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Linear Kinematics Part 1: Back and Forth Motion https://www.youtube.com/watch?v=QzWbwC7MkuY Velocity is defined at the first derivate of position with respect to time, here for the x-direction Circularly, we can retrieve the change in position from the velocity by integration. Equation (1) As acceleration is the first derivative of velocity with respect to time, here for the x-direction we retrieve the change in velocity through integration. Equation (2) When comparing two experimental values that are measure via different means, the Percent Difference can show you how close the values are… Equation (3) In our first experiment we will confirm the integral relationship between velocity and position, as well as between acceleration and velocity. Procedure https://www.youtube.com/watch?v=QzWbwC7MkuY 1. Plug dongle into computer and turn the device on. Make sure the device is oriented with the wheels pointed down. 2. Select the Wheel sensor so that three graphs, one being Wheel-Postion, next being the Wheel- Velocity, and finally Wheel-Acceleration, are present. Start collecting data. 3. Hit the iOLab device between your hands multiple times so that it rolls back and forth, quickly changing direction. Do your best to keep the motion only in the y-hat direction. Stop the data collection. 4. Zoom in on a piece of the graph that shows the motion of 2 hits (one back and forth). 5. By highlighting the proper section, find the slope of one section of the displacement plot and the average velocity during the same time interval. Remember that the slope is the value indicated by the value following the “s” and the average is the value indicated by the value following the “μ” 6. Find the area under one section of a velocity square wave. This is indicated by the value following the “a”. This video should help, https://www.youtube.com/watch?v=fLNaK4CXRtI 7. Find the starting and ending position of the wheel during this interval as well (just by using the cursor to find the values at the starting and ending points of the highlighted section). 8. Find the area under one acceleration spike. Find the average velocity both before and after this spike and calculate the difference. 9. Compare the slope of the displacement curve to the corresponding average velocity value. 10. Compare the change in position to the area under the velocity curve for the same time interval as in Equation 1. Calculate the percent difference, i.e. 11. Compare the change in velocity to the area under the acceleration curve as in Equation 2. Calculate the percent difference. For your report: Answer questions #9, 10, and 11 and screen capture your version of Figure 2. Compare your results to #9-11 with your lab partners and comment on the percent differences between group members in your report. https://www.youtube.com/watch?v=fLNaK4CXRtI Figure 1. Figure 2. Part 2: Acceleration Due to Gravity and Inclined Planes Figure 3. DIY Ramp Setup You will learn in later sections that along the surface of an incline, Equation (4) where g=9.8 m/s2. When comparing an experimentally derived value to a know Equation (5) For this part of the experiment you will estimate the acceleration due to gravity. By rolling your mass up and down an incline you construct at home you can calculate the acceleration along the surface of the ramp. By estimating the angle made by your ramp you can calculate an experimental measurement of the acceleration due to gravity. You will calculate the percent error for your experiment and compare your results with your lab partners. 1. Construct a ramp at home. The larger the better, however a simple ramp using the iOLab box and a hardcover book should allow for enough data to be taken. Each member of your group will construct a ramp of their own. 2. Measure the length L and height H of your ramp. Through trigonometry, calculate the angle your ramp makes with the ground by, You will use this angle later in the experiment. *Don’t have a ruler or tape measure? Use anything! The only constraint is that H and L need to be measured in the same units, e.g. “My ramp is approximately 3.5 phones long and 1 phone high.” 3. Select the Wheel sensor so that three graphs, one being Wheel-Postion, next being the Wheel- Velocity, and finally Wheel-Acceleration, are present. Start collecting data. 4. With your iOLab (wheels down) on the ramp (not starting on the floor), push the device up the ramp with one quick push or tap. Repeat your experiment until you get data that looks similar to below. A good cue being a velocity graph that has a linear component, and your acceleration has a flat horizontal component that is non-zero. Figure 4. 5. In order to measure the acceleration along the incline, you first need to note exactly where on your graphs the mass was moving along the incline after your push and before your catch. When you interact with the mass you affect the acceleration of the system. Our goal today is to isolate where gravity is dictating the motion. Zoom into your graph as in Figure 5 and highlight the area on one of your graphs that fit this criteria, and do not be afraid to be conservative. Note that if acceleration is constant, your velocity should have a linear slope, and your position should look quadratic. 6. On your Wheel-Acceleration graph, the μ value indicates the average value of acceleration on that interval, write this down as ax,experimental. Figure 5. 7. Using the angle measured in #2 calculate your experimental acceleration from Equation (4) as follows 8. Calculate the percent error Equation (5), where gknown=9.8 m/s2. 9. Compare your results with your lab partners. What range of values did you measure (the different between the highest and lowest values). Calculate the average gexperimental value with everyone in your group. Using this value, calculate the percent error again. Do you have better agreement with gknown? For your report: Include your results for #2, 7, 8, and 9 as well as a screen grab of your version of Figure 5.