Suppose a pendulum with length L (meters) has angle (radians) from the vertical. It can be shown that as a function of time satisfies the differential equation: d²0 9 + sin 0 = 0 dt2 L where g = 9.8 m/sec/sec is the acceleration due to gravity. For small values of we can use the approximation sin(0) 0, and with that substitution, the differential equation becomes linear. A. Determine the equation of motion of a pendulum with length 0.5 meters and initial angle 0.3 radians and initial angular velocity de/dt 0.3 radians/sec. B. At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer) seconds C. What is the maximum angle (in radians) from vertical?
Suppose a pendulum with length L (meters) has angle (radians) from the vertical. It can be shown that as a function of time satisfies the differential equation: d²0 9 + sin 0 = 0 dt2 L where g = 9.8 m/sec/sec is the acceleration due to gravity. For small values of we can use the approximation sin(0) 0, and with that substitution, the differential equation becomes linear. A. Determine the equation of motion of a pendulum with length 0.5 meters and initial angle 0.3 radians and initial angular velocity de/dt 0.3 radians/sec. B. At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer) seconds C. What is the maximum angle (in radians) from vertical?
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 with length L (meters) has angle (radians) from the vertical. It can be shown that
d²0 9
+ sin = 0
dt² L
where g = 9.8 m/sec/sec is the acceleration due to gravity. For small values of we can use the approximation sin(0)~ 0, and with that substitution, the differential equation becomes
linear.
as a function of time satisfies the differential equation:
A. Determine the equation of motion of a pendulum with length 0.5 meters and initial angle 0.3 radians and initial angular velocity de/dt 0.3 radians/sec.
B. At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer)
seconds
C. What is the maximum angle (in radians) from vertical?
D. How long after reaching its maximum angle until the pendulum reaches maximum deflection in the other direction? (Hint: where is the next critical point?)
seconds.
E. What is the period of the pendulum, that is the time for one swing back and forth?
seconds
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