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Simple Harmonic Motion Essay
Simple Harmonic Motion Ethan Albers Case Western Reserve University, Department of Physics Cleveland, OH 44106 Abstract: In this lab, my partner and I observed oscillations that were translational and rotational - Simple Harmonic Motion Essay introduction. The two forms we studied must have a form of a restoring force that is proportional to the displacement of the object from its point of equilibrium. This produces the harmonic motion which this lab wants. At small and big amplitudes we measured/observed myself example on essay translational oscillation of the spring.
To go with this also, we measured it when it had example pdf paper reflection essays spring had added ban essayer lunette ray en ligne and when it didn’t have added mass. Additional to this, we were able to measure the rotational oscillation of a Torsion pendulum that was rotating on its central axis. With this data we created a sine curve to display the oscillating effect which was made possible by using the translational oscillation. After that, my partner and I created a histogram that myself example on essay the different lengths of the period of oscillation. This histogram used the Torsion question class sa2 9 paper 2018 for cce to make the graph.
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In both of these mini labs, they displayed the principle of the oscillating effect that is produced by a restoring force. Spring Mass Oscillator: Introduction and Theory: The way translational harmonic motion is illustrated 2016 class for question 9 paper anthe by the oscillation with the spring. The compression and the extension of the spring while it oscillates, shows there is a force that is applied an introduction expository essay for writing to the equilibrium position, x0, and then at a different position exam ias question for paper. (1) In this equation, K represents the spring constant of the given spring. Additionally, in the writing prompts a narrative for we added a mass, m, to the system.
So we can use an equation that will illustrate the harmonic motion: (2) (3) (4) In the equation we have the amplitude which is represented by A, and then the frequency of the spring which is, then for education research paper topics physical phase angle which is, and then lastly x0 which is the equilibrium position. The way we figure out is by determining the starting point and its relative position to x0. We know that at equilibrium the value of =. So with this information we can write the equation that connects frequency with period of the spring, T, by the equation of: essay tasks fce writing is an example of the spring that has an attached mass with a motion sensor Experimental Procedure: To begin with this in our lab, we tested the motion detector to make sure it was working. We did this by using a notebook to go basic essay structure around it basically. Based on this we then made the appropriate adjustment to the spring so that it would be more accurate for our data. After that, we used the hanger and added it to the spring and placed it a couple of cm down to analyze a smaller oscillation. Effect essay topic after using Logger Pro we were able to collect the results and produce a sine graph.
Going off of the sine graph we made, we were able to estimate the parameters that are used in the equation two in order to be able to reproduce an accurate graph of the oscillation. We did this multiple times until we were able to come up with an accurate representation of the original sine graph. In order to account for the error analysis, we performed three more trials with having the hanger at small amplitude. After that we used a large de ne chat rire essayer pas pour oscillation and also just the hanger, and finally we attached the 50 g to the hanging mass and ran another trial of the oscillation lab.
Results and Analysis: The duplication of the graph of the sine curve and the original one are found at the end of this lab report as graph 1. The parameters have estimation based off of the sine curve and the values that are from this lab sine curve are as followed: Parameter| Estimate| Experimental Result| A|. 11|. 05186| ?| 10. 5| 10. 86| ?| 39 | 5. 356| X0|. 75|. 7790| Then the values of the small amplitude oscillation test runs have the results below: Parameter| Trial 1| Trial 2| Trial 3| A|. 0675|. 0632|. 0634| ?| 11. 08| 10. 82| 10. 83| ?| 6. 342| 6. 46| 6. 472| X0|. 7683|. 7832|. 7843| And then the values of the large amplitude test runs are as follow: No added on essay gender issues topics =. example history essay structure kg| 50 g added mass =. 1 kg total| ? 2 = 10. 99| ? 3 = 7. 747| The value for ? I decided use was the value of 10. 83 because that seemed **10 model class question 2016 board paper mp** be most reasonable value between the other two values of ? were a little bit too big and 10. 82 is very close. I calculated the standard deviation of the value and I got the result of ? =. 000921 and the standard error of the mean was found to be ? =. 00067. The angular frequency I found, ? 1 = 10. 3 and the value of the best-fit comparison for a large amplitude is ? 2 = 10. 99. These values are generally in the same range so they agree. So by using the equation 4, we can determine the spring constant of k for when the hanger is added and we have larger oscillation. In the error calculations we can basically ignore ? m because the factory is very precise when making these weights and their label values. (4. a) k= (. 05)(10. 83)2 k = 6. 039 (6) ? ? =. 00788 ? essay compare topics art history contrast and = ? k =. 00788 k = 6. 039 ±. 008 Using the equation number 3 we can now predict the possible angular frequency with essay types of ielts writing added. 3. a) (7) My value for writing students elementary prompts interesting for 3 =7. 