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,

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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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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