Proposal: Measurement of the Acceleration Due to Gravity at the Earth’s Surface using a Simple Pendulum John Belz February 18, 2008 1 Introduction; History and Background The acceleration due to gravity at the Earth’s surface — commonly represented by the symbol g — is a quantity with great signiﬁcance in our daily lives. We cannot throw a baseball, drive a car, or walk down the street without being eﬀected by the particular value of the gravitational pull of the Earth. Galileo  ﬁrst asserted that, in the absence of air resistance, all objects fall with the same constant acceleration near the surface of the Earth. Modern experiments have conﬁrmed this assertion is true to an accuracy of a few parts per billion [2, 3]. We propose to carry out a new messurement of g using the novel technique of timing the oscillations of a simple pendulum. By this technique we will measure g with a precision of better than 1%, and in so doing conﬁrm basic Newtonian dynamics. 2 Theory; Model and Assumptions The ideal simple pendulum is shown schematically in Figure 1. A pointlike bob of mass m swings on the end of an inextensible and massless string. The pendulum experiences constant force of magnitude mg in the downward direction, as well as the varying tension in the string. We assume that the force due to air resistance on the bob and string is negligible. If we apply Newton’s Second Law F = ma to the system in the ﬁgure we ﬁnd that ¨ mg sin θ = −mlθ (1) ¨ g (2) θ + sin θ = 0 l In the small-angle approximation we obtain the equation for a simple harmonic oscillator: ¨ g θ+ θ=0 l resulting in the relations T = 2π T2 = l g (4) (5) (3) 4π 2 ×l g between the length l, period of oscillation T , and the acceleration due to gravity g. Figure 1: A simple pendulum, consisting of a pointlike bob of mass m on the end of an inextensible string. The bob experiences constant force mg in the downward direction, as well as the force of tension in the string. 3 Plan of Analysis We will perform a series of measurements in which we vary the length l of the pendulum and record the period T . According to Equation 5, the square of the period and the length of the pendulum are linearly related with a slope dependent on g. Therefore, we will extract g by performing a χ2 minimization ﬁt to the linear form in order to compute the slope. This analysis is illustrated for a hypothetical set of measurements in Figure 2. Here, we assume that g has the accepted value of 980 cm/sec2 . In the actual measurement, we will perform a linear least-squares ﬁt to determine the slope of this graph and hence g according to Equation 5. 4 Numerical Estimates In this section we perform several numerical estimates to demonstrate the feasibility of our experiment as well as the validity of the experimental results. 1. What are our likely uncertainties in experimental quantities? How will these propagate to the uncertainty in our ﬁnal result? Our measured quantities are the length l of the pendulum and its period of oscillation T . Applying the error propagation equation to Figure 2: Set of hypothetical measurements for a simple pendulum experiment, for the case in which g = 980 cm/sec2 . The estimated experimental uncertainties in these measurements, as described in the text, are smaller than the data points shown. A linear least squares ﬁt will be used to extract the slope of this graph and hence the acceleration due to gravity g. Equation 5, we ﬁnd σg = 4π 2 l T2 σl l σl l 2 2 +4 σT T 2 1/2 = g σT +4 T 2 1/2 (6) The uncertainty in the length of the pendulum results primarily from the ruler we will be using, which has tickmarks every 1 mm. To do a better than 1% accuracy measurement of g we therefore need a pendulum of length ≥ 10 cm. We will be measuring T with a stopwatch, and “human reﬂex” error of about ±0.5 sec will likely be a major problem if we try to measure a single period. We propose therefore to average over many periods, a procedure which should enable us to make the uncertainty in the period arbitrarily small. According to Equation 6, we therefore need to average over ≥ 160 oscillations in order to determine the period to better than 1%, for a l = 10 cm long pendulum. 2. Model: Validity of small angle approximation. The solution to Equation 2 in the general (non-small angle) case features a period T which is a function of the maximum oscillation angle θm : T (θm ) = T0 1 + 1 2 θm 9 θm sin + sin4 +··· 4 2 64 2 (7) where T0 is the period in the limit of zero angle of oscillation. We can solve this for θm such that the total systematic error in the measured period is less than 1%: T (θm ) ≈ 1.01 ≈ T0 θm 1 2 θm sin 4 2 √ ≈ 2 sin−1 0.04 ≈ 0.403 radians ≈ 23.1◦ 1+ (8) We will keep θm less than 23◦ for our intended 1% measurement. 3. Model: Validity of neglecting air resistance. The main eﬀect of air resistance will be to gradually damp out the amplitude of the pendulum’s oscillations. Since each oscillation will be a little shorter than the one preceding it, the actual “damped” period will be given by : T0 Td = √ (9) 1 − ξ2 where ξ = T0 /2πτ and τ is the time for the oscillations to decrease in magnitude by a factor of 1/e. In preliminary studies, we found τ ∼ 8 sec for a typical simple pendulum. This corresponds to a relative systematic error of 1 Td −1 = √ −1 T0 1 − ξ2 = 8 × 10−5 (10) for a 10 cm long pendulum. We thus consider the eﬀects of air resistance negligible for the proposed experiment. References  See any basic physics textbook.  S. Carusotto et al., Phys. Rev. Lett (1992).  S. Chu et al., Nature (1999).  R. J. Stephenson, Mechanics and Properties of Matter, Wiley, New York (1952).