Simple Harmonic Motion Laith Alshaikh

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Feb 20, 2024

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LAB 10 2019 Purpose: The purpose of this lab is to examine the behavior of a mass oscillating on a spring in order to study some basic properties of Simple Harmonic Motion (SHM). Procedure : we have measured its period (the time it takes for one oscillation) and therefore determine its frequency (the number of oscillations per unit time, or the inverse of the period). PRELIMINARY QUESTIONS 1. Attach the 200 g mass to the spring and hold the free end of the spring in your hand, so the mass and spring hang down with the mass at rest. Lift the mass about 10 cm and release. Observe the motion. Sketch a graph of position vs . time for the mass. 2. Just below the graph of position vs . time, and using the same length time scale, sketch a graph of velocity vs . time for the mass. Physics with Computers 15 - 1

Experiment 15 7. Compare the position graph to your sketched prediction in the Preliminary Questions. How are the graphs similar? How are they different? Also, compare the velocity graph to your prediction The fact that both graphs—one showing position and the other showing velocity—are sinusoidal in nature explains their striking similarities. The distinction is that, if one pays close attention, the velocity is at its highest when the position is at its average value (which is the equilibrium position). DATA TABLE Run Mass y 0 A T f (g) (cm) (cm) (s) (Hz) 1 200 50 1.1 0.762 1.31 2 300 36 11 0.935 1.07 3 CACULATIONS: WEXPECTED: √k/m= 2pif w=2pi*1.31=8.23 rad/s w=2pi*1.07=6.72 rad/s Frequency: 1/T= 1/0.762=1.31 1/s ANALYSIS 1. View the graphs of the last run on the screen. Compare the position vs . time and the velocity vs . time graphs. How are they the same? How are they different? Ans. 15 - 2 Physics with Computers

Simple Harmonic Motion Similarities: Here both graphs are sinusoidal, both graphs have same time period, both graphs have same frequency and both started from negative side. Differences: Here position graph is more smooth as compare to velocity graph, initial value for position graph is zero but initial value for velocity graph is maximum, when the position graph has its maximum value that time velocity graph has its minimum value. 2. Turn on the Examine mode by clicking the Examine button, . Move the mouse cursor back and forth across the graph to view the data values for the last run on the screen. Where is the mass when the velocity is zero? Where is the mass when the velocity is greatest? Ans. when v = 0 position is 0.2775 and 0.4785 ,time is at peak/trough when v = 0 , but when v is maximum position is 0 time t is when the curve crosses the neutral axis. 3. Does the frequency, f , appear to depend on the amplitude of the motion? Do you have enough data to draw a firm conclusion? the frequency f does not depend on the amplitude of the motion We dont have enough data to draw firm conclusion, but from the same experimetn we can just add a few more data entries to conclude the same 4. Does the frequency, f , appear to depend on the mass used? Did it change much in your tests? the frequency appears to depend on mass used in the experiment In out test on changing the mass from 200 to 300g, frequency changed from 1.31 to 1.07 Hz hence, 100*100/200 % change in mass caused 0.24*100/1.31 % change in frequency hence, 50% increase in mass decreases frequency by 17.39% 5. You can compare your experimental data to the sinusoidal function model using the Manual Curve Fit feature of Logger Pro . Try it with your 300 g data. The model equation in the introduction, which is similar to the one in many textbooks, gives the displacement from equilibrium. However, your Motion Detector reports the distance from the detector. To compare the model to your data, add the equilibrium distance to the model; that is, use y = A sin ( 2 π ft + φ ) + y 0 where y 0 represents the equilibrium distance. a. Click once on the position graph to select it. b. Choose Curve Fit from the Analyze menu. c. Select Manual as the Fit Type. d. Select the Sine function from the General Equation list. e. The Sine equation is of the form y=A*sin(Bt +C) + D. Compare this to the form of the equation above to match variables; e.g. ,  corresponds to C, and 2  f corresponds to B. f. Adjust the values for A, B and D to reflect your values for A , f and y 0 . You can either enter the values directly in the dialog box or you can use the up and down arrows to adjust the values. g. The phase parameter  is called the phase constant and is used to adjust the y value reported by the model at t = 0 so that it matches your data. Since data collection did not necessarily begin when the mass was at the equilibrium position,  is needed to achieve a good match. Physics with Computers 15 - 3

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