Measurement of the Acceleration Due to Gravity Phys 303 Lab Experiment 0 Justin M. Sanders January 12, 2004 Abstract Near the surface of the earth, all objects freely fall downward with the same acceleration g called the acceleration due to gravity. The acceleration due to gravity has been obtained from the period of pendulums of various lengths, and it was found that in Mobile, AL g = 9.7 ± 0.2 m/s 2. 1
1 Introduction & Theory Near the surface of a planet, all objects experience the same acceleration due to the force of gravity. In Mobile, Alabama, this acceleration is 9.793394 m/s. 1 This constant acceleration due to gravity appears in the relationship of the period of a simple pendulum to its length. Therefore, by measuring the period of a simple pendulum as a function of its length, the acceleration due to gravity may be determined. A simple pendulum consists of a point mass m suspended from a pivot by a massless, unstretchable string of length l as shown in Figure 1. If, at Figure 1: Schematic diagram of the simple pendulum. some point in the pendulum s swing, the string makes an angle θ with the vertical, the torque on the pendulum is given by τ = mgl sin θ. (1) Using Newton s Second Law for rotation, and taking the moment of inertia of the simple pendulum to be I = ml 2, we obtain ml 2 d2 θ = mgl sin θ. (2) dt2 Rearranging and making the approximation sin θ θ for small angles results in the differential equation for a simple harmonic oscillator d 2 θ dt + g 2 l θ = 0 (3) which has an angular frequency ω = g/l. Therefore, the period T of a simple pendulum (for small angles) is l T = 2π g. (4) If the period and length of a simple pendulum are known, Eq. 4 can be solved for the acceleration due to gravity g: g = 4π2 l T 2. (5) 1 Physics Experiments for PH201 and PH202, (USA Dept. of Physics, Mobile, AL, 1998), p. 41 2
This will be referred to as Method A. Alternatively, if the period T is plotted versus the square root of the length, a straight line should result with slope 2π g. The slope can then be solved for g. This will be referred to as Method B. Both of these techniques will be used in this work. 2 Procedure A small metal annulus was suspended by a thin thread from a support which allowed the length of the thread to be varied. The resulting pendulum was drawn aside by an angle of approximately 10 (judged by comparison with a paper template attached to the support) and released. Simultaneous with the pendulum s release, a stopwatch was started. The stopwatch (least count 0.01 s) was stopped when the pendulum had completed ten oscillations. The length of the pendulum was measured using a standard meter stick (least count 1 mm) between the point of support and the center of the metal annulus. The period of the the pendulum was taken to be the one-tenth of the time required for ten oscillations. For one pendulum length, the measurements were repeated ten times in order to obtain and estimate of the variation of the period times. The resulting standard deviation was used for the uncertainty in the period. 3 Data For all length measurements, the uncertainty is taken to be 3 mm, since there was some variation in determining the position of the point of support and the center of the annulus. For the 0.090 m length, the average period was 0.591 s and the standard deviation was 0.005 s. This standard deviation was used as the uncertainty in the period for all the lengths. 3
Length (m) Period (s) Calculated g (m/s 2 ) 0.387 1.231 10.082 ± 0.1 0.353 1.194 9.775 ± 0.1 0.268 1.031 9.954 ± 0.1 0.201 0.888 10.063 ± 0.1 0.174 0.818 10.266 ± 0.1 0.130 0.713 10.095 ± 0.1 0.090 0.587 10.312 ± 0.1 0.090 0.591 0.090 0.594 0.090 0.591 0.090 0.588 0.090 0.600 0.090 0.597 0.090 0.591 0.090 0.584 0.090 0.590 Table 1: Pendulum Lengths and Periods 4 Analysis and Results 4.1 Method A: Sample Calculation Using Eq. 5: g = (4π2 )(0.387 m) = 10.1 m/s 2 (6) (1.231 s) 2 and propagating the errors ) 2 ( ( δl δg = g + 2 δt ) 2 (7) l T and evaluating (0.003 ) m 2 ( δg = (10.1 m/s 2 ) + 2 0.005 s ) 2 = 0.1 m/s 2 (8) 0.387 m 1.231 s The results of these calculations are given in Table 1, and the average of the values of g given in the table is 10.1 ± 0.1 m/s 2. 4
Figure 2: Plot of period versus square root of length for a simple pendulum. The solid points are the present measurements, the dashed line is a leastsquares fit using Eq. 9 4.2 Method B The data in Table 1 used to produce a plot of T is plotted versus l. The plot is shown in Figure 2. A least-squares fit to the data was performed using the fitting routine in the program GNUPLOT (ver. 3.71) and the model chosen was a two-parameter straight line: T = a l + b (9) The fit yielded the following values for the parameters: a = 2.0204±0.02486 s/m 1/2 and b = 0.0171951 ± 0.0119 s. The slope a can be solved for g: and the uncertainty is g = 4π2 a 2 = 4π 2 (2.0204 s/m 1/2 ) 2 = 9.6773 m/s2 (10) δg = g(2 δa a ) = (9.6773 m/s2 )(2 0.02486 2.0204 ) = 0.238 m/s2 (11) Thus the result of Method B is that g = 9.7 ± 0.2 m/s 2. 5 Conclusions In this work, the dependence of the period of the simple pendulum on its length was exploited to obtain the acceleration due to gravity. The period of the pendulum was determined by timing its swing for various lengths, and two methods were used to analyze the data. In Method A, Eq. 5 was used for each length to obtain a value for g, then all the g s were averaged together resulting in g A = 10.1 ± 0.1 m/s 2. Since g g A δg A = 9.793394 m/s2 10.1 m/s 2 0.1 m/s 2 = 3.1, (12) 5
the analysis using Method A disagrees with the true value of g. However, by performing a linear fit in Method B, the deviation from the true value was smaller: g g B δg A = 9.793394 m/s2 9.7 m/s 2 0.2 m/s 2 = 0.5, (13) The primary difference between these two methods is that the parameter b in Method B can account for systematic errors in the period. It has been determined that time intervals measured by stopwatch from visual clues tend to be biased toward short times. 2 The effect of a bias toward short periods in Eq. 5 would be to shift the calculated value of g A upward, and Method A did indeed yield a high value of g. From Eq. 4, it can be seen that for the 0.387 m pendulum, the period should be 0.387 m T = 2π = 1.249 s (14) 9.793394 m/s2 The time for ten periods would be 12.49 s, which means that the measured interval of 12.31 s was 0.18 s short. This is comparable to the 0.14 s reaction time found previously. The value of parameter b implies that the ten-period measurements are consistently short by 0.17 s, and again this is consistent with the reaction-time findings. The results of these measurements show that the fitting method for obtaining g is superior to more straight-forward method (A), since it reduces the effect of one type of systematic error. 2 J.M. Sanders, Measurement of Human Reaction Times, unpublished. 6