In the small angle approximation, the total energy of a simple pendulum is 1 E = Show analytically that E monotonically increases with time when the Euler method is used to compute the motion. Do this by using the Euler algorithm to write wn+1 and fn+1 in terms of wn and On so you can related En+1 to En.
In the small angle approximation, the total energy of a simple pendulum is 1 E = Show analytically that E monotonically increases with time when the Euler method is used to compute the motion. Do this by using the Euler algorithm to write wn+1 and fn+1 in terms of wn and On so you can related En+1 to En.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![In the small angle approximation, the total energy of a simple pendulum is
\[ E = \frac{1}{2} m \ell^2 \omega^2 + \frac{1}{2} mg \ell \theta^2 - mg \ell . \]
Show analytically that \( E \) monotonically increases with time when the Euler method is used to compute the motion. Do this by using the Euler algorithm to write \( \omega_{n+1} \) and \( \theta_{n+1} \) in terms of \( \omega_n \) and \( \theta_n \), so you can relate \( E_{n+1} \) to \( E_n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde0f25ae-2205-4929-95df-5a1976df6d11%2F95a1f158-81c4-4029-87f9-09153cb9c0b6%2Fajlniia_processed.png&w=3840&q=75)
Transcribed Image Text:In the small angle approximation, the total energy of a simple pendulum is
\[ E = \frac{1}{2} m \ell^2 \omega^2 + \frac{1}{2} mg \ell \theta^2 - mg \ell . \]
Show analytically that \( E \) monotonically increases with time when the Euler method is used to compute the motion. Do this by using the Euler algorithm to write \( \omega_{n+1} \) and \( \theta_{n+1} \) in terms of \( \omega_n \) and \( \theta_n \), so you can relate \( E_{n+1} \) to \( E_n \).
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