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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.

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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 energy 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 constant 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 experiment. 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]