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1.1: Background Material

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  • Average Position

    In this lab we will be doing several trials that all produce dots on a piece of paper that measure the position of a marble as it strikes the floor. These dots can all be thought of as existing at the heads of position vectors, \(\overrightarrow r_1\), \(\overrightarrow r_2\), \(\overrightarrow r_3\), and so on. As is typically the case for experiments, we will be interested in an average quantity over many trials – in this case the average position at which the marble lands. Finding an average vector is no different from finding an average number, namely:

    \[\text{average position} = \left<\overrightarrow r\right> = \dfrac{\overrightarrow r_1 + \overrightarrow r_2 +\dots+\overrightarrow r_n}{n}\]

    We will not want to actually choose an origin and draw all these vectors, so it is helpful to come up with some way to find an average position directly from the positions of the dots. It isn't hard to show that the average position for two trials is just the point that is located halfway between the positions of the two trials.

    Figure 1.1.1 – Average of Two Position Vectors


    Finding the average position of more than two dots does not have quite as simple of a method, but we will use a trick to keep things from getting too complicated. Notice that if we have four dots, we can write the average position vector this way:

    \[\left<\overrightarrow r\right> = \dfrac{\overrightarrow r_1 + \overrightarrow r_2 +\overrightarrow r_3 + \overrightarrow r_4}{4} = \dfrac{\dfrac{\overrightarrow r_1 + \overrightarrow r_2}{2} +\dfrac{\overrightarrow r_3 + \overrightarrow r_4}{2}}{2}\]

    This shows that we can get the average position of four dots by first finding the average positions of two pairs of dots, and then finding the average of those averages. This allows us to just use a ruler to locate the halfway points between pairs of dots to find the average position of all the dots. Notice that this procedure requires that we have some power-of-2 number of dots (2, 4, 8, 16, etc.). We could do it for a different number, but then we lose the "halfway between points" simplicity, and since we have control over the number of trials, we will stick with this method.

    It should also be noted that it doesn't matter how we pair-off the points – in the end we end up with the same average position:

    \[\left<\overrightarrow r\right> = \dfrac{\dfrac{\overrightarrow r_1 + \overrightarrow r_2}{2} +\dfrac{\overrightarrow r_3 + \overrightarrow r_4}{2}}{2} = \dfrac{\dfrac{\overrightarrow r_1 + \overrightarrow r_3}{2} +\dfrac{\overrightarrow r_2 + \overrightarrow r_4}{2}}{2}\]

    Figure 1.1.2 – Average Position of Four Dots Found Two Ways


    Estimated and Statistical Uncertainty

    When we perform an experiment, we are interested in more than just the average value we obtain from many trials, we want to know to what extent this average can be trusted. That is, we want to know how uncertain we are that what have measured can be applied to any conclusions we might wish to draw. In the experiment we will perform, we will be "aiming" the marbles at a particular point on the paper, and the scatter of the dots is a result of uncertainty in our aim. Let's consider two different methods of aiming.

    Suppose we drop the marble into a funnel, with a neck that has a diameter measurably larger than the diameter of the marble. When the marble emerges from the bottom of the funnel, it can come out in a spread of positions that is even a bit greater than the circular area of the funnel neck, since the marbles could come out with some component of horizontal velocity.

    Figure 1.1.3 – Spread of Dots from Marbles Dropped Through a Funnel


    With this method of dropping, we have some sense of what the uncertainty in our results will be before we even drop a single marble. We can use what we know about the funnel (i.e. the diameter of its neck) to estimate the uncertainty, and then use that estimate to see if our experimental results justify certain conclusions. We will use this method for many of our experiments throughout the quarter, because we often have a good handle on the degree to which we know our inputs.

    A second way of dropping the marbles (and the one we will be using for this experiment) is to simply release them directly from our hands. In this case, it is significantly harder to make an approximation of the uncertainty before we ever drop a marble. In this case, we can determine the uncertainty statistically. This consists of computing what is called the standard deviation, which goes as follows:

    • compute the average of all the data points

    \[\left<x\right> = \dfrac{x_1 + x_2 + \dots + x_n}{n}\]

    • compute how far each data point deviates from the average

    \[\Delta x_1 = x_1-\left<x\right>,\;\Delta x_2 = x_2-\left<x\right>,\;\dots\]

    • square all the deviations from the average

    \[\Delta x_1^2,\;\Delta x_2^2,\;\dots\]

    • average the square deviations

    \[\left<\Delta x^2\right> = \dfrac{\Delta x_1^2 + \Delta x_2^2 + \dots + \Delta x_n^2}{n} \]

    • compute the square root of the average

    \[\sigma_x = \sqrt{\dfrac{\Delta x_1^2 + \Delta x_2^2 + \dots + \Delta x_n^2}{n}}\]

    This description of the computation of standard deviation makes it easy to remember, as we are just computing averages (first of the data points, then of the deviation of the data point values from the mean), but technically in these situations where we compute a mean from the actual data, there is a actually a slightly more accurate formula for standard deviation.  It involves dividing the sum of the square deviations by \(n-1\), rather than by \(n\).  We won't go into the technical details of why this is so, but it is important to note that the difference between these can become significant when \(n\) is quite small, as it often will be in our experiments.  We will therefore henceforth use the so-called "unbiased" version of the standard deviation:

    \[\sigma_x = \sqrt{\dfrac{\Delta x_1^2 + \Delta x_2^2 + \dots + \Delta x_n^2}{n-1}}\]

    The way this method of measuring uncertainty works for our present experiment should be clear: First use the method described above to determine the place on the paper that is the average landing point. Second, measure the distance from each dot to the average landing point.  This is the "deviation from the mean" (\(\Delta x\), measured in centimeters) of each data point. Then do the math from there.

    While many of the experiments we do this quarter will involve estimated uncertainties, whenever there is a human element involved, this statistical method is more effective. The most common instance of this comes whenever a human is starting and stopping a timing device. Without going into the details that you may have encountered in a statistics class (like the nuances of the central limit theorem and the assumption of a normal distribution for our measurements), we will say that the range of uncertainty is such that we will expect that roughly two-thirds of the data points will land within one standard deviation of the average. While this works out automatically when we do the uncertainty statistically, we will use this as the standard for making our uncertainty estimates when we use the estimation method.

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