Overview
The purpose of this lab is to determine the amount of uncertainty (\error") in a set of measurements, to propogate uncertainty to a calculation based on that measurement, and to determine if measurements are consistent with each other and with the expected value. 1 Set-up 1. Set up a pendulum and adjust its length until the period is roughly 1.5 s. First make the pendulum an arbitrary length. Estimate the period mentally; do not measure it with a stopwatch. Adjust the length until the period is roughly 1.5 s. Note that a period of 1.5 s means that it will return to its starting point twice in three seconds. 2. Measure the length of the pendulum string, excluding the bob, using the following method: Largest possible length: Lmax = Smallest possible length: Lmin = Best estimate of the length: L = Uncertainty in length ( LmaxLmin 2 ): L = 3. Report the length of the pendulum string in the format L L. Use this brief procedure whenever you are asked to report a value. Round the uncetainty (L) to one non-zero digit. Round the most likely value (L) to the same decimal place as the rounded uncertainty. 4. How many signi cant gures does your measurement of the length of the pendulum string have? 2 Measured Period Each person in your group will choose a di erent number of consecutive periods to count (N) and measure the time it takes to complete N periods. This measurement (t) will be repeated for a total of 5 trials. Dividing this time interval by N will yield the period (T = t N ). For example, if N = 2, then you would measure the time for the pendulum to complete 2 periods (t), meaning it will return to its starting point twice. This would be repeated 4 more times, each time measuring the time for 2 periods. Your measured values of t should be similar, but won't all be the same. Do not start the pedulum and the stopwatch at the same time. Rather, start the pendulum and let it swing. Start your stopwatch when you see the pendulum reach one end of its motion and stop it when it returns to this position N periods later. Do not stop the pendulum between each measurement; let it keep swinging. Everyone in your group can and should make measurements simultainesously as the pendulum swings. 1. Number of consecutive periods counted by you (1, 2, 5, 10, or 20): N = This will be di erent for each member of your group. 2. Measure the time for your number of periods (t) ve times and complete the table below. do not round anything. Record each measured time interval (t). Divide each measured time interval by your N to get the time for 1 period (T). Find the averge of all 5 periods (T). Find the absolute error in each measurement (T = T T), including the sign (+ or ). Find the mean error by adding the absolute value of each error and dividing the sum by N. Trial t (s) T (s) T (s) 1 2 3 4 5 Averages 3. Report the measured value of the period and its uncertainty in the format T T, rounded appropriately. 4. How many signi cant gures does your measurement of the period have? 5. What is the percent error in the precision of your measured period? Round the result to 2 digits. p = T Tmeas 100% 3 Expected Period An expected value may sometimes be referred to as a calculated value or a true value. I do not refer to values as \actual values" because it is ambiguous. Some people may interpret this as a measured value because it is the value that was actually measured. Other people may interpret this as an expected value because it is what the value actually should be. I always refer to values as being either measured or expected. 1. Calculate the expected value of the pendulum using the following equation. You will learn how to derive this equation near the end of the semester. Note that the equation depends on L and g, but not L or g. Use the length of the string, excluding the bob, for L and g = 9:80 0:03 m s 2 . Show your work. Do unit conversions as needed. Do not round your answer. T = 2 s L g 2. Because L and g are uncertain, T will also be uncertain. Estimate the uncertainty in the period (T) in the following way: (a) Largest possible length (L + L): Lmax = (b) Smallest possible length (L L): Lmin = (c) Largest possible value of g (gmax = g + g): gmax = (d) Smallest possible value of g (gmin = g g): gmin = (e) In order to get the largest possible expected period, should you use Lmax or Lmin ? (Circle one). (f) In order to get the largest possible expected period, should you use gmax or gmin ? (Circle one). (g) Calculate the largest possible expected period (Tmax). Show your work. Do not round your answer. (h) In order to get the smallest possible expected period, should you use Lmax or Lmin ? (Circle one). (i) In order to get the smallest possible expected period, should you use gmax or gmin ? (Circle one). (j) Calculate the smallest possible expected period (Tmin). Show your work. Do not round your answer. (k) What is uncertainty in the expected period? Do not round your answer. T = T = TmaxTmin 2 3. Report the expected period of the pendulum in the format T T. 4 Conclusions Here, you will compare your measured value with the expected value and with the measured values of everyone else in your group. 1. What is the percent error in the accuracy of your measured period? Round the result to 2 digits. This may be + or . Include the sign, even if it's +. a = TmeasTexp Texp 100% 2. Is your measured period accurate? This is not determined by the % error in accuracy; no matter how small a is, the measured value may still be inaccurate. The measured value is accurate if the measured value is consistent with the expected value, and values are considered consistent if their ranges overlap. 3. Record the reported measured period for each member of your group. Then, shade in the range from Tmin to Tmax, including yours and the expected value. Then, determine whether each value is consistent with your measured value by writing a Y or an N in the \Consistent?" column. Values are consistent if their ranges overlap. Initials T T (s) Shaded Range of Values Consistent? Yours 1.2 s 1.3 s 1.4 s 1.5 s 1.6 s 1.7 s 1.8 s 1.2 s 1.3 s 1.4 s 1.5 s 1.6 s 1.7 s 1.8 s 1.2 s 1.3 s 1.4 s 1.5 s 1.6 s 1.7 s 1.8 s 1.2 s 1.3 s 1.4 s 1.5 s 1.6 s 1.7 s 1.8 s 1.2 s 1.3 s 1.4 s 1.5 s 1.6 s 1.7 s 1.8 s Expected 1.2 s 1.3 s 1.4 s 1.5 s 1.6 s 1.7 s 1.8 s 4. Which of your group members' measurements was most precise? Explain brielfy. 5. Which of your group members' measurements was most accurate? Explain brielf