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the title is ( SimpleHarmonicMotion—MassonaSpring )


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208 Physics Laboratory Manual Loyd 20 Physics Laboratory Manual Loyd LABORATORY Simple Harmonic Motion, Mass on a Spring Original Equilibrium Position Yo New Equilibrium Position A = y-yo = Amplitude of the motion M OBJECTIVES Directly determine the spring constant k of a spring by measuring the elongation versus applied (a) (b) (c) Figure 20-1 New equilibrium position with mass M placed on a spring. force. with angular frequency o related to the frequency fand the period T by Determine the spring constant k from measurements of the period T of oscillation for different values of mass. 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. O- k VM 0 = 2nf T = 1/f T =21 M k (Eq. 3) EQUIPMENT LIST • Spring, masking tape, laboratory timer, meter stick, table clamps, and rods • Right-angle clamps, laboratory balance, and calibrated hooked masses A mass M placed on the end of a spring hangs vertically as shown in Figure 20-1. The original equilibrium position of the lower end of the spring is shown in Figure 20-1(a). The position of the lower end of the spring when the mass is applied, shown in Figure 20-1(b), can be considered as the new equilibrium position. In Figure 20-1(c) the mass is pulled down to a displacement A from this new equilibrium position. When released, the mass will oscillate with amplitude A and period T given above. Equation 3 for the period is strictly true only if the spring is massless. For real springs with finite mass, a fraction of the spring mass m, must be included along with the mass M. If C stands for the fraction of the spring mass to be included, the period is 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 F=-ky (Eq. 1) M + Cm, T = 29V- (Eq. 4) You will be challenged to discover what fraction of the spring mass should be included from your analysis of the data that you will take in the laboratory. COPYRIGHT 2008 Thomson Brooks Cole 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=A and y=-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= A cos(cot+) (Eq. 2) EXPERIMENTAL PROCEDURE Spring Constant 1. Attach the table clamp to the edge of the laboratory table and screw a threaded rod into the clamp vertically as shown in Figure 20-2. Place a right-angle clamp on the vertical rod and extend a hori- zontal rod from the right-angle clamp. Hang the spring on the horizontal rod and attach it to the horizontal rod with a piece of tape. Screw a threaded vertical rod into a support stand, which rests on the floor. Place a right-angle clamp on the vertical rod and place a meter stick in the clamp so that the meter stick stands vertically. Adjust the height of the clamp on the vertical rod until the zero mark of the meter stick is aligned with the bottom of the hanging spring as shown in Figure 20-2. 2. Place a hooked mass Mof 0.050 kg on the end of the spring. Slowly lower the mass M until it hangs at rest in equilibrium when released. Carefully read the position of the lower end of the spring on the meter THOMSON 2008 Thomson Brooks/Cole, a part of the Thomson Corporation. Thomson, the Staroga and Brooks/Colas Tademarks used herein under hense. ALL RIGHTS RESERVED. No part of this work covered by the copyright herect may be reprodused or used in any timor byany mersi electronic, or me ric, ar nedarica, including photocopying sounding taping,web disabation, information storage and retrieval systems, or in any other manne without writen permission of the publisher BROOKS/COLE 207 Laboratory 20 Simple Harmonic Motion-Mass on a Spring 209 210 Physics Laboratory Manual Loyd CALCULATIONS Spring Constant 1. Calculate the force Mg for each mass and record the values in Data and Calculations Table 1. Use the value of 9.80 m/s for g. 2. Perform a linear least squares fit to the data with Mg as the vertical axis and y as the horizontal axis. Record in Data and Calculations Table 1 the slope of the fit as the spring constant kand the correlation coefficient r. Table Table 8 Amplitude Variation 1. Calculate the mean At and standard error of the three trials for each amplitude. Record the results in Calculations Table 2. 2. Calculate the period T from T=Af/10. Record the results in Calculations Table 2. -4 Floor Floor Mass Variation 1. Calculate the mean At and standard error for the three trials for each mass. Record the results in Calculations Table 3. 2. Calculate the period T from T=Af/10. Record the results in Calculations Table 3. 3. If both sides of Equation 4 are squared the result is Figure 20-2 Arrangement to measure displacement of spring caused by mass M. T? 472 -(M + Cm.) (Eq. 5) stick scale. Record the value of the mass M and the value of the displacement x in Data and Calculations Table 1. 3. Repeat Step 2, placing in succession 0.100, 0.200, 0.300, 0.400, and 0.500 kg on the spring and measuring the displacement y of the spring. Record all values of M and y in Data and Calculations Table 1. Amplitude Variation 1. We will investigate dependence of the period T on the amplitude A for a fixed mass of 0.500 kg. Place the mass on the end of the spring and slowly lower the mass until it hangs at rest when released. Record this position of the lower end of the spring as yo 2. Displace the mass downward to y=yo +0.0200 m as shown in Figure 20-1, which will produce A=0.0200 m. Release the mass, and let it oscillate. Measure the time for 10 complete periods and record it in Data Table 2 as At. Repeat the procedure two more times for a total of three trials at this amplitude. 