Suppose a pendulum of length L meters makes an angle of radians with the vertical, as in the figure. It can be shown that as a function of time, satisfies the differential equation where g = 9.8 m/s² is the acceleration due to gravity. For near zero we can use the linear approximation sin(0) ≈ 0 to get a linear differential equation d²0 dt² 0(t) = Use the linear differential equation to answer the following questions. = (a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.1 radians and initial angular velocity = 0.1 radians per second. de dt 9 + sin 0 = 0, L 0.1cos(4.427t)+0.023sin(4.427t) Period= 1.419 d²0 g + 0 = 0. dt² L seconds (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? radians 0
Suppose a pendulum of length L meters makes an angle of radians with the vertical, as in the figure. It can be shown that as a function of time, satisfies the differential equation where g = 9.8 m/s² is the acceleration due to gravity. For near zero we can use the linear approximation sin(0) ≈ 0 to get a linear differential equation d²0 dt² 0(t) = Use the linear differential equation to answer the following questions. = (a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.1 radians and initial angular velocity = 0.1 radians per second. de dt 9 + sin 0 = 0, L 0.1cos(4.427t)+0.023sin(4.427t) Period= 1.419 d²0 g + 0 = 0. dt² L seconds (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? radians 0
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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Transcribed Image Text:Suppose a pendulum of length I meters makes an angle of radians with the vertical, as in the
figure. It can be shown that as a function of time, satisfies the differential equation
d²0 9
dt² L
where g = 9.8 m/s² is the acceleration due to gravity. For near zero we can use the linear
approximation sin(0) ≈ 0 to get a linear differential equation
d²0 9
+ 0 = 0.
dt² L
Use the linear differential equation to answer the following questions.
+ -sin 0 = = 0,
(a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.1
de
radians and initial angular velocity
0.1 radians per second.
dt
e(t) = 0.1cos(4.427t)+0.023sin(4.427t)
Period = 1.419
(b) What is the period of the pendulum? That is, what is the time for one swing back and forth?
seconds
radians
0
L
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