Simple Pendulum

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 sinΘ Percent Difference

1 0.017453293 0.017452406 0.005077009

2 0.034906585 0.034899497 0.020308653

3 0.052359878 0.052335956 0.045696789

4 0.06981317 0.069756474 0.081244509

5 0.087266463 0.087155743 0.126956143

6 0.104719755 0.104528463 0.182837258

7 0.122173048 0.121869343 0.248894656

8 0.13962634 0.139173101 0.325136376

9 0.157079633 0.156434465 0.411571693

10 0.174532925 0.173648178 0.508211115

11 0.191986218 0.190808995 0.615066387

12 0.20943951 0.207911691 0.732150485

13 0.226892803 0.224951054 0.85947762

14 0.244346095 0.241921896 0.997063235

15 0.261799388 0.258819045 1.144924001

16 0.27925268 0.275637356 1.303077823

17 0.296705973 0.292371705 1.471543832

18 0.314159265 0.309016994 1.650342388

19 0.331612558 0.325568154 1.839495076

20 0.34906585 0.342020143 2.039024705

21 0.366519143 0.35836795 2.248955309

22 0.383972435 0.374606593 2.469312139

23 0.401425728 0.390731128 2.70012167

24 0.41887902 0.406736643 2.941411589

25 0.436332313 0.422618262 3.1932108

26 0.453785606 0.438371147 3.455549418

27 0.471238898 0.4539905 3.728458767

28 0.488692191 0.469471563 4.011971379

29 0.506145483 0.48480962 4.306120986

30 0.523598776 0.5 4.610942522

31 0.541052068 0.515038075 4.926472116

32 0.558505361 0.529919264 5.25274709

33 0.575958653 0.544639035 5.589805953

34 0.593411946 0.559192903 5.937688399

35 0.610865238 0.573576436 6.296435299

36 0.628318531 0.587785252 6.666088699

37 0.645771823 0.601815023 7.046691814

38 0.663225116 0.615661475 7.438289019

39 0.680678408 0.629320391 7.840925847

40 0.698131701 0.64278761 8.254648983

41 0.715584993 0.656059029 8.679506251

42 0.733038286 0.669130606 9.115546613

43 0.750491578 0.68199836 9.562820158

44 0.767944871 0.69465837 10.02137809

45 0.785398163 0.707106781 10.49127274

Table