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

Simple Harmonic Motion (SHM) of a Simple Pendulum Essay Objectives: To study the simple harmonic motion (SHM) of a simple pendulum and to investigate the phase relationship between the displacement, velocity and acceleration, and to investigate how acceleration is related to displacement in a simple harmonic motion. Apparatus: * half metre rule * a light string * pendulum bob * video camera with tripod stand * computer with Motion Video Analysis (MVA) software and Microsoft Excel installed Experimental design: Fig. 0 Theory: For an object or mass 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 position and must always be directed towards the equilibrium point. Mathematically it can be expressed as a = -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 obtained. Theoretically, the following equations are true for S.H.M.: When the motion starts at the equilibrium position (point Q) x = A sin ?t where ? is angular velocity v = ? A cos ?t a = ?2A sin ?t Period T = 2 ? / ? Fig. 1 When the motion starts at the position where the amplitude is obtained (point P or R) x = A cos ?t where ? is angular velocity v = ? A sin ?t a = ?2A cos ?t Period T = 2 ? / ? In theory, the displacement-time, velocity-time and acceleration-time graphs should be in a sine or cosine curve. Moreover, the velocity graph should lead the displacement by a quarter of the cycle (? = 90à ¯Ã‚ ¿Ã‚ ½), and the acceleration graph should lead the velocity by also a quarter of the cycle. This can be illustrated 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 time Fig.2(c): Graph of acceleration a against time Procedure: 1. The set-up was assembled in the following procedures: (a) One end of the spring 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 marked 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 few 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 suitable 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 open the exported files and plot the graphs for displacement, velocity and acceleration against time respectively. Also plot 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-02 Values 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-02 E+00= 100=1; E-01=10-1; E-02=10-2 E-03=10-3 (See the bolded ones as reference) BR/-Index Time(s) x-coordinate (displacement in x-direction)/m y-coordinate (displacement in y-direction)/m Velocity(v)/ms-1 Acceleration(a)ms-2 BR /0.00E+00 -5.80E+00 0.00E+00 0.00E+00 BR /1.00E+00 0.00E+00 -7.54E-02 1.02E-02 -1.30E-02 BR /2.00E+00 6.67E-02 -4.08E-02 0.00E+00 5.20E-01 7.99E+00 BR /3.00E+00 1.33E-01 -1.63E-02 0.00E+00 3.67E-01 -2.29E+00 BR /4.00E+00 2.00E-01 1.43E-02 -2.04E-03 4.59E-01 1.38E+00 BR /5.00E+00 2.67E-01 3.87E-02 -4.08E-03 3.67E-01 -1.38E+00 BR /6.00E+00 3.33E-01 6.73E-02 -4.08E-03 4.28E-01 9.17E-01 BR /7.00E+00 4.00E-01 8.97E-02 -4.08E-03 3.36E-01 -1.38E+00 BR /8.00E+00 4.67E-01 1.08E-01 -8.15E-03 2.75E-01 -9.17E-01 BR /9.00E+00 5.33E-01 1.30E-01 -4.08E-03 3.36E-01 9.17E-01 BR /1.00E+01 6.00E-01 1.45E-01 -6.11E-03 2.14E-01 -1.83E+00 BR /1.10E+01 6.67E-01 1.57E-01 -6.11E-03 1.83E-01 -4.59E-01 BR /1.20E+01 7.33E-01 1.57E-01 -8.15E-03 0.00E+00 -2.75E+00 BR /1.30E+01 8.00E-01 1.63E-01 -1.02E-02 9.17E-02 1.38E+00 BR /1.40E+01 8.67E-01 1.57E-01 -1.02E-02 -9.17E-02 -2.75E+00 BR /1.50E+01 9.33E-01 1.53E-01 -1.02E-02 -6.11E-02 4.59E-01 BR /1.60E+01 1.00E+00 1.39E-01 -1.02E-02 -2.14E-01 -2.29E+00 BR /1.70E+01 1.07E+00 1.22E-01 -8.15E-03 -2.45E-01 -4.59E-01 BR /1.80E+01 1.13E+00 1.04E-01 -1.02E-02 -2.75E-01 -4.59E-01 BR /1.90E+01 1.20E+00 8.56E-02 -6.11E-03 -2.75E-01 6.64E-06 BR /2.00E+01 1.27E+00 6.11E-02 -2.04E-03 -3.67E-01 -1.38E+00 BR /2.10E+01 1.33E+00 3.87E-02 2.04E-03 -3.36E-01 4.59E-01 BR /2.20E+01 1.40E+00 8.15E-03 -2.04E-03 -4.59E-01 -1.83E+00 BR /2.30E+01 1.47E+00 -1.63E-02 0.00E+00 -3.67E-01 1.38E+00 BR /2.40E+01 1.53E+00 -4.69E-02 2.04E-03 -4.59E-01 -1.38E+00 BR /2.50E+01 1.60E+00 -8.36E-02 8.15E-03 -5.50E-01 -1.38E+00 BR /2.60E+01 1.67E+00 -1.08E-01 1.43E-02 -3.67E-01 2.75E+00 BR /2.70E+01 1.73E+00 -1.32E-01 1.63E-02 -3.67E-01 -7.58E-06 BR /2.80E+01 1.80E+00 -1.59E-01 1.63E-02 -3.97E-01 -4.59E-01 BR /2.90E+01 1.87E+00 -1.81E-01 1.83E-02 -3.36E-01 9.17E-01 BR /3.00E+01 1.93E+00 -2.02E-01 2.04E-02 -3.06E-01 4.59E-01 BR /3.10E+01 2.00E+00 -2.26E-01 2.65E-02 -3.67E-01 -9.17E-01 BR /3.20E+01 2.07E+00 -2.41E-01 2.85E-02 -2.14E-01 2.29E+00 BR /3.30E+01 2.13E+00 -2.47E-01 2.85E-02 -9.17E-02 1.83E+00 BR /3.40E+01 2.20E+00 -2.59E-01 3.06E-02 -1.83E-01 -1.38E+00 BR /3.50E+01 2.27E+00 -2.65E-01 3.06E-02 -9.17E-02 1.38E+00 BR /3.60E+01 2.33E+00 -2.71E-01 3.06E-02 -9.17E-02 1.86E-07 BR /3.70E+01 2.40E+00 -2.55E-01 3.06E-02 2.45E-01 5.04E+00 BR /3.80E+01 2.47E+00 -2.43E-01 2.85E-02 1.83E-01 -9.17E-01 BR /3.90E+01 2.53E+00 -2.32E-01 2.85E-02 1.53E-01 -4.59E-01 BR /4.00E+01 2.60E+00 -2.20E-01 2.65E-02 1.83E-01 4.59E-01 BR /4.10E+01 2.67E+00 -1.98E-01 2.45E-02 3.36E-01 2.29E+00 BR /4.20E+01 2.73E+00 -1.79E-01 2.24E-02 2.75E-01 -9.17E-01 BR /4.30E+01 2.80E+00 -1.57E-01 2.24E-02 3.36E-01 9.17E-01 BR /4.40E+01 2.87E+00 -1.30E-01 1.83E-02 3.97E-01 9.17E-01 BR /4.50E+01 2.93E+00 -1.06E-01 1.22E-02 3.67E-01 -4.59E-01 BR /4.60E+01 3.00E+00 -8.36E-02 1.02E-02 3.36E-01 -4.59E-01 BR /4.70E+01 3.07E+00 -4.28E-02 6.11E-03 6.11E-01 4.13E+00 BR /4.80E+01 3.13E+00 -1.83E-02 6.11E-03 3.67E-01 -3.67E+00 BR /4.90E+01 3.20E+00 1.43E-02 4.08E-03 4.89E-01 1.83E+00 BR /5.00E+01 3.27E+00 3.87E-02 2.04E-03 3.67E-01 -1.83E+00 BR /5.10E+01 3.33E+00 6.93E-02 -4.08E-03 4.59E-01 1.38E+00 BR /5.20E+01 3.40E+00 9.17E-02 -2.04E-03 3.36E-01 -1.83E+00 BR /5.30E+01 3.47E+00 1.14E-01 -2.04E-03 3.36E-01 9.00E-07 BR /5.40E+01 3.53E+00 1.32E-01 -2.04E-03 2.75E-01 -9.17E-01 BR /5.50E+01 3.60E+00 1.43E-01 -4.08E-03 1.53E-01 -1.83E+00 BR /5.60E+01 3.67E+00 1.53E-01 -4.08E-03 1.53E-01 2.12E-14 BR /5.70E+01 3.73E+00 1.63E-01 -4.08E-03 1.53E-01 1.19E-06 BR /5.80E+01 3.80E+00 1.55E-01 -4.08E-03 -1.22E-01 -4.13E+00 BR /5.90E+01 3.87E+00 1.47E-01 -4.08E-03 -1.22E-01 1.21E-14 BR /6.00E+01 3.93E+00 1.41E-01 -6.11E-03 -9.17E-02 4.59E-01 BR /6.10E+01 4.00E+00 1.35E-01 -6.11E-03 -9.17E-02 -1.86E-07 BR /6.20E+01 4.07E+00 1.26E-01 -6.11E-03 -1.22E-01 -4.59E-01 BR /6.30E+01 4.13E+00 1.08E-01 0.00E+00 -2.75E-01 -2.29E+00 BR /6.40E+01 4.20E+00 8.56E-02 -4.08E-03 -3.36E-01 -9.17E-01 BR /6.50E+01 4.27E+00 6.11E-02 -4.08E-03 -3.67E-01 -4.59E-01 BR /6.60E+01 4.33E+00 3.26E-02 -2.04E-03 -4.28E-01 -9.17E-01 BR /6.70E+01 4.40E+00 6.11E-03 2.04E-03 -3.97E-01 4.59E-01 BR /6.80E+01 4.47E+00 -2.65E-02 2.04E-03 -4.89E-01 -1.38E+00 BR /6.90E+01 4.53E+00 -5.30E-02 8.15E-03 -3.97E-01 1.38E+00 BR /7.00E+01 4.60E+00 -8.36E-02 8.15E-03 -4.59E-01 -9.17E-01 BR /7.10E+01 4.67E+00 -1.10E-01 8.15E-03 -3.97E-01 9.17E-01 BR /7.20E+01 4.73E+00 -1.43E-01 1.83E-02 -4.89E-01 -1.38E+00 BR /7.30E+01 4.80E+00 -1.59E-01 1.43E-02 -2.45E-01 3.67E+00 BR /7.40E+01 4.87E+00 -1.88E-01 2.04E-02 -4.28E-01 -2.75E+00 BR /7.50E+01 4.93E+00 -2.06E-01 2.04E-02 -2.75E-01 2.29E+00 BR /7.60E+01 5.00E+00 -2.16E-01 2.04E-02 -1.53E-01 1.83E+00 BR /7.70E+01 5.07E+00 -2.38E-01 2.65E-02 -3.36E-01 -2.75E+00 BR /7.80E+01 5.13E+00 -2.49E-01 2.85E-02 -1.53E-01 2.75E+00 BR /7.90E+01 5.20E+00 -2.55E-01 2.85E-02 -9.17E-02 9.17E-01 BR /8.00E+01 5.27E+00 -2.59E-01 2.85E-02 -6.11E-02 4.59E-01 BR /8.10E+01 5.33E+00 -2.57E-01 2.85E-02 3.06E-02 1.38E+00 BR /8.20E+01 5.40E+00 -2.51E-01 2.85E-02 9.17E-02 9.17E-01 BR /8.30E+01 5.47E+00 -2.43E-01 3.06E-02 1.22E-01 4.59E-01 BR /8.40E+01 5.53E+00 -2.30E-01 2.65E-02 1.83E-01 9.17E-01 BR /8.50E+01 5.60E+00 -2.16E-01 2.65E-02 2.14E-01 4.59E-01 BR /8.60E+01 5.67E+00 -1.96E-01 2.65E-02 3.06E-01 1.38E+00 BR /8.70E+01 5.73E+00 -1.71E-01 2.24E-02 3.67E-01 9.17E-01 BR /8.80E+01 5.80E+00 -1.63E-01 2.04E-02 1.22E-01 -3.67E+00 BR /8.90E+01 5.87E+00 -1.22E-01 1.63E-02 6.11E-01 7.34E+00 BR /9.00E+01 5.93E+00 -9.78E-02 1.22E-02 3.67E-01 -3.67E+00 Experimental results in graphical representation: Analysis: (a) Shape of displacement-time, velocity-time and acceleration-time graphs From 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, due to errors in marking, may 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, ?à ¯Ã‚ ¿Ã‚ ½3.0s (c) The phase relationship between the displacement, velocity and acceleration By 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 motion From 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 directly proportional to displacement in a simple harmonic motion and is in an opposite direction to x. Error and Accuracy: Errors Systematic Error Random Error The 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 two 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 air resistance. The spring may not be perfectly elastic (1) The motion of the spring system is not entirely vertical No 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, causing deviations in displacement obtained. (2) The half-metre rule is not clamped vertically The 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 marked very accurately in the MVA software The inaccuracy of the centre 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 software As 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 intervals It 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 resistance Air resistance exists, hence 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 elastic As 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. Conclusion The 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 experiment 1. 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.