PHYS_0212_Oscillatory_Motion_fall_2007

# PHYS_0212_Oscillatory_Motion_fall_2007 - Oscillatory Motion...

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Oscillatory Motion Abstract Oscillatory motion was studied by measuring the gravitational acceleration of a falling body, determining the gravitational acceleration of a simple pendulum by observing the period of the pendulum at various lengths, determining the spring constant of an inertial balance by observing the period at different masses, and by measuring the period of the pendulum motion of arms and legs during walking and relating them to their moments of inertia. The average gravitational acceleration of the free falling body was found to be 9.7 m/s. The simple pendulum experiment involved using a small bob and a large bob. The average gravitational acceleration of the simple pendulum with the small bob was 9.9 m/s 2 , and 9.5 m/s 2 with the large bob. The spring constant of the inertial balance was 60661.4 g/s 2 . The pendulum of the motion of arms and legs was 1.02 s. The moment of inertia of the arms was 0.69 and for the legs was 0.58. Introduction and Theory A free falling body falls in the vertical direction without any direction in the horizontal direction; therefore, is has an acceleration due to gravity that can be found by the equation a n = (v n+1 - v n ) / ((1/2)(T n+1 + T n )) 1 where a is the acceleration, v is the velocity, and T is the time. A simple pendulum consists of a bob (a point mass) suspended by a string of a particular length which exhibits simple harmonic motion. Simple harmonic motion is found when a restoring force is proportional to the displacement from the equilibrium position which is known as Hooke’s law. Hooke’s law can be applied to the simple pendulum by the equation

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τ = -(mg)x = -(mg)Lsin θ Equation 4.1 2 where x=Lsin θ is the displacement of the bob from the equilibrium position. If the angle is small, then we can apply the small angle approximation which would approximate sin θ = θ . This allows for the equation to be simplified by τ = -mgL θ = -k θ Equation 4.2 2 where k = mgL. Torque, in addition, is equal to τ = I α = − mgL θ Equation 4.3 2 where I is the moment of inertia and α is the angular acceleration. The angular acceleration = -(A/L) ω 2 cos ω t, where ω is the angular velocity. θ is also defined as θ = (A/L)cos ω t. I(-(A/L) ω 2 cos ω t) = -mgL((A/L)cos ω t) Equation 4.4 2 Solving for angular velocity, the equation is ω = (mgL/I) = 2 п f = 2 п /T Equation 4.5 2 and rearranging for the period gives T = 2 п (I/mgL) Equation 4.6 2 To determine the gravitational acceleration of the simple pendulum, the period of the pendulum will be observed at various lengths. The period of a simple pendulum is
T = 2 п (L/g) Equation 4.7 2 because the moment of inertia is I = mL 2 and a simple pendulum’s period depends only on its length. The relationship between T and L are not linear. If both sides of equation 4.7 are squared, L and T have a linear form. T

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## PHYS_0212_Oscillatory_Motion_fall_2007 - Oscillatory Motion...

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