Simple Harmonic Motion (SHM) of a Simple Pendulum Essay Example for Free

Simple Harmonic Motion (SHM) of a Simple Pendulum EssayObjectivesTo study the aboveboard benevolent bowel movement (SHM) of a mere(a) pendulum and to investigate the manikin relationship between the shimmy, stop number and speedup, and to investigate how acceleration is related to displacement in a simple sympathetic motion.Apparatus* half metre find out* a light string* pendulum bob* video photographic camera with tripod stand* computer with Motion movie Analysis (MVA) software and Microsoft Excel installedExperimental designFig. 0TheoryFor an object or plurality moving in a simple harmonic motion, the displacement, velocity and acceleration change periodically in both magnitude and direction. The acceleration in particular is always proportional to its displacement from the equilibrium opinion and must always be directed towards the equilibrium point. Mathematically it can be expressed asa = -kx, where k is a constant and x is the displacement from the equilibrium point.Also for a simple harmonic oscillation, the period or frequency of oscillation is independent of the amplitude of the motion.In Figure 1, x is the displacement of the pendulum bob from the equilibrium point Q. Points P and R are points where the maximum displacement (amplitude A) can be seeed. Theoretically, the sideline equatings are true for S.H.M.When the motion starts at the equilibrium plaza (point Q)x = A dark ?t where ? is angular velocityv = ? A cos ?ta = ?2A sin ?tPeriod T = 2 ? / ?Fig. 1When the motion starts at the positionwhere the amplitude is obtained (point P or R)x = A cos ?t where ? is angular velocityv = ? A sin ?ta = ?2A cos ?tPeriod T = 2 ? / ?In theory, the displacement-time, velocity-time and acceleration-time interprets should be in a sine or cosine curve. Moreover, the velocity graph should lead the displacement by a quarter of the cycle (? = 90), and the acceleration graph should lead the velocity by also a quarter of the cycle. This can be ill ustrated by the fig. 2(a), (b) and (c).Fig. 2(a) Graph of displacement x against time (Suppose the motion starts at the point where lower amplitude is obtained)Fig. 2(b) Graph of velocity v against timeFig.2(c) Graph of acceleration a against timeProcedure1. The develop-up was assembled in the following procedures(a) unmatched end of the skip over is clamped firmly on the stand.(b) The ringed mass was attached to the other end of the spring.(c) A half-metre rule was clamped on the stand beside the spring and mass such that the top of the half-metre rule corresponds to the top of the spring. (Refer to Fig. 0)(d) The equilibrium position was tag by a sticker.2. Take readings by using the apparatus in the following procedures(a) Student holding the white foam board (Student A)(i) Hold the white foam board behind the set up so that the movement of the spring system is not disturbed by any other backgrounds.(b) Student conducting the experiment (Student B)(i) Stand beside the set up. Make sure that the spring system at equilibrium is in a steady and stable condition.(ii) When the video taking was on, pull down the spring some distance (e.g. about 5 cm) and set the spring moving.(iii) Make sure the spring is mostly moving in a vertical direction and not swinging to and fro.(iv) After a some oscillations, ask student C to stop the video.(c) Student conducting the video-taking (Student C)(i) Set up the video camera and fix it on the tripod stand firmly.