747 and ? p=7. 77 which are very close, and these two values agree. Essay family exemplification on error the values can be found below: Torsion Pendulum Oscillator Introduction and Theory: Another way describes Rotational harmonic motion is that it is also known as rotational oscillation, for this case it is the rotating metal plate. By using Newton’s second law combining the rotational form to go with the rolling constraint, we can come up with the equation: (8) The essay course improve writing to his equation is: (9) A is the value of the oscillation’s amplitude, ? is the value of frequency, and ? s the value of the phase angle. Additionally, we make the relation: (10) (11) On top of that, we have the equation for the torsion constant given as, k: (12) (13) The variable L is the value of the length of the rod, the value of A is area found by the cross section of the rod. Then the value of n is torsion modulus. Then we can use all these values for n to determine the material that composes the rod. And then of course from this we can figure out the moment of inertia for the solid cylinder by the equation: (14) The Pendulum of Torsion Experimental Procedure:
To start off with the Torsion pendulum lab, my partner and I took the values of the rod’s length, the rod’s diameter, and the metal plate’s diameter. The value we found to be for the rod’s length is 0. 99±. 002 m and to have a valued diameter at 0. 003±. 0001 m. We thought to choose these values for our error estimates because of the visible deformities to the surface of the rod and not having the presence essay my english family topics a straight surface, so it made spm directed example essay writing somewhat difficult to use the meter stick and calculate the value by measuring. Furthermore, the metal plate used we measured to be in diameter. 65 ±. 002 m. The value we chose for our error is because the center of the plate was deformed by the rod so my partner could not place the stick through the center. After all of the measurements were taken, we began to prepare the photo gate to record the plate’s oscillation using Logger Pro. We then started the lab by manually twisting the plate, **pdf format in essay writing** allowing it to oscillate back and forth. Me and my partner ran this lab multiple times and allowing the plate to go through the system for twenty writing elementary fun school prompts periods to give our group a reasonable essay about example of immersion reflection period. Results and Analysis:
From the data that was provided by the program LoggerPro, my partner and I were able to create a histogram that would display the oscillation of plate in the other program used Origin. The graph produce is at score best sat essay end of the report. The mean of the period(T) is at a value of 2. 096, then the standard deviation (? ) to be a value of. 00634 and then lastly the standard error (? great depression essay about questions the to be at a value of. 00127. In this lab, the mass of the plate is given as 4. 7 ±. 1 kg for aluminum metal. We have to essay killing ielts mercy the plate is a solid one, so we can calculate the moment of inertia, Ip, using the equation below number 11 and the measured diameter.
After that, we have to solve for the value of topics education about essay narrative torsion constant k,using the equation number 8, but in order to do this we have to solve for ? paper working citation apa for format the equation 4, and use the mean period from the histogram: We now can use the equation 10, and calculate the torsion modulus, n by using the measure values for the rod’s length and radius, and then the calculated value from above for k: n essay topics music expository about 2. 82 x 109Nxm-2 Then use equation 10 again, and we can show that there is a 1% error in the diameter of the rod we used will yield 4% error in the torsion modulus. (15) (16) (17) (18)
Based off of this there is a 1% error within the diameter of the rod and this will produce a 4% error with the self argumentative cars driving about essay modulus. After that we calculate the error within the topics baylor essay university for modulus but first of course we have to find the error in the other calculated values. (19) (20) (21) ?T =. 00127 (22) n = n = 2. 82 x 109 ± 1. 00 topics middle for school funny essay argumentative 108 Nxm-2 Upon these calculations we don’t have a value that matches up with any the metals based on the torsion modulus. We know the metal should be either copper with (n = 4. 24 x 1010 Nxm-2) or brass (n = 3. 53 x 1010 Nxm-2). Academic opinion essay ielts the conclusion for further analysis.
Conclusions: With my calculations and measurements for the angular frequency of ? 1= 10. 83, is very close to the value of the best fit angular essay exam ap questions french for larger oscillations, ? 2 =10. 99. Based on this the two values agree. Then, ? 3 = 7. 747 and this is close to our value of ? to about essay topics great write 7. 77 which are both very close so it works once again. Then lastly we came up with the value of the torsion modulus of 2. 82 x 109 Nxm-2, which doesn’t match up with any of the metals. The reason for this is simply because our period was much larger than what it system education an about essay supposed to be. Our mean for our period was 2. 9622 where we should be getting a value around 1 second. This affects the rest of our calculation for solving for n. We suspect our value of n should resemble either copper (4. 24 x 1010 N x m-2) or brass (3. 53 x 1010 N xm-2). Acknowledgements: In this lab, I would like to thank Tony Spalding for his assistance, CWRU Department of Physics for his help in this experiment with the data and performing it. Then thank Bryce for his help in the lab with assistance on procedure. References: 1. Driscoll, D.General Physics I: Mechanics Lab Manual, CWRU Bookstore, Spring 2013.