# The Simple Pendulum - City University of New York The The Simple Simple Pendulum Pendulum Students Sandra Foster Alex ProtyagovSusan Bethel-Jame Presented by Alex Protyagov SC 441 Honors Class Fall 2001, Dr. Roman Kezerashvili Objectives Objectives of of the the experiment: experiment: To investigate the properties of a pendulum To make an experimental determination of the acceleration due to gravity. The The theory theory The motion of a simple pendulum is periodic motion. The mass starts at point A, moves to point B where this mass starts moving in opposite direction, back to point A. For the simple pendulum all the mass is considered to be concentrated at the L center. The vibration of the simple pendulum is not (in general) a SHM, but approaches SHM when the amplitude of the vibration is small compared to the length of the pendulum. Pivot point m m B

m A Here length, L of the pendulum is distance from a pivot point to the center of a bob. is the angular displacement through which the pendulum swings and m is mass of the pendulum bob. The time, which is required for one complete vibration of oscillation, from position A to position B & back to A is period of the pendulum, T. There are two forces acting on the bob: weight W=mg & tension T in the string. When the string makes an angle with the vertical, the weight has the components mgcos, along the string, and mgsin perpendicular to the string. L The bob oscillates because there is restoring force. This force is resultant of the gravity and the torsion in the string. The restoring force F is the tangential component 1 of the net force: F=-mgsin T x m m F=mgsin S W=m g mgcos From math we know than the sin when the angle is small. Therefore 2 F=-mg From figure & the definition of an angle in radians (= arc length/radius) 3

s L Here s is path length followed by the bob. This path s is curved, but when pretty small, the arc length s is approximately equal to a chord x. Therefore 4 x L 5 ewtons second law is applied to the motion of the pendulum bob, we get: 6 mg F ma L d 7 x 2 g a 2 L x g 2 x x where Acceleration is also L d t2 this Equation 8 shows that the small oscillations of a simple pendulum can be considered as simple harmonic motion, The constant represents the angular frequency of simple harmonic motion and thus the period of oscillation of the simple pendulum is 8 1

0 2 T Henc e 9 1 1 L T 2 g The period is independent of the bobs mass. Really pendulum will have some depends on the amplitude and therefore it may be shown as 1 2 T 2 L g 1 1 4 sin 2 9 2 64 sin 4 2 .... T 2 L g This series is going smaller as in continues.

n=0 sin 2 2n ( 4n ) 1 n 2n The pendulum can be used as a very simple device to measure the acceleration due to gravity at a particular location. We may use equation 1 to find g. Solving for g we obtain: 2 4 L 1 0 g T 2 The period dependence on length Period T, s Length L, m % error for T Square of period T2, s2 g from equatio n 13, m/s2 Experi mental period T, s Theoret ical period T from

eq 11, s 0.30 1.095 1.099 0.39 1.20 9.88 0.40 1.275 1.269 0.44 1.63 9.71 0.60 1.557 1.555 0.15 2.42 9.77 0.80 1.786 1.795 0.51 3.19 9.90 1.00 2.008 2.007 0.05 4.03

9.79 Mean value for g from equatio n 13, m/ s2 g from equatio n 14, m/ s2 Standar d value of g, m/ s2 % error for g from equatio n 13 % error for g from equatio n 14 9.811 9.834 9.807 0.04 0.28 The period vs. length The length vs. square of period y =2.0054x0.4995 1.00 2.00 1.50 1.00 0.50 0.00 0.00 y =0.2491x - 0.0012 1.20 The length L, m

The period T, s 2.50 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 The l ength L, m 1.00 1.20 0.00 1.00 2.00 3.00 Square of period T 2 , s 2 4.00 5.00 The small angle approximation Period T, s Theoretical Experimental Angle , value for value for degree period T period from eq 12, T, s s 10.0 20.0 30.0 40.0 50.0 60.0

1.095 1.097 1.104 1.120 1.135 1.162 1.101 1.107 1.118 1.133 1.153 1.179 1 2 % error for T T 2 L g n=0 0.55 0.94 1.26 1.18 1.60 1.42 The period dependence on mass Period T, s Theoretic Experiment Length L , Mass of a bob al value al value for m m , kg for period period T from eq T, s 11, s 0.30 0.068 0.024 1.095 1.077 1.099 1.099

% error for T % difference for T 0.39 2.03 0.197 1.025 sin 2 2n 2n ( 4n ) 1 n Summary At this experiment we found out that the period of the simple pendulum is independent from the mass of the bob. At the first part of the experiment we find acceleration due to gravity by different ways, we compare our results with standard value, found the percent error and those numbers were pretty small. We plotted two graphs; there are The period vs. length &The length vs. square of period. The first graph looks like square root, it is understandable, it is following from equation 11th. The second graph looks like straight line, which is following from equation 11th too. As we had seen at the second part of the experiment, for small angles, smaller 10 degree, we may neglect of sin and we may use formula 11th to find the period. But for angles, which are greater formula 11 is not as good as for small angles. It is follow from second table. When we increase the angle the percent error is increasing too. Therefore for big angles is better to use formula 12. Thank you for your attention