4AL
Lab Physics for Science and Engineering Mechanics
Experiment 5: Simple and Damped Harmonic Motion Lab Section: 8 Name: Christine Probst UID: 303458589 Date: February 21, 2008 TA: Yong WANG
Partner: Jimmy Wang Lab Station: 9
Introduction
Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator that is neither driven nor damped. SHM is a very useful way to explain many physical phenomenons throughout the universe. The very foundation of physics, quantum theory, rests on the belief that electrons and other subatomic particles behave as waves. Therefore, it is of utmost importance to understand the behavior of waves in a quantitative manner. The goal of this experiment is to calculate some basic components of a wave in simple and damped harmonic motion. This experiment uses a spring and a force transducer to calculate the period and frequency of damped and undamped systems. Equations F=-kx
Mx''=-bx'-kx
Q=(km)^/b
W=(k/m)^.5
Wo=Wfree(1-1/(4Q))^ Q=pi/2ln(Rn) Rn=A2/A1
Hooke's Law F= force (N) K=spring constant (Nm) X=distance displaced from equilibrium (m) The motion of mass under damping X= Position (m) M=mass(kg) B=damping due to magnetic resistance K=spring constant (Nm) The Quality factor M=mass (kg) B=damping due to magnetic resistance K=spring constant (Nm) W is the frequency of a free oscillating mass W=frequency (1/s) K=spring constant M=mass (kg) Wo is the frequency in damped harmonic motion Calculate Quality factor experimentally Rn=ratio of peaks Ratio of peaks calculated from amplitudes
Procedure First, measure the spring constant k by weighing a series of five small masses as well as a zero mass and record the change in position. Then, using excel, graph Force vs. change in equilibrium. The spring constant is the slope of this line, which can be recorded by adding a trend line. Knowing the spring constant, the next step is to create a system undergoing simple harmonic motion and measure its frequency. Attach the spring to a firmly secured force transducer (FT) and make sure it can move freely. The coils of the spring should not touch at any point during its natural movement. Next, hang a known mass from the spring, and, again make sure the coils do not touch during normal motion. Don't forget to weigh the spring an include 1/3 of its total weigh as a mass additive. Now raise the weight vertically a few centimeters from equilibrium and release, while simultaneously instructing PSW to begin reading FT data. After an interval of 20
seconds, stop data recording and transfer the output to excel for further analysis. Plot the data on excel and observe the peaks. Calculate the period and the angular frequency of this system. For the damped system, use a weight with embedded magnets. Hang the weight from the spring and make sure that the coils do not touch during oscillation. Lift the weight and position the metal tube directly beneath it, so that the entirety of the magnets motion is within the walls of the metal cylinder. Carefully grasp the weight and raise it to just beneath the top edge of the tube. Release the weight and begin recording data. Make sure that the weight does not collide with the tube and that it is cleanly oscillating. This magnetic resistance of the magnet with the magnetic material will create resistance, which is considered the damping force. Again, after about 20 seconds, import the data into excel. Plot and calculate peak to peak rations, and compare predicted verse calculated period and quality factor values. Data
SUMMARY OUTPUT Regression Statistics Multiple R 0.999826 R Square 0.999651 Adjusted R Square 0.999477 Standard Error 0.019142 Observations 4 ANOVA df Regression Residual Total 1 2 3 SS 2.100142 0.000733 2.100875 Standard Error 0.018591 0.041546 MS 2.100142 0.000366 F 5731.806 Significance F 0.000174
Intercept X Variable 1
Coefficients 0.349155 -3.1454
t Stat 18.78089 -75.7087
P-value 0.002823 0.000174
Lower 95% 0.269165 -3.32416
Upper 95% 0.429146 -2.96664
Lower 95.0% 0.269165 -3.32416
Upp 95. 0.42 -2.9
Data Analysis
Spring Constant y = -3.1454x + 0.3492 0 0 -0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Force (N)
-1 -1.5 -2 -2.5 Distance (m)
Here is force graphed verse distance, illustrating Hooke's law where F=-kx. We can take k as the slope of the line, so k=3.1454. Using regression, we find the standard error to be 0.041546126.
Simple Harmonic motions (expanded scale)
0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.6 1.6 2.6 Time (seconds) 3.6 4.6
FT volts
Series1
Here is the motion of a simple harmonic oscillator. Notice that the amplitude remains constant over time. Following is a table of the predicted and calculated values for the waves. The predicted angular frequency is w=(k/m)^. From this, T and f can be calculated as w=2(pi)f and T=1/f. In order to calculate experimental w, T, and f, we first find T by taking the amount of oscillations and dividing it by the time elapsed. This value can be used to calculate both w and f using the same equations as above.
