Lab 2: Pendulum I Conceptual Focus: Uncertainty All measurements have an inherent amount of uncertainty; no measurement is perfect. The existence of some uncertainty is not a sign that the

experiment is flawed or that someone made a measurement error. Rather, uncertainty is a fundamental part of experimental data. That said, there are legitimate ways to reduce uncertainty, and we will see that this is an essential part of conducting a successful experiment. Background Uncertainty in Repeated Measurements Recall from the previous lab that uncertainty is an estimated quantity describing how sure you are in the value of a given measurement. When you make a single measurement, the quantifiable uncertainty related to that measurement comes from the resolution of the measurement device. This, however, is not the only possible source of uncertainty. When you repeat measurements more than once, you can see the spread of the data and identify if it is greater than the sum of your identifiable uncertainties (hint: it usually is). For example, suppose you measure a person's height seven times and get the following values: 1.83m, 1.72m, 1.88m, 1.76m, 1.92m, 1.85m, and 2.40m. Height is a two-point measurement, so we can assume that the resolution of our measurement tool was 0.01 m (1 cm). Just by looking at these values, we can see that they vary far more than the measurement resolution! Evidently, there are other uncertainties present in the measurement process. What contributes to uncertainty? Recall how we explored different methods of measurement in lab last week. There are many potential sources of uncertainty in a measurement. One primary source of uncertainty is the resolution of a measurement device. Another source is error induced by the method of measurement. In the single-point measurement described above in Figures 1 and 2, the uncertainty was estimated based on the resolution of the ruler. However, in experimental measurements, there are more sources of uncertainty than just the resolution of the measurement device. For example, suppose you are trying to measure the time it takes for a mass on a string to swing back and forth on a fulcrum point (i.e. the period of a pendulum). Figure 3: Period of a simple pendulum If you are using a stopwatch, it likely has a resolution of 0.01 s. But for each measurement, it is highly unlikely that you can start the watch at the exact moment that you release the mass and stop it exactly when the mass swings back to the starting position. Your brain must first notice that the mass was released and then tell your fingers to start the watch. This “signal travel” time is called reaction time, and it is often longer than the resolution of the stopwatch. Given that, it would be unreasonable to state Lab 2: Pendulum I that the resolution is the only source of uncertainty. Therefore, it is impossible to estimate the uncertainty in a measurement from just one or two readings. Only by looking at the spread of values over many measurements can you account for the various known and unknown sources of uncertainty in an experiment. Average and Standard Deviation of Repeated Measurements You can make repeated measurements and calculate uncertainty using a formula. When you make repeated measurements, the average (x) of the readings is considered the measured value. The average is calculated as the sum of the collected measurements divided by the number of measurements (Equation 2). x = 0 = Σῇ xi n Equation 2: Average of a measurement, where i is incremental counter, and n is the total number of measurements The standard deviation is a metric describing the spread of values in a set. This is a useful statistical tool for repeated measurements, because as we've already discussed, repeated measurements inherently vary. Mathematically, we define the standard deviation o for a set of n values as: |Σ{(x − x)2 n- 1 Equation 3: Standard deviation, where xi is each measurement, is the average measurement, and n is the total number of measurements You will likely encounter two formulas for standard deviation. The one shown above is called the sample standard deviation which has n 1 in the denominator. This differs from the population standard deviation which has only the total number of measurements n. For this class, we will always use the sample standard deviation. While they are almost identical, especially for large data sets, the sample standard deviation is a more conservative estimate. Keep in mind that the sum in the numerator of Equation 3 includes every measurement in the set of data you want to analyze. When you have a large set of data, doing the calculation by hand can be tedious, so use your tools wisely! Excel is an excellent tool for data analysis, and has a built in function to calculate the standard deviation =STDEV.S() — the “.S” at the end stands for "sample". Do not use the formula =STDEV.P(), which will calculate the population standard deviation. Once the standard deviation has been calculated, it can then be used to calculate the uncertainty in the mean (8x). The uncertainty in the mean is defined as: 8x = 0 √n Equation 4: Uncertainty