# SPENDLAB - UTC Physics 183 Simple Pendulum THE SIMPLE...

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UTC Physics 183 – Simple Pendulum THE SIMPLE PENDULUM Objective: To investigate the relationship between the length of a simple pendulum and the period of its motion. Apparatus: String, pendulum bob, stopwatch, meter stick, logarithmic graph paper, a computer with ULI interface, and a photogate. Theory: A simple pendulum consists of a small bob of mass m suspended by a light (massless) string of length L fixed at its upper end. When released from point P, the mass swings through point O and to a point of equal height to point P, and back to point P over the same path (Figure 1). The force that restores the pendulum to its equilibrium position, point O, after it is disturbed is the component of the weight of the bob, mg, that is tangential to the circular path, denoted as F t : F t = - m g sin θ (eq. 1), where the negative sign denotes acceleration due to gravity is in the downward direction. For small angles, the value for the angle θ (measured in radians) is equal to the sine of that angle, or sin θ = θ . Substituting into equation 1 yields F t = - m g θ . Furthermore, the arc length that the bob swings through in traveling between points O and P is denoted as S and is related to the length of the string L (or radius of circular motion) by θ = S / L. Thus, S L mg F t = (eq. 2). For small angles, the arc length is adequately approximated by the straight line distance between points O and P. The period, T, of an object in simple harmonic motion is defined as the time for one complete cycle. For the simple pendulum, this is specifically given by: g L T π 2 = or 2 1 2 L g T = (eq. 3), where g is the acceleration due to gravity, 9.8 m/s 2 . Equation 3 indicates that the period and length of the pendulum are directly proportional; that is, as the length, L, of a pendulum is increased, so will its period, T, increase. However, it is not a linear relationship. The period increases as the square root of the length. Thus, if the length of a pendulum is increased by a factor of 4, the period is only doubled. This is a logarithmic relationship. A more general form of equation 3 is: T = k L n (eq. 4), 1

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UTC Physics 183 – Simple Pendulum where k = g π 2 and the exponent n is ½. Rearranging this equation yields: 2 2 4 k g = (eq. 5). Taking the log
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## This note was uploaded on 10/25/2010 for the course FOSEE CVL1040 taught by Professor None during the Spring '09 term at Multimedia University, Cyberjaya.

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SPENDLAB - UTC Physics 183 Simple Pendulum THE SIMPLE...

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