A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e ¯p' cos (@,t-a,). Also, find the undamped position function u(t) = Co cos (@ot – ¤0) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). c = 3, k = 8, ×o = 7, vo = 0 m = ..... x(t) = 14 e -4t -7 e -8t , which means the system is overdamped. (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2n.) The undamped position is u(t) = (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2r.)
A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e ¯p' cos (@,t-a,). Also, find the undamped position function u(t) = Co cos (@ot – ¤0) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). c = 3, k = 8, ×o = 7, vo = 0 m = ..... x(t) = 14 e -4t -7 e -8t , which means the system is overdamped. (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2n.) The undamped position is u(t) = (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or equal to 2r.)
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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![**Title: Analysis of Damped Harmonic Motion in a Spring-Mass System**
**Introduction:**
In this exploration, we analyze the dynamics of a mass-spring system attached to a dashpot (or damper) characterized by specific constants. The study involves identifying the position function x(t) and determining the damping nature of the system—whether it is overdamped, critically damped, or underdamped. Additionally, we calculate the position function for the undamped case and compare the outcomes graphically.
**Given Data:**
- Mass (m) = \( \frac{1}{4} \)
- Damping constant (c) = 3
- Spring constant (k) = 8
- Initial position (\( x_0 \)) = 7
- Initial velocity (\( v_0 \)) = 0
**Problem Statement:**
1. Determine the position function x(t).
2. Classify the system as overdamped, critically damped, or underdamped.
3. For the underdamped case, express the position function as:
\[
x(t) = C_1 e^{-pt} \cos(\omega_1 t - \alpha_1)
\]
4. Calculate the undamped position u(t), assuming the dashpot is disconnected (c = 0), using:
\[
u(t) = C_0 \cos(\omega_0 t - \alpha_0)
\]
5. Illustrate the effects of damping by graphically comparing x(t) and u(t).
**Solution:**
**Position Function:**
The position function is determined as:
\[
x(t) = 14 e^{-4t} - 7 e^{-8t}
\]
The calculation indicates that the system is **overdamped**.
**Undamped Position Function:**
To find the undamped position function, calculate:
\[
u(t) = \boxed{\phantom{u}}
\]
(Note: Box for user input or further calculation needed.)
**Graphical Illustration:**
Create a graph to visually compare the functions x(t) and u(t), highlighting how damping affects motion over time.
**Instructions for Analysis:**
- Use integers or decimals for numerical expressions.
- Round to four decimal places.
- Express angle measures in radians, using values between 0 and \(2\pi\).
**Conclusion:**
This study guides users through understanding different damping](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F811e9e09-f827-40da-a830-8949fa4f6658%2F86b9bf87-11b2-4eb7-aa44-457cd67482db%2Fx37mvlp_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Analysis of Damped Harmonic Motion in a Spring-Mass System**
**Introduction:**
In this exploration, we analyze the dynamics of a mass-spring system attached to a dashpot (or damper) characterized by specific constants. The study involves identifying the position function x(t) and determining the damping nature of the system—whether it is overdamped, critically damped, or underdamped. Additionally, we calculate the position function for the undamped case and compare the outcomes graphically.
**Given Data:**
- Mass (m) = \( \frac{1}{4} \)
- Damping constant (c) = 3
- Spring constant (k) = 8
- Initial position (\( x_0 \)) = 7
- Initial velocity (\( v_0 \)) = 0
**Problem Statement:**
1. Determine the position function x(t).
2. Classify the system as overdamped, critically damped, or underdamped.
3. For the underdamped case, express the position function as:
\[
x(t) = C_1 e^{-pt} \cos(\omega_1 t - \alpha_1)
\]
4. Calculate the undamped position u(t), assuming the dashpot is disconnected (c = 0), using:
\[
u(t) = C_0 \cos(\omega_0 t - \alpha_0)
\]
5. Illustrate the effects of damping by graphically comparing x(t) and u(t).
**Solution:**
**Position Function:**
The position function is determined as:
\[
x(t) = 14 e^{-4t} - 7 e^{-8t}
\]
The calculation indicates that the system is **overdamped**.
**Undamped Position Function:**
To find the undamped position function, calculate:
\[
u(t) = \boxed{\phantom{u}}
\]
(Note: Box for user input or further calculation needed.)
**Graphical Illustration:**
Create a graph to visually compare the functions x(t) and u(t), highlighting how damping affects motion over time.
**Instructions for Analysis:**
- Use integers or decimals for numerical expressions.
- Round to four decimal places.
- Express angle measures in radians, using values between 0 and \(2\pi\).
**Conclusion:**
This study guides users through understanding different damping
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