Error Propagation Exponential Fit
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How to calculate uncertainties of a natural exponential function? up vote 2 down vote favorite (I apologize if this should be posted in mathematics, however I chose to post it here as it's technically about physics) I conducted an experiment in which position of items were shifted on an object, either on the ends of wings of it, or on the base (I'd rather not get too much into what it's exponential standard deviation about), and the effect on its fall rate over a certain distance was measured. The result was a decay model of the form: $T(N)=Ae^{-bN}+c$, where $A=1.44,b=0.132,c=0.303$ and $T =$ Time,$N =$ Number of items added to wings. However, for each of the times there is an uncertainty of between 0.08 and 0.09 seconds. So, I asked my teacher for assistance and he explained the following: First you remove the 0.303, and then you can rearrange it as follows: $T = 1.44*e^{-0.132N}$ $\ln{T} = \ln(1.44*e^{-0.132N})$ $\ln{T} = \ln{1.44} + \ln{e^{-0.132N}}$ $\ln{T} = 0.365 + -0.132N$ $\ln{T} = -0.132N + 0.365$ And thus you have a linear equation. Then I calculated $\ln{T}$ and $-0.132N + 0.365$ for each value of N, and graphed it in a graphic software, and made error bars of $±((\ln(T+\delta T)-\ln{(T-\delta T))/2})$, and thereby can get a best-fit gradient, a maximum possible gradient, and a minimum possible gradient, all in terms of $(ln(T)/(-0.132N + 0.365))$ if I'm not mistaken. But now for the questions: Why could the (+0.303) simply be removed, and how can that be justified? What do I do with my newly acquired values for the max. and min. gradients? I'd truly appreciate any help on this! homework-and-exercises measurement error-analysis share|cite|improve this question edited Jan 8
Origin • Power Laws • Dimensional Analysis Fitting Data A common and powerful way to compare data to a theory is to search for a theoretical curve that matches the data as closely as possible. You may suspect, for example, that friction causes a uniform deceleration exponential regression analysis excel of a spinning disk, so you have gathered data for the angular velocity of the
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disk as a function of time. If your hypothesis is correct, then these data should lie approximately on a straight line when
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angular velocity is plotted as a function of time. They won't be exactly on the line because your experimental observations are inevitably uncertain to some degree. They might look like the data shown in the figure below. Our http://physics.stackexchange.com/questions/48629/how-to-calculate-uncertainties-of-a-natural-exponential-function task is to find the best line that goes through these data. When we have found it, we would like answers to the following questions: What is the best estimate of the deceleration caused by friction? That is, what is the slope of the line. What is the uncertainty in the value of deceleration? What is the likelihood that these data are in fact consistent with our hypothesis? That is, how probable is it that http://www.physics.hmc.edu/analysis/fitting.php the disk is uniformly accelerated? What do you mean, "best line"? Associated with each data point is an error bar, which is the graphical representation of the uncertainty of the measured value. We assume that the errors are normally distributed, which means that they are described by the bell-shaped curve or Gaussian shown in the discussion of standard deviation. The height between the data point and the top or bottom of the error bar is σ, so about 2/3 of the time, the line or curve should pass within one error bar of the data point. Sometimes the uncertainty of each data point is the same, but it is just as likely (if not more likely!) that the uncertainty varies from datum to datum. In that case the line should pay more attention to the points that have smaller uncertainty. That is, it should try to get close to those "more certain" points. When it can't, we should grow worried that the data and the line (or curve) fundamentally don't agree. A pretty good way to fit straight lines to plotted data is to fiddle with a ruler, doing your best to get the line to pass close to as many data points as possible, taking care to count more heavily the points with smaller uncertainty. This method is quick and intuitive, a
uncertainty of an answer obtained from a calculation. Every time data are measured, there is an uncertainty associated with that measurement. (Refer to guide to Measurement and http://physics.appstate.edu/undergraduate-programs/laboratory/resources/error-propagation Uncertainty.) If these measurements used in your calculation have some uncertainty associated with them, then the final answer will, of course, have some level of uncertainty. For instance, in lab you might measure an object's position at different times in order to find the object's average velocity. Since both distance and time measurements have uncertainties associated with them, those uncertainties error propagation follow the numbers throughout the calculations and eventually affect your final answer for the velocity of that object. How would you determine the uncertainty in your calculated values? In lab, graphs are often used where LoggerPro software calculates uncertainties in slope and intercept values for you. In other classes, like chemistry, there are particular ways to calculate uncertainties. In fact, since error propagation for uncertainty calculations are based on statistics, there are as many different ways to determine uncertainties as there are statistical methods. The error propagation methods presented in this guide are a set of general rules that will be consistently used for all levels of physics classes in this department. In the following examples: q is the result of a mathematical operation δ is the uncertainty associated with a measurement. For example, if you have a measurement that looks like this: m = 20.4 kg ±0.2 kg Thenq = 20.4 kg and δm = 0.2 kg First Step: Make sure that your units are consistent Make sure that you are using SI units and that they are consistent. If you are converting between unit systems, then you are probably multiplying your value by a constant. Please see the following rule on how to use constants. Logger Pro If you are using a curve fit generated by Logger Pro, please use the uncertainty associated with the parameters that Logger Pro give you. In the above linear fit, m = 0.9000 andδm = 0.0
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