Simple Pendulum
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Simple Harmonic Motion (II):
Simple Pendulum
Introduction
When a mass is attached to a spring is displaced from its vertical position, and then released, it oscillates. Such motion is known in physics as simple harmonic motion, while the string mass system is known as a simple pendulum. In this experiment, simple harmonic motions, simple pendulum and the small angle approximation are studied. A spreadsheet is used to compare the values of θ and sin θ, for θ measured in radians for 0 ≤ θ ≤ 45o. The small angle approximation and how it is not valid as θ ≈ 1, is demonstrated in the experiment. Furthermore, the effect of changing the mass and the length on the period are further analyzed and demonstrated. In addition, the increase in the angle size with its effect on the period, along with the demonstration of how well would the equation for the period would hold as the angle gets larger are demonstrated.
Experimental Procedure
Small angle approximation
A spread sheet is used in order to calculate and compare the values of θ and sin θ, for θ measured in radians for 0 ≤ θ ≤ 45o.
Period of a Simple Pendulum
Components and test equipments: mass, string, protractor and meter stick.
A mass is attached to a string; the other end of the string is attached to a post and the length of the string is measured. The mass is displaced an angle θ0=5o from the vertical, using a protractor. The mass is released and the time it takes for it to complete 10 oscillations is measured, using this value the period of the pendulum is calculated, and the percent difference is calculated using the values obtained in the first part of the experiment, and the percent difference is obtained. The same procedures are repeated using θ0=10o. Furthermore, those two trials are repeated one time using a different length of a string and another using a different mass. Finally, the procedures are done at θ = 50o.
Results
Small angle approximation
Using a spreadsheet, the following values are obtained, those were calculated every 1o. The values of sin θ, for θ measured in radians for 0 ≤ θ ≤ 45o.
Θ in degrees
Θ in radians
Percent Difference
0.017453293
0.017452406
0.005077009
0.034906585
0.034899497
0.020308653
0.052359878
0.052335956
0.045696789
0.06981317
0.069756474
0.081244509
0.087266463
0.087155743
0.126956143
0.104719755
0.104528463
0.182837258
0.122173048
0.121869343
0.248894656
0.13962634
0.139173101
0.325136376
0.157079633
0.156434465
0.411571693
0.174532925
0.173648178
0.508211115
0.191986218
0.190808995
0.615066387
0.20943951
0.207911691
0.732150485
0.226892803
0.224951054
0.85947762
0.244346095
0.241921896
0.997063235
0.261799388
0.258819045
1.144924001
0.27925268
0.275637356
1.303077823
0.296705973
0.292371705
1.471543832
0.314159265
0.309016994
1.650342388
0.331612558
0.325568154
1.839495076
0.34906585
0.342020143
2.039024705
0.366519143
0.35836795
2.248955309
0.383972435
0.374606593
2.469312139
0.401425728
0.390731128
2.70012167
0.41887902
0.406736643
2.941411589
0.436332313
0.422618262
3.1932108
0.453785606
0.438371147
3.455549418
0.471238898
0.4539905
3.728458767
0.488692191
0.469471563
4.011971379
0.506145483
0.48480962
4.306120986
0.523598776
4.610942522
0.541052068
0.515038075
4.926472116
0.558505361
0.529919264
5.25274709
0.575958653
0.544639035
5.589805953
0.593411946
0.559192903
5.937688399
0.610865238
0.573576436
6.296435299
0.628318531
0.587785252
6.666088699
0.645771823
0.601815023
7.046691814
0.663225116
0.615661475
7.438289019
0.680678408
0.629320391
7.840925847
0.698131701
0.64278761
8.254648983
0.715584993
0.656059029
8.679506251
0.733038286
0.669130606
9.115546613
0.750491578
0.68199836
9.562820158
0.767944871
0.69465837
10.02137809
0.785398163
0.707106781
10.49127274
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