Physics Laboratory Manual n L ABOR AT ORY Loyd 20 Simple Harmonic Motion— Mass on a Spring OBJECTIVES o Directly determine the spring constant k of a spring by measuring the elongation versus applied force. o Determine the spring constant k from measurements of the period T of oscillation for different values of mass. o Investigate the dependence of the period T of oscillation of a mass on a spring on the value of the mass and on the amplitude of the motion. EQUIPMENT LIST . Spring, masking tape, laboratory timer, meter stick, table clamps, and rods . Right-angle clamps, laboratory balance, and calibrated hooked masses THEORY A mass that experiences a restoring force proportional to its displacement from an equilibrium position is said to obey Hooke’s law. In equation form this relationship can be expressed as COPYRIGHT ª 2008 Thomson Brooks/Cole F ¼ "ky ðEq: 1Þ where k is a constant with dimensions of N/m. The negative sign indicates that the force is in the opposite direction of the displacement. If a spring exerts the force, the constant k is the spring constant. A force described by Equation 1 will produce an oscillatory motion called simple harmonic motion because it can be described by a single sine or cosine function of time. A mass displaced from its equilibrium position by some value A, and then released, will oscillate about the equilibrium position. Its displacement y from the equilibrium position will range between y ¼ A and y ¼ "A with A called the amplitude of the motion. For the initial conditions described above, the displacement y as a function of time t is given by y ¼ A cosðot þ fÞ ðEq: 2Þ ª 2008 Thomson Brooks/Cole, a part of TheThomson Corporation.Thomson,the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTSRESERVED.No part of this work covered by the copyright hereon may be reproduced or used in any form or by any meansçgraphic, electronic, or mechanical,including photocopying, recording, taping,web distribution, information storage and retrievalsystems,or in any other mannerçwithout the written permission of the publisher. 207 214 Physics Laboratory Manual n Loyd ms ¼ _______________kg Data and Calculations Table 3 M (kg) Dt1 (s) Dt2 (s) Dt3 (s) Dt (s) at (s) T (s) T2 ( s2 ) 0.050 0.100 0.200 0.300 0.400 0.500 Slope ¼ k¼ N/m Intercept ¼ r¼ C¼ % Diff ¼ SAMPLE CALCULATIONS 1. Mg ¼ 2. T ¼ Dt/10 3. T2 ¼ 4. k ¼ 4p2/(Slope) 5. C ¼ k(Intercept)/(4p2ms) ¼ QUESTIONS 1. Do the data for the displacement of the spring y versus the applied force Mg indicate that the spring constant is constant for this range of forces? State clearly the evidence for your answer. 2. How is the period T expected to depend upon the amplitude A? State how your data do or do not confirm this expectation. Laboratory 20 n Simple Harmonic Motion—Mass on a Spring 215 3. Consider the value you obtained for C. If you express that fraction as a whole number fraction, which of the following would best fit your data? ( ½ ˆ!¯ ¼ ˆ!˙ ) 4. Calculate T predicted by Equation 3 for M ¼ 0.050 kg. Calculate T predicted by Equation 4 with the same M and your value of C. What is the percentage difference between these two values of T? Do the same calculations for M ¼ 0.500 kg. For which case are the percentage differences greater and why are they greater? COPYRIGHT ª 2008 Thomson Brooks/Cole 5. The determination of T was done by measuring for 10 periods. Why was the time for more than one period measured? If there is an advantage to measuring for 10 periods, why not measure for 1000 periods? 772 76 Lab Outline * Conclusion * Cover Sheet * Introduction (theory) - 4-5 sentences -Summery title Phys 2052.005 Name * Data -charts/tables Proff * Analysis Questions
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