(ii) Adjust the position of the camera so that the spring system and the movement of the spring is shown clearly.(iii) Watch out for Student B to start or stop the video-taking.3. Convert the video into qualified format.4. Use the MVA software to record the positions and times for 2 complete oscillations of the mass. Save the project and export the data to a text file.5. Use Microsoft Excel to pay the exported files and plot the graphs for displacement, velocity and acceleration against time respectively. Also pl ot a graph of acceleration against time.Results and Measurements(Copied from the data of MVA software)Values of velocity is found by the equation (x2-x1)/(t2/t1),whereas x1=-7.54E-02 x2= -4.08E-02, t1=0.00E+00 ,t2=6.67E-02Values of acceleration is found by the equation (v2-v1)/(t2-t1)Whereas v1= -1.30E-02,v2= 5.20E-01, t1=0.00E+00 ,t2=6.67E-02E+00= 100=1 E-01=10-1 E-02=10-2 E-03=10-3(See the bolded ones as reference)BR/-IndexTime(s)x-coordinate(displacement in x-direction)/my-coordinate(displacement in y-direction)/mVelocity(v)/ms-1Acceleration(a)ms-2BR /0.00E+00-5.80E+000.00E+000.00E+00BR /1.00E+000.00E+00-7.54E-021.02E-02-1.30E-02BR /2.00E+006.67E-02-4.08E-020.00E+005.20E-017.99E+00BR /3.00E+001.33E-01-1.63E-020.00E+003.67E-01-2.29E+00BR /4.00E+002.00E-011.43E-02-2.04E-034.59E-011.38E+00BR /5.00E+002.67E-013.87E-02-4.08E-033.67E-01-1.38E+00BR /6.00E+003.33E-016.73E-02-4.08E-034.28E-019.17E-01BR /7.00E+004.00E-018.97E-02-4.08E-033.36E-01-1.38E+00BR /8.00E+004.67E-011.08E-01-8.15E-0 32.75E-01-9.17E-01BR /9.00E+005.33E-011.30E-01-4.08E-033.36E-019.17E-01BR /1.00E+016.00E-011.45E-01-6.11E-032.14E-01-1.83E+00BR /1.10E+016.67E-011.57E-01-6.11E-031.83E-01-4.59E-01BR /1.20E+017.33E-011.57E-01-8.15E-030.00E+00-2.75E+00BR /1.30E+018.00E-011.63E-01-1.02E-029.17E-021.38E+00BR /1.40E+018.67E-011.57E-01-1.02E-02-9.17E-02-2.75E+00BR /1.50E+019.33E-011.53E-01-1.02E-02-6.11E-024.59E-01BR /1.60E+011.00E+001.39E-01-1.02E-02-2.14E-01-2.29E+00BR /1.70E+011.07E+001.22E-01-8.15E-03-2.45E-01-4.59E-01BR /1.80E+011.13E+001.04E-01-1.02E-02-2.75E-01-4.59E-01BR /1.90E+011.20E+008.56E-02-6.11E-03-2.75E-016.64E-06BR /2.00E+011.27E+006.11E-02-2.04E-03-3.67E-01-1.38E+00BR /2.10E+011.33E+003.87E-022.04E-03-3.36E-014.59E-01BR /2.20E+011.40E+008.15E-03-2.04E-03-4.59E-01-1.83E+00BR /2.30E+011.47E+00-1.63E-020.00E+00-3.67E-011.38E+00BR /2.40E+011.53E+00-4.69E-022.04E-03-4.59E-01-1.38E+00BR /2.50E+011.60E+00-8.36E-028.15E-03-5.50E-01-1.38E+00BR /2.60E+011.67E+00-1.08E-011.43E-02-3.67E-012.75E+00BR /2.70E+011.73E+00-1.32E-011.63E-02-3.67E-01-7.58E-06BR /2.80E+011.80E+00-1.59E-011.63E-02-3.97E-01-4.59E-01BR /2.90E+011.87E+00-1.81E-011.83E-02-3.36E-019.17E-01BR /3.00E+011.93E+00-2.02E-012.04E-02-3.06E-014.59E-01BR /3.10E+012.00E+00-2.26E-012.65E-02-3.67E-01-9.17E-01BR /3.20E+012.07E+00-2.41E-012.85E-02-2.14E-012.29E+00BR /3.30E+012.13E+00-2.47E-012.85E-02-9.17E-021.83E+00BR /3.40E+012.20E+00-2.59E-013.06E-02-1.83E-01-1.38E+00BR /3.50E+012.27E+00-2.65E-013.06E-02-9.17E-021.38E+00BR /3.60E+012.33E+00-2.71E-013.06E-02-9.17E-021.86E-07BR /3.70E+012.40E+00-2.55E-013.06E-022.45E-015.04E+00BR /3.80E+012.47E+00-2.43E-012.85E-021.83E-01-9.17E-01BR /3.90E+012.53E+00-2.32E-012.85E-021.53E-01-4.59E-01BR /4.00E+012.60E+00-2.20E-012.65E-021.83E-014.59E-01BR /4.10E+012.67E+00-1.98E-012.45E-023.36E-012.29E+00BR /4.20E+012.73E+00-1.79E-012.24E-022.75E-01-9.17E-01BR /4.30E+012.80E+00-1.57E-012.24E-023.36E-019.17E-01BR /4.40E+012.87E+00-1.30E-011.83E-023.97E-019.17E-01BR /4.50E+012.93E+00-1.06E-0 11.22E-023.67E-01-4.59E-01BR /4.60E+013.00E+00-8.36E-021.02E-023.36E-01-4.59E-01BR /4.70E+013.07E+00-4.28E-026.11E-036.11E-014.13E+00BR /4.80E+013.13E+00-1.83E-026.11E-033.67E-01-3.67E+00BR /4.90E+013.20E+001.43E-024.08E-034.89E-011.83E+00BR /5.00E+013.27E+003.87E-022.04E-033.67E-01-1.83E+00BR /5.10E+013.33E+006.93E-02-4.08E-034.59E-011.38E+00BR /5.20E+013.40E+009.17E-02-2.04E-033.36E-01-1.83E+00BR /5.30E+013.47E+001.14E-01-2.04E-033.36E-019.00E-07BR /5.40E+013.53E+001.32E-01-2.04E-032.75E-01-9.17E-01BR /5.50E+013.60E+001.43E-01-4.08E-031.53E-01-1.83E+00BR /5.60E+013.67E+001.53E-01-4.08E-031.53E-012.12E-14BR /5.70E+013.73E+001.63E-01-4.08E-031.53E-011.19E-06BR /5.80E+013.80E+001.55E-01-4.08E-03-1.22E-01-4.13E+00BR /5.90E+013.87E+001.47E-01-4.08E-03-1.22E-011.21E-14BR /6.00E+013.93E+001.41E-01-6.11E-03-9.17E-024.59E-01BR /6.10E+014.00E+001.35E-01-6.11E-03-9.17E-02-1.86E-07BR /6.20E+014.07E+001.26E-01-6.11E-03-1.22E-01-4.59E-01BR /6.30E+014.13E+001.08E-010.00E+00-2.75E-01-2.29E+00BR / 6.40E+014.20E+008.56E-02-4.08E-03-3.36E-01-9.17E-01BR /6.50E+014.27E+006.11E-02-4.08E-03-3.67E-01-4.59E-01BR /6.60E+014.33E+003.26E-02-2.04E-03-4.28E-01-9.17E-01BR /6.70E+014.40E+006.11E-032.04E-03-3.97E-014.59E-01BR /6.80E+014.47E+00-2.65E-022.04E-03-4.89E-01-1.38E+00BR /6.90E+014.53E+00-5.30E-028.15E-03-3.97E-011.38E+00BR /7.00E+014.60E+00-8.36E-028.15E-03-4.59E-01-9.17E-01BR /7.10E+014.67E+00-1.10E-018.15E-03-3.97E-019.17E-01BR /7.20E+014.73E+00-1.43E-011.83E-02-4.89E-01-1.38E+00BR /7.30E+014.80E+00-1.59E-011.43E-02-2.45E-013.67E+00BR /7.40E+014.87E+00-1.88E-012.04E-02-4.28E-01-2.75E+00BR /7.50E+014.93E+00-2.06E-012.04E-02-2.75E-012.29E+00BR /7.60E+015.00E+00-2.16E-012.04E-02-1.53E-011.83E+00BR /7.70E+015.07E+00-2.38E-012.65E-02-3.36E-01-2.75E+00BR /7.80E+015.13E+00-2.49E-012.85E-02-1.53E-012.75E+00BR /7.90E+015.20E+00-2.55E-012.85E-02-9.17E-029.17E-01BR /8.00E+015.27E+00-2.59E-012.85E-02-6.11E-024.59E-01BR /8.10E+015.33E+00-2.57E-012.85E-023.06E-021.38E+00BR /8.20E+015.40E+00-2. 51E-012.85E-029.17E-029.17E-01BR /8.30E+015.47E+00-2.43E-013.06E-021.22E-014.59E-01BR /8.40E+015.53E+00-2.30E-012.65E-021.83E-019.17E-01BR /8.50E+015.60E+00-2.16E-012.65E-022.14E-014.59E-01BR /8.60E+015.67E+00-1.96E-012.65E-023.06E-011.38E+00BR /8.70E+015.73E+00-1.71E-012.24E-023.67E-019.17E-01BR /8.80E+015.80E+00-1.63E-012.04E-021.22E-01-3.67E+00BR /8.90E+015.87E+00-1.22E-011.63E-026.11E-017.34E+00BR /9.00E+015.93E+00-9.78E-021.22E-023.67E-01-3.67E+00Experimental results in graphical representationAnalysis(a) Shape of displacement-time, velocity-time and acceleration-time graphsFrom the experimental results and the graphs plotted above, it appears clearly that the displacement-time, velocity-time are in the form of sine and cosine curves respectively. For acceleration-time, out-of-pocket to errors in marking, whitethorn not appear as clear as sine curves. It can be seen more clearly after drawing a trend line.(b) Value of amplitude A and ?it can be read from the graph of x against time that the amplitude is within the range of 0.15-0.25m.Also, read from the graph, ?3.0s(c) The phase relationship between the displacement, velocity and accelerationBy comparing the graphs of displacement-time, velocity-time and acceleration-time, it can be seen that the velocity leads the displacement by a quarter of the cycle, and the acceleration leads the velocity also by a quarter of the cycle.(d) The relationship between acceleration and displacement in a simple harmonic motionFrom the graph of acceleration against displacement x, the points tend to form a straight line going through the origin with a negative slope. It can be deduced that acceleration is at once proportional to displacement in a simple harmonic motion and is in an opposite direction to x.Error and AccuracyErrorsSystematic Errorrandom ErrorThe motion of the spring system is not entirely vertical.The half-metre rule is not clamped vertically.The origin is not marked very accurately in the MVA software.The 2 ends of the half-metre rule are not marked accurately in the MVA software.The position of mass marked for each time interval may not be the same for all time intervals.There may be a damping effect by transmit resistance.The spring may not be short elastic(1) The motion of the spring system is not entirely verticalNo matter how carefully we set the motion off, the spring may not be moving vertically all throughout the motion. It may swing to and fro instead, hence the motion may not be entirely a simple harmonic motion, cause deviations in displacement obtained.(2) The half-metre rule is not clamped verticallyThe half-metre rule is not entirely vertical, so the marked points on the MVA software do not indicate an actual distance of 0.5 m. As the MVA software requires the setting of the end points of the half-metre rule as a reference to locate the displacement, the displacement at each time interval does not reflect the true value of the displacement.(3) The origin is not mark ed very accurately in the MVA softwareThe inaccuracy of the center on of mass marked in the MVA software will result in the shifting up or down of the graphs of displacement, velocity and acceleration against time.(4) The two ends of the half-metre rule are not marked accurately in the MVA softwareAs the two ends of the half-metre rule may not be marked accurately in the MVA software, the distance marked may not be exactly 0.5 m. Same as error (2), as the MVA software requires the setting of the end points of the half-metre rule as a reference to locate the displacement, the displacement at each time interval does not reflect the true value of the displacement.(5) The position of mass marked for each time interval may not be the same for all time intervalsIt is difficult to locate the mass at the same position for each time interval, therefore the displacement obtained is not accurate for each time interval.(6) There may be a damping effect by air resistanceAir resistance exists, h ence a damping force acts on the mass in motion, resulting in smaller and smaller amplitude obtained and also causing deviations in displacement.(7) The spring may not be perfectly elasticAs the spring provided may not be perfect, the whole motion may not be entirely a simple harmonic motion. The graphs obtained from the experimental results may not truly reflect the characteristics of a simple harmonic motion. terminationThe velocity leads the displacement by a quarter of the cycle, and the acceleration leads the velocity also by a quarter of the cycle.Also, the acceleration is directly proportional to displacement in a simple harmonic motion and is in an opposite direction to x.Possible improvements of the experiment1. A heavier mass could be used to obtain a smoother motion.2. If possible, more trials can be done to average out the random errors and obtain a better result.