1. Measurements and error calculus

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Hector Parrish
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1 EV 1 Measurements and error cacuus 11 Introduction The goa of this aboratory course is to introduce the notions of carrying out an experiment, acquiring and writing up the data, and finay anayzing the resuts By means of a simpe mathematica penduum the gravitationa acceeration g sha be determined Various measurement methods and the corresponding error sources wi be discussed Some keywords reated to the present experiment are: precision and accuracy of reading of measurement devices, book keeping and anaysis of measured data, systematic and statistica errors, histograms of data, cacuation of mean vaues and corresponding errors 12 Theoretica background F Fa ϕ tangentia m at rest G norma Figure 11: The penduum A mass m be suspended by means of a thread, the ength of which be much greater than the diameter of the mass We assume that friction may be negected For sma osciation ampitudes the period of time T is independent of mass and ampitude; the period soey depends on the ength of the penduum and on the gravitationa acceeration g (see Appendix): T = 2π g or g = 4 π2 T 2 (11) Thus, the measurement of g can be reduced to simpe measurements of the ength and of the period of osciation of a penduum 1

2 2 1 Measurements and error cacuus 13 Experiment 131 Goa of the experiment determination of the gravitationa acceeration g from the ength and the period of osciation of a penduum, comparison of severa ways to measure time, discussion of error propagation and finay the cacuation of the error on g 132 Measurements Measure the ength of the penduum using the ruer and estimate the accuracy of the measurement Time measurement (cf appendix on error cacuus): at constant ength of the penduum and for sma ampitudes (ϕ = 5 10 corresponding to ϕ = radians) measure the period of osciation in the foowing ways: (a) Measure severa times the tempora period T over a singe cyce of osciation using a standard stopwatch Cacuate the mean vaue T (b) The same as in (a) but using an eectronic stopwatch (c) Using the standard mechanica stopwatch, measure severa times the time duration of five cyces of osciation Cacuate T and the mean vaue T (d) The same as in (c) but using the eectronic stopwatch Cacuate again T and T Enter the resuts into the tabes in Section 135 Cacuate the statistica errors on the mean vaues m T and the reative error r T Discuss the different resuts under the foowing aspects: In which case does the cacuation of the statistica errors make sense? In which case does an estimation of the measurement error yied more reasonabe resuts? Give the reasons of your caims Usuay, methods (c) or (d) are chosen for measuring the time of osciation Why shoud (c) and (d) be preferred over (a) and (b)? 133 Anaysis of the measurements Cacuate g from the mean vaue of T and the ength using Eq 11: g = 4 π2 T 2 (12)

3 13 EXPERIMENT 3 Cacuate the error on g from the reative errors on T and using the corresponding aws of error propagation (cf Appendix on Error Cacuus) Verify the reation T : Measure the time of osciation of the penduum for different engths Pot T as a function of Which kind of function do you expect? Determine the gravitationa acceeration g from the sope of the graph 134 If there is enough time eft Dependence of the period on the ampitude of the osciation: You are about to measure a sma effect: which method is best suited for this kind of precise measurements? Measure the time of osciation for three different initia ampitudes and enter the vaues into the corresponding tabe

4 4 1 Measurements and error cacuus 135 Data and resuts Determination of g ength: = uncertainty: m = period of time for one singe cyce: stopwatch T (s) eec watch T (s) T = m T r T = m T T measurement over five cyces of osciation: mech watch 5T (s) mech watch T (s) eec watch 5T (s) eec watch T (s) T = XXX XXX m T XXX XXX r T XXX XXX

5 13 EXPERIMENT 5 Dependence of T on the ength of the penduum ength (m) 5T (s) T (s) Dependence of T on the initia ampitude ampitude 5T (s) T (s)

6 6 1 Measurements and error cacuus 14 Appendix m at rest ϕ G t F Fa tangentia G n norma Figure 12: The penduum The mass m issuspended by means of a thread, the ength of which be much greater than the diameter of the mass We consider the mass to be a point mass Moreover, we wi negect friction The ange ϕ(t) is chosen as coordinate describing the state of the penduum We separate the horizonta and vertica components of the forces, and the equation of motion reads: equation of motion: m a = F F a + G tangentia : norma : m d2 ϕ = mg sin ϕ (13) dt 2 m [ dϕ dt ]2 = F Fa mg cos ϕ (14) Both equations contain the force aong the thread F F a (t), which a priori is unknown, as we as the coordinate ϕ(t), which is to be determined The atter can be deduced from Eq 13: since this differentia equation is transcendenta and cannot be soved for ϕ, we repace the sin-term in Eq 13 by ϕ, the approximation being vaid for sma anges (ϕ π/2): sin ϕ ϕ + and therefore m d2 ϕ dt 2 = mg ϕ or d 2 ϕ dt 2 + g ϕ = 0 (15) According to experience, the motion of the penduum is periodic Therefore, the foowing ansatz is chosen for describing this periodic motion: ϕ(t) = ϕ 0 cos(ωt + δ) ϕ 0 = ampitude ω = 2πν = 2π/T = anguar frequency δ = constant phase offset (16) ϕ 0 and δ depend on the initia state of the penduum, and ω is determined by inserting the ansatz Eq 16 into the differentia equation Eq 15 Differentiating twice Eq 16 yieds: We insert this into Eq 15 and get: d 2 ϕ dt 2 = ϕ 0 ω 2 cos(ωt + δ) (17) ϕ 0 ω 2 cos(ωt + δ) + g ϕ 0 cos(ωt + δ) = 0 ω = g = 2π T (18) The period of osciation is for sma ampitudes and point masses independent of mass and ampitude For arger ampitudes, the motion sti is periodic in time but becomes anharmonic; the period of osciation increases with increasing ampitude due to higher orders appearing in Eq 15

Module 22: Simple Harmonic Oscillation and Torque

Module 22: Simple Harmonic Oscillation and Torque Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque