Simple Harmonic Motion

Physics Laboratory Report Simple Harmonic Motion: Determining the force constant Aim of experiment: The objective of this experiment is: 1. To study the simple harmonic motion of a mass-spring system 2. To estimate the force constant of a spring Principles involved: A horizontal or vertical mass-spring system can perform simple harmonic motion as shown below. If we know the period (T) of the motion and the mass (m), the force constant (k) of the spring can be determined. [pic] Consider pulling the mass of a horizontal mass-spring system to an extension x on a table, the mass subjected to a restoring force (F=-kx) stated by Hooke’s Law.If the mass is now released, it will move with acceleration (a) towards the equilibrium position. By Newton’s second law, the force (ma) acting on the mass is equal to the restoring force, i. e. ma = -kx a = -(k/m)x ————————-(1) As the movement continues, it performs a simple harmonic motion with angular velocity (? ) and has acceleration (a = -? 2x). By comparing it with equation (1), we have: ? = v(k/m) Thus, the period can be represented as follows: T = 2? /? T = 2? x v(m/k) T2 = (4? 2/k) m —————————(2)From the equation, it can be seen that the period of the simple harmonic motion is independent of the amplitude. As the result also applies to vertical mass-spring system, a vertical mass-spring system, which has a smaller frictional effects, is used in this experiment. Apparatus: Slotted mass (20g)x 9 Hanger (20g)x 1 Springx 1 Retort stand and clampx 1 Stop watchx 1 G-clampx 1 Procedure: 1. The apparatuses were set up as shown on the right. 2. No slotted mass was originally put into the hanger and it was set to oscillate in moderate amplitude. 3.The period (t1) for 20 complete oscillations was measured and recorded. 4. Step 3 was repeated to obtain another record (t2). 5. Steps 2 to 4 were repeated by adding one slotted mass to the hanger each time until all of the nine given masses have been used. 6. A graph of the square of the period (T2) against mass (m) was plotted. 7. A best-fitted line was drawn on the graph and its slope was measured. Precaution: 1. The oscillations of the spring were of moderate amplitudes to reduce errors. 2. The oscillations of the spring were carefully initiated so that the spring did not swing to ensure accurate results. . The spring used was carefully chosen that it could perform 20 oscillations with little decay in amplitude when the hanger was put on it, and it was not over-stretched when all the 9 slotted masses were put on it. This could ensure accurate and reliable results. 4. The experiment was carried out in a place with little air movement (wind), in order to reduce swinging of the spring during oscillations and errors of the experiment. 5. The spring was clamped tightly so that the spring did not slide during oscillation. It reduced ene rgy loss from the spring and ensured accurate results. . A G-clamp was used to attach the stand firmly on the bench.This reduced energy loss from the spring and ensured accurate results. Results |Hanger and slotted mass |20 periods / s |One period (T) |T2 / s2 | |(m) / kg | |/ s | | | |t1 |t2 |Mean | | | | |( ±0. 1s) |( ±0. s) |= (t1 + t2) / 2 | | | |0. 02 |5. 0 |5. 4 |5. 2 |0. 26 |0. 0676 | |0. 04 |6. 0 |6. 0 |6. 0 |0. 3 |0. 09 | |0. 06 |7. 0 |7. 0 |7. 0 |0. 35 |0. 1225 | |0. 08 |7. 8 |7. 8 |7. 8 |0. 9 |0. 1521 | |0. 10 |8. 6 |8. 6 |8. 6 |0. 43 |0. 1849 | |0. 12 |9. 4 |9. 5 |9. 45 |0. 4725 |0. 22325625 | |0. 14 |10. 1 |10. 1 |10. 1 |0. 505 |0. 255025 | |0. 16 |10. 5 |10. 4 |10. 45 |0. 5225 |0. 27300625 | |0. 8 |11. 1 |11. 3 |11. 2 |0. 56 |0. 3136 | |0. 20 |11. 9 |12. 0 |11. 95 |0. 5975 |0. 35700625 | Calculations and Interpretation of results [pic] From equation (2), the slope of the graph is equal to (4? 2/k), i. e. 1. 5968 = 4? 2/k k = 4? 2/1. 5968 ? 24. 723 Nm-1 ?The force c onstant of the spring is 24. 723 Nm-1. Sources of error 1. The spring swung during oscillations in the experiments. 2.As the amplitudes of oscillations were small, there was difficulty to determine whether an oscillation was completed. 3. Reaction time of observer was involved in time-taking. 4. Energy was lost from the oscillations of the spring to resonance of the spring. Order of Accuracy Absolute error in time-taking =  ±0. 1s |Hanger and slotted mass (m) / kg |20 periods / s |Relative error in time-taking | | |t1 |t2 | | | |( ±0. s) |( ±0. 1s) | | | | | |t1 |t2 | |0. 02 |5. 0 |5. 4 |2. 00% |1. 85% | |0. 04 |6. 0 |6. 0 |1. 67% |1. 67% | |0. 06 |7. 0 |7. 0 |1. 3% |1. 43% | |0. 08 |7. 8 |7. 8 |1. 28% |1. 28% | |0. 10 |8. 6 |8. 6 |1. 16% |1. 16% | |0. 12 |9. 4 |9. 5 |1. 06% |1. 05% | |0. 14 |10. 1 |10. 1 |0. 990% |0. 990% | |0. 6 |10. 5 |10. 4 |0. 952% |0. 962% | |0. 18 |11. 1 |11. 3 |0. 901% |0. 885% | |0. 20 |11. 9 |12. 0 |0. 840% |0. 833% | Improvement 1. The spring should be initiated to oscillate as vertical as possible to prevent swinging of the spring, which would cause energy loss from the spring and give inaccurate results. 2.Several observers could observe the oscillations of the spring and determine a more accurate and reliable result that whether the spring has completed an oscillation. 3. The time taken for oscillations should be taken by the same observer. This allows more reliable results as error-error cancellation of reaction time of the observer occurs. 4. The spring used should be made of a material that its resonance frequency is difficult to match. Discussion In this experiment, several assumptions were made. First, it is assumed that the spring used is weightless and resonance does not occur.Furthermore, it is assumed that no energy is lost from the spring to overcome the air resistance. Besides, it is assumed that no swinging of the spring occurs during the experiment. In addition, there were difficulties in carrying out the exper iment. For timing the oscillation, as the spring oscillates with moderate amplitude, it was hard to determine if a complete oscillation has been accomplished. Added to this, in drawing the best-fitted line, as all the points do not join to form a straight line, there was a little difficult encountered while drawing the line.Nevertheless, they were all solved. Several observers observed the oscillations of the spring and determined a more reliable result that whether the spring has completed an oscillation. For the best-fitted line, computer was employed to obtain a reliable graph. Conclusion The mass-spring system performs simple harmonic motion and the force constant of the spring used in this experiment is 24. 723 Nm-1. ———————– A graph of T2 against m Square of the period (T2) [pic] Simple Harmonic Motion Shanise Hawes 04/04/2012 Simple Harmonic Motion Lab Introduction: In this two part lab we sought out to demonstrate simple harmonic motion by observing the behavior of a spring. For the first part we needed to observe the motion or oscillation of a spring in order to find k, the spring constant; which is commonly described as how stiff the spring is. Using the equation Fs=-kx or, Fs=mg=kx; where Fs is the force of the spring, mg represents mass times gravity, and kx is the spring constant times the distance, we can mathematically isolate for the spring constant k.We can also graph the data collected and the slope of the line will reflect the spring constant. In the second part of the lab we used the equation T=2? mk, where T is the period of the spring. After calculating and graphing the data the x-intercept represented k, the spring constant. The spring constant is technically the measure of elasticity of the spring. Data: mass of weight | displacement| m (kg)| x (m)| 0. 1| 0. 12| 0 . 2| 0. 24| 0. 3 | 0. 36| 0. 4| 0. 48| 0. 5| 0. 60|We began the experiment by placing a helical spring on a clamp, creating a â€Å"spring system†. We then measured the distance from the bottom of the suspended spring to the floor. Next we placed a 100g weight on the bottom of the spring and then measured the displacement of the spring due to the weight . We repeated the procedure with 200g, 300g, 400g, and 500g weights. We then placed the recorded data for each trial into the equation Fs=mg=kx. For example: 300g weight mg=kx 0. 30kg9. 8ms2=k0. 36m 0. 30kg 9. 8ms20. 36m=k 8. 17kgs=kHere we graphed our collected data. The slope of the line verified that the spring constant is approximately 8. 17kgs. In the second part of the experiment we suspended a 100g weight from the bottom of the spring and pulled it very slightly in order to set the spring in motion. We then used a timer to time how long it took for the spring to make one complete oscillation. We repeated this for the 20 0g, 300g, 400g, and 500g weights. Next we divided the times by 30 in order to find the average period of oscillation. We then used the equation T2=4? mk to mathematically isolate and find k. Lastly we graphed our data in order to find the x-intercept which should represent the value of k. Data Collected: Derived Data: mass of weight | time of 30 osscillation | avg osscilation T| T2| | m (kg)| t (s)| t30 (s)| T2 s2| | 0. 10| 26. 35| 0. 88| 0. 77| | 0. 20| 33. 53| 1. 12| 1. 25| | 0. 30| 39. 34| 1. 31| 1. 72| | 0. 40| 44. 81| 1. 49| 2. 22| | 0. 50| 49. 78| 1. 66| 2. 76| | Going back to our equation T2=4? 2mk .We found the average period squared and the average mass and set the equation up as T2m=4? 2k. Since T2 is our change in y and m is our change in x, this also helped us to find the slope of our line. We got T2m equals approximately 4. 98s2kg. We now have 4. 98s2kg= 4? 2k. Rearranging we have k=4? 24. 98s2k= 7. 92N/m. Plotting the points and observing that the slope of our line is indeed approximately 4. 98 we see that the line does cross the x-axis at approximately 7. 92. Conclusion Prior to placing any additional weight onto our spring we measured the length of spring to be 0. 8m. So if we hooked an identical spring and an additional 200g the elongation of our total spring would be approximately 0. 8m; accounting for twice our spring and the . 24m the additional weight added. However, I believe the additional weight of the second spring would slightly elongate the initial spring; bringing it roughly over a meter. Since our spring elongation has almost tripled I believe that an effective spring constant would be triple that of what we found it to be initially, making a new spring constant of 24. 51kgs