3. Repeat Step 2 above for A of 0.0400, 0.0600, 0.0800, 0.1000, and 0.1200 m. Make three trials for each amplitude and measure the time for 10 periods for each trial. Record all results in Data Table 2. 4. Equation 5 states that Tº is proportional to M with 4r+/k as the slope and 4x?Cm./k as the intercept. Calculate and record the values of T in Calculations Table 3. Perform a linear least squares fit with T? as the vertical axis and M as the horizontal axis. Record the values of the slope, intercept, and r in Calculations Table 3. 5. Equate the value of the slope determined in Step 4 to 4z"/k and solve for the value of k in the resulting equation. Record this value of k in Calculations Table 3. 6. Calculate the percentage difference between the value of k determined in Step 5 and the value of k determined earlier and record it in Calculations Table 3. 7. Equate the value of the intercept determined in Step 4 to 4n%Cm-/k and solve for the value of C in the resulting equation. In the equation, use the value of k determined in Step 5. Record the value C in Calculations Table 3. COPYRIGHT © 2008 Thomson Brooks Cole Mass Variation 1. Place a hooked mass of 0.050 kg on the spring and let it hang at rest. Displace the mass 0.0500 m below the equilibrium (A=0.0500 m), release it, and let the system oscillate. Measure the time for 10 periods of the motion and record it in Data Table 3 as At. Repeat the procedure two more times for a total of three trials with this mass. 2. Repeat the procedure of Step 1 with the same A for values of the mass M of 0.100,0.200,0.300,0.400, and 0.500 kg. Perform three trials of the time for 10 periods for each mass and record the results in Data Table 3. 3. Determine the mass of the spring me and record it in Data Table 3. GRAPHS 1. Graph the data from Calculations Table 1 for force Mg versus displacement y with Mg as the vertical axis and y as the horizontal axis. Show on the graph the straight line obtained from the fit to the data. 2. Graph the data from Calculations Table 2 for the period T versus the amplitude A with T as the vertical axis and A as the horizontal axis. 3. Graph Tº versus M with Tº as the vertical axis and M as the horizontal axis. Also show on the graph the straight line obtained from the linear least squares fit to the data. 214 Physics Laboratory Manual Loyd Nam Section Date Data and Calculations Table 3 m = kg Lab Partners M(kg) At (s) At(s) Atz (s) At(s) T(s) T(S) 0.050 0.100 0.200 20 LABORATORY 20 Simple Harmonic Motion-Mass on a Spring 0.300 0.400 0.500 LABORATORY REPORT Slope = Intercept = Data and Calculations Table 1 k= N/m C= % Diff M(kg) y (m) k(N/m) 0.050 0.127 0.100 0.200 Mg (N) 0.490 0.980 1.96 2.94 3.92 4.90 0.150 0.191 0.230 0.272 0.312 SAMPLE CALCULATIONS 1. Mga 2. T=At/10 0.300 3. T 0.400 4. k= 4x+/(Slope) 5. C=k(Intercept)/(4nºm)= 0.500 Data and Calculations Table 2 Yo = m 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. A (m) At, (s) Atz (s) At(s) x(s) T(s) Atz(s) 9.0 0.0200 8.76 8.26 9.12 0.0400 0.0600 0.0800 2. How is the period T expected to depend upon the amplitude A? State how your data do or do not confirm this expectation. 0.1000 COPYRIGHT © 2008 Thomson Brooks Cole 0.1200 213 Laboratory 20 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? (4 % 14 %) 4. Calculate T predicted by Equation 3 for M=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=0.500 kg. For which case are the percentage differences greater and why are they greater? This page intentionally left blank 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? COPYRIGHT © 2008 Thomson Brooks Cole
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Running head: LABORATORY REPORT

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Laboratory Report
Name of Student
Name of Professor
Course Title
Date

LABORATORY REPORT

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Introduction

Normally, Hooke’s law is obeyed by all masses experiencing restoring forces proportional
to the spring extension. This statement may be presented in equation form as Force (F)= - Ky,
whereby k is the spring constant and y is the spring displacement or extension. As shown above,
the negative sign is used to show that the spring is extending in the opposite direction. The
displacement above results to a motion termed as simple harmonic motion. This means that, the
motion can be expressed in a simple sine or cosine function of time. If we take a mass, A being
displaced from its initial position y by an extension of A, the equilibrium position will be expressed
as; y-A and y=-A.

In this case, A is defined as amplitude of motion. The displacement y can be expressed as a function
of time t as:

As per our case, a relationship between angular frequency ω, frequency f and time period T can be
represented as shown below:

LABORATORY REPORT

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Apparatus/ materials

1. Spring
2. Masking tape
3. Laboratory timer
4. Meter stick
5. Table clamps
6. Rods
7. Right angled clamps
8. Laboratory balance
9. Calibrated hooked masses.
Procedure
1. The table clamp was attached to the edge of laboratory table and a threaded rod was placed
in the clamp vertically as indicated in Figure 20-2. The spring was then ha...


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