Partner 1: Partner 2: Section: PLEASE NOTE: You will need this particular lab report later in the semester again for the homework of the Rolling Motion Experiment. When you get back this graded report, keep it and only return it to your TA after the mentioned homework is submitted. Purpose: This lab will show you how to obtain a scientific result. You will measure the motion of an object traveling freely through space, and from these measurements assert that a particular mathematical equation describes the motion to a certain degree of accuracy and precision. To do this you will need to obtain and analyze your data in a way that you understand and can explain. From the perspective of measurement and data analysis, this laboratory experience is the basis upon which all remaining laboratory activities are built. Paying special attention to the measurement and analysis details of this lab will help you in future lab activities, as well as in other laboratories that you may encounter in your professional career. Materials: Mini-launcher with photogate attachment 16 mm steel ball Stand with O-ring attachment Projectile stopping device Computer Microsoft Excel for data plotting WARNING! The mini-launchers used in the following activity can be DANGEROUS if used inappropriately. When at a table with a mini-launcher, you must wear protective eyewear! Do not aim launcher at computers, windows, lab partners or rival lab groups. At the instructor s discretion, you will be given a zero for this activity if your lab group misuses the mini-launcher. University of Utah Department of Physics & Astronomy 43
Introduction: From your first few weeks of physics, you learned that the position of a projectile can be easily found. The first step is to separate the motion carefully into its x and y components. (We assume the projectile cuts a path in a two-dimensional space -- otherwise we would also consider a z-component of the motion.) See Figure 1. Figure 1 Projectile path when fired from the mini-launcher Q1. Although it is not explicitly indicated, the origin of an x, y coordinate system is implied by the drawing. Where is it? Q2. In the space below write down (do not derive) the kinematic equations that give x and y as function of time, i.e., x(t) and y(t). These equations should include x o, y o, v ox, v oy a x, a y and t. You should be able to state the meaning of each one of these, and in the case of x o, a x and a y, give explicit numerical values. University of Utah Department of Physics & Astronomy 44
Procedure: Activity 1: The Determination of Velocity, The objective of the first part of this experiment is to obtain the value of, the launch velocity of the steel ball projectile at the moment it has just separated from the firing spring. Q3. Derive a formula that will allow you to calculate from the quantities measured and one or more known constants. You will need to use the equations you wrote down for Q2. This derivation is part of the homework and the result should be: 2 Q4. In the space provided below, describe in words the measurements you will need to make in order to calculate. University of Utah Department of Physics & Astronomy 45
Some Experimental Issues to Consider: A. It would be useful to fire the steel ball numerous times in order to produce an average x distance,. B. Be sure to include all uncertainties: is the uncertainty in, is the uncertainty in, and is the uncertainty in the launch angle. Since you will measure at multiple times from the various launches, the uncertainty of this quantity should be calculated from the STANDARD DEVIATION OF THE MEAN. In contrast, you will only measure the launch angle and y once. Therefore, you can just make a good estimate of the uncertainty of that measurement. C. The calculation of should, in principle, be accompanied by a calculation for the uncertainty in. However, that requires differentiation with respect to the three measured quantities, and it becomes quite tedious very quickly. In real life, one would try to use a program like Maple or Mathematica to do these calculations reliably and efficiently. Instead of doing the entire error calculation, make the (potentially wrong) assumption that the error in the quantity with the highest % error matters the most and that the other errors can be neglected. This will simplify the error calculation somewhat, but it will still be a bit tedious. Give it a try! D. To make life a bit simpler, set the launch angle as close to 30 as you can and use the intermediate launch position, i.e., push the projectile into the launcher until you hear the second click. E. In the space below, build a table that contains all your measurements and their uncertainties as well as all the results of the calculations. University of Utah Department of Physics & Astronomy 46
Activity 2: Experimental Challenge. (Part of your grade will depend upon how successful you and your partner are.) Place the launcher at the end of the table so that it will fire the projectile away from the table. Set the launch angle to 20 (as close as you can). Do the following by prediction based upon calculation, and NOT by trial and error. Use the stand with the large O ring held by the clamp. Set the position of the clamp so that the center of the O ring is 30 cm vertically above the floor (or as close as you can make it). Q5. Now calculate where you have to place this stand on the floor so that the launched projectile will pass through the center of the O-ring on its way to the floor. Show all this work below. When you are confident you have done the calculation correctly, call your TA. Set the stand at the calculated position and fire the projectile. You might try a couple of firings. Your TA will first check to see if the calculated position is where you set your stand and then how close you came to accomplishing the task. Distance Check Successful (TA Initials) Unsuccessful How far off? University of Utah Department of Physics & Astronomy 47