in the mean, where o is the standard deviation, and n is the total number of measurements To use Excel to calculate the uncertainty in the mean, you can use =STDEV.S()/SQRT(n). Outliers It's expected that repeated measurements will always have some spread, but sometimes you may take a measurement that seems to be wildly different from the general trend followed by the rest of the Lab 2: Pendulum I collected data. These points are called outliers, and generally indicate that something went awry with that particular trial. At the start of this lab guide, we described a data set of height measurements: 1.83m, 1.72m, 1.88m, 1.76m, 1.92m, 1.85m, and 2.40m. While the majority of the data points are fairly close to ~1.8m, one of the data points (2.4m) is very far from this - around six times further from 1.8 than the rest. This huge difference is an indication that something might be off with this measurement. Outliers require close examination before they are 'thrown out' of the data set (or before someone gets too excited, thinking they made some groundbreaking discovery!). The most common reason you will encounter outliers in this class is due to a minor measurement error. It is valuable to pay attention while collecting data - if you notice an error in the method (e.g. your thumb slipped and you failed to stop the stopwatch at the right time), then you can flag the data point as an outlier at that moment, and note the reason. Later, during data analysis, you can omit the outliers so that they are not skewing the trend or average value. If you notice outliers in your data after data collection, and no one can remember making a mistake during the experiment (i.e. the reason for the outlier is unknown), what then? This is where you may be most tempted to make up a reason for the outlier and cut it out of your data set. But how can you be sure the outlier is an invalid data point? This would not be an ethical course of action. In this case, you have two options: keep the outlier, or entirely retake the set of measurements. Make sure to retain any omitted data in your lab notebook (mark or separate these data to distinguish them from the rest of your data) and describe the reason behind your decision to exclude them. If you've chosen to omit an outlier, make sure to exclude it from any calculations as well. Lab 2: Pendulum I In-Lab Activities Measuring the Period of a Simple Pendulum In this lab, you will measure the period of oscillation of a simple pendulum and calculate the average and uncertainty for the data you collect. A simple pendulum is simply (pun intended) a mass suspended by a flexible string attached at a pivot point. When the mass is pulled to one side and released, it oscillates back and forth. The time to complete one full cycle is called the period. Figure 1 shows one complete cycle of a pendulum indicated by dashed arrows - the period will be the same no matter where you decide to start your timer, be it at an extreme angle, a zero angle, or an in-between angle. The important things are these: the pendulum must go through one complete cycle for you to accurately measure the period, and you must use the same timing strategy for every measurement you make. A) Maximum angle B) Zero angle C) In-between angle Figure 1: Different cycles of a pendulum ➤ This activity will begin with a full-class discussion. Take notes on the outcomes of this discussion, including the pendulum parameters (variables) that you can control. Following the class discussion, you and one other group will be assigned a variable to compare. Decide on the experimental values you will use. Note these values and any reasoning for why you chose them. As a group, determine how you are going to measure the period of a pendulum - that is, determine a procedure. Be sure to discuss: O Any forms of error you think this procedure may have - be more specific than "human error"! O Ways you may reduce this error Note that digital devices, like stopwatches, have an inherent resolution based on the smallest value reported. It is convention to cite the uncertainty as ±1 on the last decimal place. Thus, a value of 34.2 seconds would be reported as 34.2 ±.1 s. ➤ Using your procedure, take at least five (5) separate measurements for the period. ➤ Compare your readings and answer the following questions in your notebook: How do the differences in your measurements compare to the resolution of the stopwatch? Can you use resolution to estimate uncertainty if the spread of your values is greater than the resolution? Why? Lab 2: Pendulum I ➤ Calculate the average and uncertainty of your periods, and report your measured value in the format x = x + 8x. Clearly show and justify your calculations in your notebook. O Write your reported period on the board. Is this a "good" measurement? How do you know? Have a class discussion about your pendulum experiments. Answer the following questions: O Were the uncertainties for each group's period similar? Why might this be? Were the values for each group's period similar? Explain what any similarities or differences in these values imply about the parameters of the pendulum. O O