Name Section Date Lab Partners 19 LABORATORY 19 The Pendulum-Approximate Simple Harmonic Motion LABORATORY REPORT Data and Calculations Table 1 L (m) At (s) Atz (s) Atz (s) At (s) a(s) T(s) VI (Vm) 1.0000 0.8000 0.6000 0.5000 0.3000 0.2000 0.1000 Slope = Sexp m/s2 r= Data and Calculations Table 2 M (kg Δff (s) At2 (s) Atz (s) At (s) Q: (s) T(s) 0.0500 0.1000 0.2000 0.5000 COPYRIGHT © 2008 Thomson Brooks/Cole 203 204 Physics Laboratory Manual Loyd Data and Calculations Table 3 A At, (s) Atz (s) Atz (s) At(s) ar(s) Texp(s) Ttheo (s) Texp(0) Texp(5) Ttheo (0) Ttheo (5) 5.0 10.0 20.0 30.0 45.0 SAMPLE CALCULATIONS 1. Texp = At/10 = 2. VL 3. Sexp = 4x+/(slope) = 4. Theo = QUESTIONS 1. In general, what is the precision of the measurements of T? Answer this question by considering what percentage is of At for the measurements as a whole. 2. Do your data confirm the expected dependence of the period T on the length L of a pendulum? Consider the correlation coefficient r for the least squares fit in your answer. 3. Comment on the accuracy of your experimental value for the acceleration due to gravity g. Laboratory 19 The Pendulum—Approximate Simple Harmonic Motion 205 4. What does the theory predict for the shape of the graph of period T versus M? Do your data confirm this expectation? Calculate the mean and standard error of the periods for the four masses and comment on how this relates to mass independence of T. 5. Do your measured values for the period T as a function of the amplitude 6 confirm the theoretical predictions? State clearly what is expected and what your data show. 6. The values of T were determined by measuring the time for 10 periods. Why is the time for more than one period measured? If there is an advantage to measuring for 10 periods, why not measure for 1000 periods? COPYRIGHT © 2008 Thomson Brooks/Cole Physics Laboratory Manual Loyd LABORATORY 19 The Pendulum- Approximate Simple Harmonic Motion OBJECTIVES Investigate the dependence of the period T of a pendulum on the length L and the mass M of the bob. Demonstrate that the period T of a pendulum depends slightly on the angular amplitude of the oscillation for large angles, but that the dependence is negligible for small angular amplitude of oscillation. Determine an experimental value of the acceleration due to gravity g by comparing the measured period of a pendulum with the theoretical prediction. EQUIPMENT LIST • Pendulum clamp, string, and calibrated hooked masses, laboratory timer • Protractor and meter stick THEORY A mass M moving in one dimension is said to exhibit simple harmonic motion if its displacement x from some equilibrium position is described by a single sine or cosine function. This happens when the particle to a force F directly proportional magnitude of the displacemen directed toward the equilibrium position. In equation form this is F = kx (Eq. 1) The period T of the motion is the time for one complete oscillation, and it is determined by the mass M and the constant k. The equation that describes the dependence of T on M and k is M T = 21 COPYRIGHT © 2008 Thomson Brooks/Cole (Eq. 2) THOMSON 2008 Thomson Brooks/Cole, a part of The Thomson Corporation Thomson, the Starlogo and Brooks/Cole are trademarks used herein under license ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, Information storage and retrieval systems, or in any other maner-without the written permission of the publisher BROOKS/COLE 197 198 Physics Laboratory Manual Loyd 2 Mg sin e Mg cos e Mg Figure 19-1 Force components acting on the mass bob of a simple pendulum. A pendulum does not exactly satisfy the conditions for simple harmonic motion, but it approximates them under certain conditions. An ideal pendulum is a point mass M on one end of a massless string with the other end fixed as shown in Figure 19-1. The motion of the system takes place in a vertical plane when the mass M is released from an initial angle with respect to the vertical. The downward weight of the pendulum can be resolved into two components as shown in Figure 19-1. The component Mg cos equals the magnitude of the tension N in the string. The component Mg sino acts tangent to the arc along which the mass M moves. This component provides the force that drives the system. In equation form the force F along the direction of motion is F = -Mg sino (Eq. 3) For small values of the initial angle 0, we can use the approximation sin 0 = tan 0 = x/L in Equation 3, which gives Mg F=- x L (Eq. 4) Although Equation 4 is an approximation, it is of the form of Equation 1 with k=Mg/L. Using that value of k in Equation 2 gives M L T = 21 = 211 (Eq. 5) Mg/L 8 Equation 5 predicts that the period T of a simple pendulum is independent of the mass M and the angular amplitude 0 and depends only on the length L of the pendulum. The exact solution to the period of a simple pendulum without making the small angle approxi- mation leads to an infinite series of terms, with each successive term becoming smaller. Equation 6 gives the first three terms in the series. They are sufficient to determine the very slight dependence of the period T on the angular amplitude of the motion. T = 24 [1+1/4 sin? (0/2) +9/64 sin*(0/2) + ....] (Eq. 6) For an ideal pendulum with no friction, the motion repeats indefinitely with no reduction in the amplitude as time goes on. For a real pendulum there will always be some friction, and the amplitude of the motion decreases slowly with time. However, for small initial amplitudes, the change in the period
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