Activity 3: Determination of the Acceleration Due to Gravity, g Exit Capstone, and open an Excel Spreadsheet. Excel, has data fields that allow you to enter data. In this part of the lab, we will plot the positions (x and y coordinates) of the steel ball. Label two adjacent fields in Excel with x and y. Underneath x and y you will enter the measured values. Make sure your ball launcher is level with the horizontal (at 0 o ), and make sure the ball clicks two times, like the earlier experiment. With your meter stick and the O-Ring apparatus (as pictured in Figure 2), you will be able to record x and y values. The O-Ring stand can be moved around the floor, and the clamp that holds the O-Ring on can me moved up and down on the metal stand. You will move the O-ring stand to at least 5 different positions on the floor where the projectile goes through the center of the ring, where you can then measure the x (distance from the table) and the y (distance the center of the ring is above the ground) values at those points. Figure 2 Projectile motion tracked by the O-ring apparatus University of Utah Department of Physics & Astronomy 48
As soon as you have entered all x- and corresponding y-values you need to graph y versus x in Excel in a Scatter-plot. (Short hint: In Excel select all x and y data by clicking and dragging. Then click on Insert -> Scatter and select the scatter plot with only Markers.) Please refer to the Excel Tutorial at the beginning of this manual on how to do this if you are not familiar with this Excel feature. The points should look like a path similar to that of the projectile. (Check your work if this is not the case). You will also need to show uncertainty on your graph in the form of error bars. Again, adding error bars is described in the Excel Tutorial. (Short Hint: Left-click once on any of the data points in the graph to select them all. Then click on Layout underneath the Chart Tools. Towards the right side you will see Error bars. Click on Error bars and select More error bars options. There you can enter a fixed error for horizontal/vertical errors. To switch between horizontal and vertical error, you can select x-error bars or y-error bars in the upper left region of your spreadsheet.) On the next page, derive the equation of motion for y in terms of the x position using the kinematic equations (Hint: The kinematic equations should be in your answer to Question 2). Note that your equations will be much easier to use than they were on your homework assignment because the angle from which the projectile is being launched is 0 o, which should cause parts of your equation to cancel out. Show you work on the next page under Q6. University of Utah Department of Physics & Astronomy 49
Q6. In the space below, derive the equation for y as a function of the variable x. When you finish this derivation, the formula should have the form:, where A and B are constants Q7. What kind of curve has this form? What would the curve look like if you plotted y versus x 2? Now plot the curve with x 2 on the horizontal axis. You can use Excel to create this plot To create this graph in Excel, insert a new column between the x and y column (Click on the top column letter of the column in which your y-data are. Then go to the Insert Cells button and select Insert Sheet Columns ). Add a label called x^2 and calculate x 2 in that column from the x-data to its left. (Hint: This can be done easily and efficiently with the spreadsheet with minimal effort. Look at the Excel Tutorial if you do not know how to do that or have someone show you.) University of Utah Department of Physics & Astronomy 50
You will now plot y versus x 2 by selecting the y and x^2 data and inserting another scatter-plot. Now, your last task will be to determine the slope, which will give you a value that allows you to calculate the gravitational constant g. Excel can determine the slope of a graph using the trendline (linear fit) see the Excel tutorial on how to do that if you do not know. You will also need to find a way to estimate the uncertainty in the slope of this graph (Hint: Look at the scatter of your data and estimate by how much the slope of the trendline could vary and still reasonably fit our data. Alternatively, you can get a the uncertainy in the slope from Excel using the Linest(.) command. This exact procedure is described in the Excel tutorial.). Use the obtained slope and its uncertainty to calculate the gravitational constant g with uncertainty. Show your work in the space given to you below. Q8. You now have all you need to obtain a value of g. In the space below, indicate how to get g from the linear plot of y versus x 2 and the do the calculation. How accurate is the value of g? How precise is it? University of Utah Department of Physics & Astronomy 51
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