Physics 191 Free Fall 2016-09-21 1 Introduction 2 2 Experimental Procedure 2 3 Homework Questions - Hand in before class! 3 4 Data Analysis 3 4.1 Prepare the data in Excel.................................. 3 4.2 Data Analysis in Kaleidagraph............................... 4 4.2.1 Quality checking of data............................... 4 5 Results and Discussion 4 1
1 Introduction An object falling freely near the surface of the Earth experiences a constant downward acceleration caused by the pull of the Earth s gravity, g. If we choose the upward direction as positive, the sign of the body s acceleration is negative, a = g. Given the acceleration a(t) is given, how do we find the velocity v(t) and the distance y(t) that the body has traveled in a time t? To derive the equations of motion we apply integral calculus. Thus, choosing the direction of motion along the y-axis only, we can write a(t) = dv y dt = d2 y = g. (1) dt2 We integrate this equation with respect to time to get the instantaneous downward velocity v(t): v v 0 t dv = g dt 0 v(t) = v 0 gt (2) where v 0 is the velocity at time t = 0. Since v(t) = dy/dt, we can integrate Eq. (2) once more to find the distance that the object has fallen in a time t: y(t) = y 0 + v 0 t 1 2 gt2 (3) where y 0 is the object s position at time t = 0. We could have arrived at the same expressions, if we just substituted a = g (and y = x) in the general equations of linear motion: v(t) = v 0 + at x(t) = x 0 + v 0 t + 1 2 at2. 2 Experimental Procedure You will measure g with the Behr free fall apparatus, which records the position of a cylinder at regularly spaced times as it drops through about 1.5 m. The apparatus is depicted in Fig. 1. You will measure g by fitting the kinematical equations (2) and (3) to your data using the least-squares technique. This is a rather precise experiment: measure carefully and you should measure g to within 1% or so. However, at this level of accuracy, the possible systematic errors can be subtle. Your instructor will demonstrate how to operate the apparatus. Before measurement, the cylinder is suspended at the top of the stand with an electromagnet. When the electromagnet is turned off, the cylinder begins to fall. Simultaneously, the spark timer starts to send high-voltage pulses between two wires. As the cylinder falls, it closes the gap between the two wires and a spark will jump from one wire to the other at the point where the cylinder passes. At the time of each pulse a spark goes through the wires and the cylinder, leaving a mark on the paper tape. The time interval t between the two adjacent sparks is 1/60th of a second (to high accuracy we have measured it to check). The appearance of the beginning portion of such a tape is indicated in Fig. 2, with time increasing to the right. You should have about 30 burn marks on your paper tape. If you cannot clearly identify 30 burn marks, you should redo the measurement. 2
Figure 1: The Behr free fall apparatus used in the lab. The picture on the right indicates where the cylinder will fall between the wires. Sparks will jump across the conducting ring around the cylinder at a rate of 60 Hz leaving marks on the carbon paper. Figure 2: This is roughly how the tape will look like. The markings will be rather faint, so be careful with the tape. 3 Homework Questions - Hand in before class! 1. How does v depend on t? Draw a diagram. 2. How does y depend on t? Draw a diagram. 3. What kind of systematic error might influence your experiment? 4. Explain why a 1/60 s time interval between sparks (and thus between markings) is suitable for this experiment. (Consider what would happen if you had a very short or very long interval.) 4 4.1 Data Analysis Prepare the data in Excel Create a table in Excel with your data. Fill in the following values: 1. Use a meter stick to measure the positions of the burn marks on the paper. Make sure to only include burn marks that were created while the cylinder was in free fall. 3
2. Assign uncertainties to the measurements, stating in your report how you arrived at these values. These uncertainties represent how much you trust your measurement. 3. Calculate the distances y i between marks. 4. Calculate the time at each interval t i = i t. 5. Compute the velocity v i and its uncertainty for each interval (you can assume that the uncertainty in the time is negligible: δt = 0). Save your file using the Excel 2003 format and close it; Kaleidagraph will not open the file if Excel still has it open. 4.2 Data Analysis in Kaleidagraph Open your Excel spreadsheet in Kaleidagraph. Remember that whenever you plot data that has an uncertainty, you have to include the errorbars! 1. Make a graph of y(t) vs. t by plotting y i vs. t i. 2. Fit an appropriate function to the graph. 3. Extract a value for g from the y(t) fit. 4. Make a graph of v(t) vs. t by plotting v i vs. t i. 5. Fit an appropriate function to the graph. 6. Extract a value for g from the v(t) fit. For Kaleidagraph to calculate the uncertainties in the curve fit parameters, you must create a general curve fit with an appropriate user-defined function. Check the reference guide for general guidelines on what should be on a plot. Remember to give all results with their uncertainties in your report; extract uncertainties of g from the fit uncertainties. 4.2.1 Quality checking of data Your data should be smooth: a point obviously high or low may well be a measurement error, or indicate a missing spark or random scratch mark. Similarly, points alternately above and below the trend may have recording problems or precision problems in your assignment of the time of the spark. Consult with your instructor if you see such anomalies. You may want to analyze a different set of data if the problems are pervasive or too hard to eliminate. If you decide to drop some points, print out your plots with and without the points dropped, circle the points you dropped, and explain why you dropped them. 5 Results and Discussion Discuss the following questions and summarize your results in a table. 1. Does your straight line fit for v vs. t pass within all error bars? The error bars are likely too small to inspect visually, and determining this visually is tricky and unreliable in any case. Instead, to examine the question of whether the data goes through your error bars in a quantitative fashion, look at the normalized residuals: that is, the residuals (= data - fit value) divided by the uncertainties z i = v i f i δv i 4
where f i is the fit value for point i. If more than 32% of your data points lie outside the uncertainties, i.e. z i > 1, suggest reasons why this may be: inappropriate error bars, badly measured data points, systematic errors; or (probably not here!) the theory may be incorrect. 2. What is the y-axis intercept value from your v vs. t graph, and what does it mean? 3. Calculate y 0 and v 0 and their uncertainties from your y vs. t plot, and v 0 from the v vs. t plot. 4. Are the values of v 0 from the two fits compatible? Should they be? Why? 5. Compare your values of g and their uncertainties from both fits. Do they agree within uncertainties? 6. For both values of g, record the fractional uncertainty δg/g of g. 7. Check the compatibility of your measurement with the accepted value of 9.804 m/s 2, that is calculate the t value and discuss the result. See Reference Guide for how to calculate and interpret t. 5