Thinking of this as a mechanical spring-mass system for the position as a function of time ¤ (t), which categories does this system fall under? (there may be multiple) a" + 3x' + 2x = 0 O unforced O overdamped O undamped O forced O under damped O critically damped
Thinking of this as a mechanical spring-mass system for the position as a function of time ¤ (t), which categories does this system fall under? (there may be multiple) a" + 3x' + 2x = 0 O unforced O overdamped O undamped O forced O under damped O critically damped
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Question:**
Thinking of this as a mechanical spring-mass system for the position as a function of time \( x(t) \), which *categories* does this system fall under? *(there may be multiple)*
\[ x'' + 3x' + 2x = 0 \]
**Answer Choices:**
- [ ] unforced
- [ ] overdamped
- [ ] undamped
- [ ] forced
- [ ] underdamped
- [ ] critically damped
---
**Explanation:**
The equation given is a second-order linear homogeneous differential equation representing a damped harmonic oscillator. It can be compared to the standard form:
\[ m x'' + c x' + k x = 0 \]
where \( m \) is mass, \( c \) is damping coefficient, and \( k \) is the spring constant. Analyzing the coefficients helps determine the damping characteristics.
1. **Unforced vs. Forced:**
- The system is unforced, as there is no external force function \( f(t) \) added to the equation.
2. **Damping Analysis:**
- Based on damping, the system could be overdamped, critically damped, or underdamped. Determine this by calculating the discriminant (\( \Delta \)) from the characteristic equation:
Characteristic equation: \( \lambda^2 + 3\lambda + 2 = 0 \)
Solving gives \( \lambda_1 = -1 \) and \( \lambda_2 = -2 \). Both roots are real and distinct.
Since the discriminant (\( (3)^2 - 4 \times 1 \times 2 \)) is positive, the system is **overdamped**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6cee183-215d-43ed-a508-06b044f7e9eb%2F780cbcca-4d6c-4693-ae97-b2c472e88f52%2F2ttkpz_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Thinking of this as a mechanical spring-mass system for the position as a function of time \( x(t) \), which *categories* does this system fall under? *(there may be multiple)*
\[ x'' + 3x' + 2x = 0 \]
**Answer Choices:**
- [ ] unforced
- [ ] overdamped
- [ ] undamped
- [ ] forced
- [ ] underdamped
- [ ] critically damped
---
**Explanation:**
The equation given is a second-order linear homogeneous differential equation representing a damped harmonic oscillator. It can be compared to the standard form:
\[ m x'' + c x' + k x = 0 \]
where \( m \) is mass, \( c \) is damping coefficient, and \( k \) is the spring constant. Analyzing the coefficients helps determine the damping characteristics.
1. **Unforced vs. Forced:**
- The system is unforced, as there is no external force function \( f(t) \) added to the equation.
2. **Damping Analysis:**
- Based on damping, the system could be overdamped, critically damped, or underdamped. Determine this by calculating the discriminant (\( \Delta \)) from the characteristic equation:
Characteristic equation: \( \lambda^2 + 3\lambda + 2 = 0 \)
Solving gives \( \lambda_1 = -1 \) and \( \lambda_2 = -2 \). Both roots are real and distinct.
Since the discriminant (\( (3)^2 - 4 \times 1 \times 2 \)) is positive, the system is **overdamped**.
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