8. Will an ideal pendulum clock that keeps perfect time on earth run slow or fast on the moon?
8. Will an ideal pendulum clock that keeps perfect time on earth run slow or fast on the moon?
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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Hi, The textbook is on the website here but there is no help for this question. I am interested in #8 and how to solve whether it is slower or faster using the
![In Exercises 4-8, we continue the study of the ideal pendulum system with bob mass \( m \) and arm length \( l \) given by
\[
\frac{d\theta}{dt} = v
\]
\[
\frac{dv}{dt} = -\frac{g}{l} \sin \theta.
\]
4.
(a) What is the linearization of the ideal pendulum system above at the equilibrium point \( (0, 0) \)?
(b) Using \( g = 9.8 \, \text{m/s}^2 \), how should \( l \) and \( m \) be chosen so that small swings of the pendulum have period 1 second?
5. For the linearization of the ideal pendulum above at \( (0, 0) \), the period of the oscillation is independent of the amplitude. Does the same statement hold for the ideal pendulum itself? Is the period of oscillation the same no matter how high the ideal pendulum swings? If not, will the period be shorter or longer for high swings?
6. An ideal pendulum clock—a clock containing an ideal pendulum that "ticks" once for each swing of the pendulum arm—keeps perfect time when the pendulum makes very high swings. Will the clock run slow or fast if the amplitude of the swings is very small?
7.
(a) If the arm length of the ideal pendulum is doubled from \( l \) to \( 2l \), what is the effect on the period of small amplitude swinging solutions?
(b) What is the rate of change of the period of small amplitude swings as \( l \) is varied?
8. Will an ideal pendulum clock that keeps perfect time on earth run slow or fast on the moon?
---
*Note: No graphs or diagrams need to be explained for this content.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37ca8490-5870-44f0-88e4-a9f89a6f9c05%2F195ffa54-79ee-4187-b883-4ee69c94bf7c%2Fd1x5spv_processed.png&w=3840&q=75)
Transcribed Image Text:In Exercises 4-8, we continue the study of the ideal pendulum system with bob mass \( m \) and arm length \( l \) given by
\[
\frac{d\theta}{dt} = v
\]
\[
\frac{dv}{dt} = -\frac{g}{l} \sin \theta.
\]
4.
(a) What is the linearization of the ideal pendulum system above at the equilibrium point \( (0, 0) \)?
(b) Using \( g = 9.8 \, \text{m/s}^2 \), how should \( l \) and \( m \) be chosen so that small swings of the pendulum have period 1 second?
5. For the linearization of the ideal pendulum above at \( (0, 0) \), the period of the oscillation is independent of the amplitude. Does the same statement hold for the ideal pendulum itself? Is the period of oscillation the same no matter how high the ideal pendulum swings? If not, will the period be shorter or longer for high swings?
6. An ideal pendulum clock—a clock containing an ideal pendulum that "ticks" once for each swing of the pendulum arm—keeps perfect time when the pendulum makes very high swings. Will the clock run slow or fast if the amplitude of the swings is very small?
7.
(a) If the arm length of the ideal pendulum is doubled from \( l \) to \( 2l \), what is the effect on the period of small amplitude swinging solutions?
(b) What is the rate of change of the period of small amplitude swings as \( l \) is varied?
8. Will an ideal pendulum clock that keeps perfect time on earth run slow or fast on the moon?
---
*Note: No graphs or diagrams need to be explained for this content.*
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