To determine the 2 probability and the fitting parameters for a straight line fit, using the method of least squares. Premise: One way to reduce the uncertainty in a measurement as described in the previous experiment, was to take a number of readings of a single variable x, N times, and then determine the mean value with the standard deviation describing the uncertainty in the measurement. However, any experiment is the study of the correlation between the independent variables and the dependent variables. The dependent variables are response of the experiment over that range of predetermined independent variables (domain). The aim is to experimentally determine the relation between y and x and the mathematical nature of y=f(x). In this experiment we will assume that the mathematical nature of y=f(x) is a straight line represented as =+ In other words, we will fit this function to describe a phenomenon that exhibits a linear response. The inherent uncertainty in any measurement can be represented on a graph by a vertical error bar. Assuming that the fluctuations of individual measurements are the experimental uncertainty only, and not operator errors, we will experimentally obtain an entire set of data (,). This represents a scatter plot about some mean with the probability that any individual measurement will be about 1 from it's mean-value at that particular point. Assuming that the experiment follows Gaussian distribution, this 1 denotes a probability of ~68%. Since both (independent variable) and (dependent variable) are being measured, there is an uncertainty in both the measurements. For this experiment, we will assume that the uncertainty/error in x (denoted by the horizontal error bar), is much smaller than the uncertainty in y (denoted by the vertical error bar. Hence, in the first order approximation we will neglect the uncertainty in x.

Method of least squares: In an experiment where the dependent variable is related to the independent variable linearly by the relationship y(x) = a+ bx The aim is to fit a straight line to this scatter plot of (x,, );), i.e., to find parameters a and b in a way that minimizes the discrepancy between measured values of y, and the calculated values of y(x) For any arbitrary values of a and b, we calculate the deviations Ayi = Vi - y(x) = Vi - a-bx; We also define a "goodness of fit parameter" x2, such that x2 = Vi - y(x;)]2 Oi [(); - a - bx;)12The aim is to find the coefficients a and b to minimize the value of x2. Hence, we take the partial derivatives of x with respect to a and b and equate them to zero ax2 = 0 da And ax2 = 0 ab And from these equations, find the values of a and b ax2 da [( - a - bx)] - -2[-[(: - a -bx,)] = 0 And ax2 a 5 -[ (): - a - bx;)12 ab ab Oi = -2> ~[(); - a - bx;)] = 0 Oi[L. - a - bx;] = 0 _x,ly: - a - bx;] = 0 Or [ v. = na + b [x [ vex. = a [ x. + b [x? We solve the above two equations simultaneously to obtain a and b that minimize the goodness of fit parameter x . Experiment instructions: Given below is a dataset that a student obtained while studying Hooke's law. She suspended masses on a spring and measured the extension as a function of the applied force in order to find the spring constant " . Her measurements are Mass (kg) 200 300 400 500 600 700 800 900 Extension 5.1 5.5 5.9 6.8 7.4 7.5 8.6 9.4 (cm)There is an uncertainty of 0.2 in each measurement of the extension. The uncertainty in the masses is negligible. For a perfect spring, the extension ALof the spring is related to the applied force by the relation F = KAL Where for this problem, F = mg, and AL = L - Lo, and Lo is the unstretched length of the spring. Use this data and the method of least squares to find a) The spring constant k b) The unstretched length of the spring Lo c) And their uncertainties (a and b constants) d) Find x2 e ) Comment upon the limitations of this method. Show your work, no points will be awarded without showing work