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A pendulum consists of a mass m attached to the end of a rod of length L. The pendulum is displaced from its equilibrium position by an angle of radians. Assume no air resistance or friction in the motion of the pendulum. See figure below. Ө Fill in the blank to write a differential equation to model the motion of the pendulum for any angle 0. d²0 dt² = Is this differential equation linear or nonlinear? nonlinear linear πT πT Now, assume the angle is small (that is,

A pendulum consists of a mass m attached to the end of a rod of length L. The pendulum is displaced from its equilibrium position by an angle of radians. Assume no air resistance or friction in the motion of the pendulum. See figure below. Ө Fill in the blank to write a differential equation to model the motion of the pendulum for any angle 0. d²0 dt² = Is this differential equation linear or nonlinear? nonlinear linear πT πT Now, assume the angle is small (that is,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A pendulum consists of a mass m attached to the end of a rod of length L. The pendulum is displaced from its equilibrium position by an angle of radians. Assume no air resistance or friction in the motion of the pendulum. See figure below. Ө Fill in the blank to write a differential equation to model the motion of the pendulum for any angle 0. d²0 dt² = Is this differential equation linear or nonlinear? nonlinear linear πT πT Now, assume the angle is small (that is, <O < 12 radians or -15° < 0 ≤ 15º), and make an appropriate substitution to simplify the equation above. 12 d²0 dt²
Transcribed Image Text:A pendulum consists of a mass m attached to the end of a rod of length L. The pendulum is displaced from its equilibrium position by an angle of radians. Assume no air resistance or friction in the motion of the pendulum. See figure below. Ө Fill in the blank to write a differential equation to model the motion of the pendulum for any angle 0. d²0 dt² = Is this differential equation linear or nonlinear? nonlinear linear πT πT Now, assume the angle is small (that is, <O < 12 radians or -15° < 0 ≤ 15º), and make an appropriate substitution to simplify the equation above. 12 d²0 dt²
Is this differential equation linear or nonlinear? nonlinear linear Now, assume the angle is small (that is, πT πT <o 12 12 radians or -15° ≤ 0 ≤ 15º), and make an appropriate substitution to simplify the equation above. d²0 dt² Is this differential equation linear or nonlinear? linear nonlinear Let (t) represent the angular displacement of the pendulum over time. Assume 0(0) = 00 is the initial displacement and that this initial displacement is a small angle and thus the small angle approximation is appropriate. Write the general solution for the pendulum's angular displacement assuming the small angle approximation (with w for angular speed) and write the period T of the oscillation with the small angle approximation in terms of and L. The general solution is: (t) = g The period of the oscillation is T =
Transcribed Image Text:Is this differential equation linear or nonlinear? nonlinear linear Now, assume the angle is small (that is, πT πT <o 12 12 radians or -15° ≤ 0 ≤ 15º), and make an appropriate substitution to simplify the equation above. d²0 dt² Is this differential equation linear or nonlinear? linear nonlinear Let (t) represent the angular displacement of the pendulum over time. Assume 0(0) = 00 is the initial displacement and that this initial displacement is a small angle and thus the small angle approximation is appropriate. Write the general solution for the pendulum's angular displacement assuming the small angle approximation (with w for angular speed) and write the period T of the oscillation with the small angle approximation in terms of and L. The general solution is: (t) = g The period of the oscillation is T =
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