Angular velocity Theoretical 7.562 Experimental 7.620 Total Error 0.058 Percent Error 0.77%
(1/s) Frequency (1/s) Period (s)
1.203 0.831
1.213 0.824
0.01 .007
0.83% 0.84%
Here are the calculations of Q for the undamped case. For the first calculation, we are given a value of infinity, meaning that no damping occurred. Q was not taken as an average averaging these values will not yield meaningful A results. peak (V) 0.94607 0.94729 0.94607 0.94668 A base (V) 0.83621 0.83743 0.83804 0.8356 A (V) 0.10986 0.10986 0.10803 0.11108 R 1 0.983342 1.028233 0.961559 Q #DIV/0! -93.4643 56.39024 -40.0519
Following is the graph for damped motion. Notice that the points fall of the main curve. This is caused by noise.
Damped Motion (expanded scale)
0.85 0.8
FT Volts
0.75 Series1 0.7 0.65 0.6 0.4 2.4 4.4 Time (s) 6.4
Here is a graph for the noise with no mass or spring attached. The standard error for the voltage 0.001003 and the standard error for the amplitude is 0.04478.
Noise
1.002 1 0.998 0.996 0.994 0.992 0.99 0.988 0 2 4 6 8 Time (s) 10 12 14 16 Series1
FT VOLTS
Following are the calculations for Q in the damped system, with the average Quality factor= -6.11.
1 2 3 4 4 A peak (V) 0.81 0.791 0.774 0.762 0.755 A low (V) 0.638 0.659 0.674 0.686 0.694 A total (V) 0.172 0.132 0.1 0.076 0.061 R 0.767442 0.757576 0.76 0.802632 Q -5.93141 -5.65497 -5.72081 -7.14092 Q avg -6.11203
Again we compare the theoretical and experimental values for period, frequency, and angular velocity in the damped system.
Theoretical Angular velocity (1/s) Frequency (1/s) Period (s) 4.194 0.66778 1.4975 Experimental 4.199 .6687 1.495 Absolute Error .005 .00092 0.0025 Percent Error 0.02% 0.01% 0.02%
The plotted data for both the simple and damped oscillations accurately coincides with the expected results. The errors for both the simple and damped harmonic motion indicate that there were no significant errors and that the data was both accurate and precise. Error Analysis Error propagation can be found on a handwritten sheet attached to the end of this report. Motion Type Damped Undamped F exp (Hz) 1.213 .6687 F th (Hz) 1.203 .6678 Sigma F .008 .008 Within error? No Yes
Post-Lab Questions 1. The force will increase linearly as expected, but will drop off once the coils touch
2. Set up the same experiment using a photo gate to measure the distance that the mass travels in one oscillation. This could be performed on a simple harmonic oscillator as the distance the mass travels will be the same no matter what period it is, as the amplitude remains constant (although depletes slightly over time). As the period can be calculated from this experiment, one could calculate the distance traveled over the period to find velocity. Graph this against the force for a series of oscillations to prove to force is indeed proportional to velocity. 3. Accuracy is the difference between a measured value and its true value. In this experiment, we can take the true value to be the theoretical and the measured value to be the experimental. We find that the deviation for simple harmonic motion to be about 0.83% and the deviation for damped harmonic motion to be about 0.2%. Through error analysis we discover that our damped system was accurate, but not the simple system. With regards to precision for the damped system, we do not take an average Q factor, but instead calculate the angular frequency for the Q factor of each period. We find that at most the angular frequency deviates by 0.2%, indicating that this experiment is reproducible. 4. The masses used in this experiment accounted for the weight of the spring. When neglecting to add 1/3 the mass of the spring, the angular frequency is increased from 7.56 to 7.93! The error for simple harmonic motion angular frequency is 3.9%! Therefore, it is very critical to include a fraction of the mass of the spring within this experiment. 5. The estimated mass of the spring is 10 grams, but not all of this weight would influence the oscillation frequency because not all of this mass is actually oscillated. Instead, a portion of the spring will remain fixed near the top of the apparatus. Observing the motion of the spring under harmonic oscillation, it is evident that less than half of the spring is actually moving. Therefore, we must include somewhere between zero and half of the mass of the spring. The handout mentions that 1/3 of the mass of the spring should be considered significant, and that is the factor used throughout this experiment. 6. The force required to compress a spring is dependent on the material of the spring, the width of the spring, and the number of coils per unit length. Assuming that the diameter and length of these two springs were the same, it would take a greater force to compress the spring with 100 coils. Using Hooke's law F=-kx, we know that k is proportional to F. Therefore, the spring with 100 coils would have a higher force constant. 7. From Giancoli's Physics for Scientists and Engineers, we know that the amount of time it takes to reduce the amplitude to 1/e is t=2m/b (page 375). We also know that from page 73 of the lab manual, that bT/4m=pi/2Q. Also, Q= pi/(2ln(Rn). Through some manipulation of these equations, we find that t=T/4ln(Rn). As Rn= .993284, and T=.824 s, we calculate the time to be 30.5 cycles. Conclusion The goal of this experiment was to demonstrate simple and damped harmonic motion. A mass hanging on a spring oscillated vertically while a force transducer
quantized the mass's movement. The data was then analyzed and various aspects of its movements were derived and defined. The test for damped motion worked well, and the frequency of the mass's motion was precisely calculated. However, the simple harmonic oscillator approached, but did not fall within the uncertainty.