Suppose a pendulum of length L meters makes an angle of 0 radians with the vertical, as in the figure. It can be shown that as a function of time, 0 satisfies the differential equation 7sin 0 = 0, L° dt² where g = 9.8 m/s² is the acceleration due to gravity. For 0 near zero we can use the linear approximation sin(8) z 0 to get a linear differential equation de 20 = 0. L dt? Use the linear differential equation to answer the following questions. de = 0.2 radians per (a) Determine the equation of motion for a pendulum of length 2 meters having initial angle 0.2 radians and initial angular velocity second. 0(t) = radians (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? Period = seconds
Suppose a pendulum of length L meters makes an angle of 0 radians with the vertical, as in the figure. It can be shown that as a function of time, 0 satisfies the differential equation 7sin 0 = 0, L° dt² where g = 9.8 m/s² is the acceleration due to gravity. For 0 near zero we can use the linear approximation sin(8) z 0 to get a linear differential equation de 20 = 0. L dt? Use the linear differential equation to answer the following questions. de = 0.2 radians per (a) Determine the equation of motion for a pendulum of length 2 meters having initial angle 0.2 radians and initial angular velocity second. 0(t) = radians (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? Period = seconds
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 L meters makes an angle of 0 radians with the vertical, as in the figure. It can be shown that as a function of time, 0 satisfies the
differential equation
+
dt?
sin 0 = 0,
where g = 9.8 m/s is the acceleration due to gravity. For 0 near zero we can use the linear approximation sin(0) z 0 to get a linear differential equation
- 0 = 0.
dt?
Use the linear differential equation to answer the following questions.
do
(a) Determine the equation of motion for a pendulum of length 2 meters having initial angle 0.2 radians and initial angular velocity
0.2 radians per
dt
second.
O(t) =
radians
(b) What is the period of the pendulum? That is, what is the time for one swing back and forth?
Period =
seconds
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