vector 𝐴𝐴 ⃗ . Drag the vector so that its tip is at the top of the workspace, as shown in the following figure. 5. Drag another arrow from the box into the graph area. 6. Move the start (tail) of the second arrow to the start (tail) of the first arrow. 7. Adjust the length and angle of the second arrow to match the magnitude and direction of vector 𝐵𝐵 �⃗ . 8. Drag a third arrow from the box into the graph area. 9. Move the start of the third arrow (tail) to the start of (tail) of the first arrow. 10. Adjust the length and angle of the third arrow to match the magnitude and direction of vector 𝐶𝐶 ⃗ . 11. Click the Show Sum button. Your simulation should now look similar to the following figure. Drag from here

Physics Lab Vectors Penn State Harrisburg 4 21. Review your list of angles for the equilibrant from steps 15 through 17. Calculate the absolute value of the difference between the best estimate value and these values for the angle. Record the greatest value of these differences as the uncertainty in the angle of the equilibrant in the template. 22. The resultant 𝑅𝑅 �⃗ has a magnitude equal to that of the equilibrant, and a direction exactly opposite (180°) to that of the equilibrant. (Recall 𝑅𝑅 �⃗ = −𝐸𝐸 �⃗ .) Record the magnitude and direction of your resultant vector, with uncertainty, on the template. Part B: Graphical Vector Addition In this part of the lab, you will find the vector sum of the same three vectors using the graphical method, which is described in further detail in your textbook. Note: like usual, each partner should make his or her own diagram and measurements in the template. However, you should discuss with each other how to perform this part correctly and compare answers (it’s a good way to catch errors). 1. Draw an x-y coordinate system on the paper, being sure to make the two axes perpendicular. (You may use the given page on the template.) 2. Choose a convenient scale for your drawing. For example, a map may have a legend that reads “1 in. = 1.5 mi.” meaning that one inch on the map means 1.5 miles of the actual ground that the map refers to. In your case, you should have something like “1 cm = X arbs,” where X is chosen so that the graphical method will work well. That is, if X is too small, then the sum of vectors will not all fit on your page. If X is too large, then the vectors will be so small that they will be very difficult to measure accurately. Be sure to write your scale somewhere on your paper. You should all use the same scale for your drawings. 3. Using a ruler and protractor, draw vector 𝐴𝐴 ⃗ starting from the origin, of the appropriate size and angle (based on your chosen scale). 4. Lightly sketch a new x-y axis at the end (tip) of vector 𝐴𝐴 ⃗ . The new x-y axis must be parallel to the original x-y axis. Then, again using ruler and protractor, draw vector 𝐵𝐵 �⃗ starting from the end of vector 𝐴𝐴 ⃗ , of the appropriate size and direction. 5. Lightly sketch a new x-y axis at the end (tip) of vector 𝐵𝐵 �⃗ . Then, again using ruler and protractor, draw vector 𝐶𝐶 ⃗ starting from the end of vector 𝐵𝐵 �⃗ , of the appropriate size and direction. 6. Draw the resultant 𝑅𝑅 �⃗ , starting at the origin of the original coordinate system, and ending at the end of vector 𝐶𝐶 ⃗ . Measure the resultant’s magnitude in cm and direction and record your results in the template. NOTE : Do not try to draw your resultant 𝑅𝑅 �⃗ from part A! The point is to employ three different ways to find the vector 𝑅𝑅 �⃗ .

Physics Lab Vectors Penn State Harrisburg 5 7. Since it is not possible to draw lines of exactly the right length, at exactly the right angle, starting from exactly the right point, it shouldn’t be surprising if your result is not perfectly accurate. Based on how well you were able to use pencil and paper, try to get a reasonable estimate of how far off the correct values may be from your measured values. This rough judgment call on your part will represent your uncertainty in this result. Record the uncertainties in magnitude in cm and direction of your resultant vector in the template. 8. Convert the magnitude and its uncertainty from cm to arbs and record these values in the template. Part C: Component Method In this part of the lab, you will calculate the vector sum of the same three vectors ( 𝐴𝐴 ⃗ , 𝐵𝐵 �⃗ , and 𝐶𝐶 ⃗ ). This step is purely mathematical: there are no measurements. NOTE : For this part, you should assume all of the given vectors are exact. However, when you report your final answers for 𝑅𝑅 �⃗ , you may round to three sig figs. 1. Draw individual triangles roughly showing each vector with its vector components. Then, using the methods described in class, calculate the 𝑥𝑥 - and 𝑦𝑦 - components of each vector ( 𝐴𝐴 𝑥𝑥 , 𝐴𝐴 𝑦𝑦 , 𝐵𝐵 𝑥𝑥 , 𝐵𝐵 𝑦𝑦 , etc.). Note that these components may be positive or negative, depending on the direction. 2. Add the three 𝑥𝑥 -components to find the 𝑥𝑥 -component of the resultant vector, 𝑅𝑅 𝑥𝑥 . 3. Add the three 𝑦𝑦 -components to find the 𝑦𝑦 -component of the resultant vector, 𝑅𝑅 𝑦𝑦 . 4. Draw a rough triangle showing the components 𝑅𝑅 𝑥𝑥 and 𝑅𝑅 𝑦𝑦 , and the resultant vector 𝑅𝑅 �⃗ . Use this triangle to calculate the magnitude and direction of the resultant vector 𝑅𝑅 �⃗ , to the nearest 0.01 arb or degree. Analysis 1. Calculate the percent error in the magnitude of the result for Part A, assuming that the result of Part C is the correct value. % Error = |measured − actual| actual × 100% 2. Calculate the percent error in the magnitude of the result for Part B, assuming that the result of Part C is the correct value. 3. Summarize your results by filling out the table with your results from each of the three parts, including uncertainties for Parts A and B, along with the percent difference and percent error you just calculated. Conclusions