Acceleration “g” Due to Gravity – Method 2
Introduction Tonight we will measure the acceleration due to gravity again. This time however, we will collect more data and the analysis will be different. We will first fit the data using a second order polynomial. Recall for a mass falling from rest, that 1 (1.1) y a yt 2 2 Suppose a mass falls through n successively greater displacements, each time starting from rest. The displacements can be expressed a 2 y y t ; 1 n . (1.2) 2 Analyzing the Data Data for y is not linear in time t. We have two unique ways we can analyze the data. The first is to simply plot the data with vertical displacement on the y-axis and time on the x-axis and perform a 2nd order polynomial curve fit. We can then extract acceleration from the coefficient of the 2nd order term. The second method involves transforming the nonlinear data into a linear form by means of the logarithm from which we can extract acceleration. We are going to use both methods because it demonstrates the power of mathematics as a data analysis tool. Fitting the Data to a 2nd Order Polynomial Free-fall data is shown in figure 1 and has the form y At2 Bt C
Figure 1. Free-fall plot (dots) and 2nd order fit (solid line).
If we fit ideal free-fall data to equation (1.3) we should find that B = 0, C = 0, and A = ay/2. If you look at the polynomial fit equation embedded in figure 1 you will see
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that B = -10-13, C = -10-14, and A = -4.905. So the data is not perfect but essentially both B and C are zero while A = -4.0905. If you compare the polynomial equation to our kinematic equation… y At 2 Bt C a y y t 2 vyit yi 2 …it becomes immediately evident that B corresponds to initial velocity, C the initial position, and A = ay/2. If dropped from rest, initial velocity and position are zero. This all boils down to the fact that fitting a second order