Oscillatory motion

Oscillatory motion - Oscillatory Motion Abstract In the...

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Oscillatory Motion Abstract In the study of Oscillatory motion, or periodic motion, experiments were performed by means of a falling body, a simple pendulum, an inertial balance, and by pendulum motion of walking. In the acceleration of a falling body experiment, the average gravitational acceleration was found to be 9.75 m/s 2 , compared to the established value of 9.80 m/s 2 . To find the gravitational acceleration using a simple pendulum, two different bobs of varying lengths were used. The small bob was found to have a gravitational acceleration of 9.98 m/s 2 while the gravitational acceleration of the large bob was 10.03 m/s 2 . Using an inertial balance and different masses, the spring constant was calculated to be 55721.12 g/s 2 . Using the spring constant, the mass of the tray and an unknown mass were found to be 115.82 g and 289.68 g, respectively. In the experiment involving the pendulum motion of walking, the mean period was found to be 1.25 s and the value of α for the arms and legs were 0.12 and 0.09, respectively. Introduction and Theory Experiment 1 A free falling body is an object that is moving under the influence of gravity only. These objects fall in the vertical direction without any horizontal component. The acceleration, due to gravity, can be determined by the equation: a n = (v n+1 – v n )/[(1/2)(t n+1 + t n )] 1
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where a is the acceleration, v is the velocity, and t is the time. Experiment 2 A simple pendulum is ideally nothing more then a point mass, a bob, suspended by a massless string of a particular length, which exhibit simple harmonic motion. Simple harmonic motion can be summed up by Hooke’s law. Hooke’s law, F = - k x,   is any system where a restoring force is directly proportional to the displacement from the equilibrium position. In the equation, k is the spring constant and x is the distance from the equilibrium position of the spring. When using a simple pendulum, Hooke’s law can be applied by the equation Equation 4.1 1 where x = L sinθ is the displacement of the bob from the equilibrium position. The small angle approximation in which sinθ=θ can be applied if the angle, θ, is less then about 20°. The previous equation could then be simplified to Equation 4.2 1 where k = mg L. Since the bob at the end of the simple pendulum has rotational motion, torque can then be applied
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Equation 4.3 1 where α is angular acceleration and I is the moment of inertia. The angular acceleration is equal to –(A/L)ω 2 cosωt, where ω is angular velocity. The displacement can also be defined as θ=(A/L)cosωt. From this the equation becomes Equation 4.4 1 When solving for angular velocity, ω, the equation is Equation 4.5 1 and for T , the period, the equation is rearranged to Equation 4.6 1 As the moment of inertia for a simple pendulum is I = mL 2 , the expression for period becomes
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Equation 4.7 1 In order to observe the change in the gravitational constant due to the change of period, T ,and length, L , the previous equation must be rearranged to give linear form. By squaring both sides
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Oscillatory motion - Oscillatory Motion Abstract In the...

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