**Title: Oscillatory Motion of a Spring-Mass System** A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 3 inches below the equilibrium position. **(a) Determining the Position of the Mass** Find the position \( x \) of the mass at the specified times: \( t = \pi/12, \pi/8, \pi/6, \pi/4, \) and \( 9\pi/32 \) seconds. (Use \( g = 32 \, \text{ft/s}^2 \) for the acceleration due to gravity.) - \( x(\pi/12) = -0.125 \, \text{ft} \) ✔️ - \( x(\pi/8) = -0.25 \, \text{ft} \) ✔️ - \( x(\pi/6) = -0.125 \, \text{ft} \) ✔️ - \( x(\pi/4) = 0.25 \, \text{ft} \) ✔️ - \( x(9\pi/32) = \frac{1}{4\sqrt{2}} \, \text{ft} \) ✔️ **(b) Velocity of the Mass** - What is the velocity of the mass when \( t = 3\pi/16 \, \text{s}? \) \[\_\_\_\_\_\_\_\_\_ \, \text{ft/s}\] - In which direction is the mass heading at this instant? - [ ] downward - [ ] upward **(c) Equilibrium Position** At what times does the mass pass through the equilibrium position? \[ n = 0, 1, 2, \ldots \] --- **Explanation:** This problem involves analyzing the simple harmonic motion of a mass-spring system. Each part of the task aims to deduce different aspects of this motion, including the position and velocity at specific times, as well as moments when the system passes through equilibrium.
**Title: Oscillatory Motion of a Spring-Mass System** A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 3 inches below the equilibrium position. **(a) Determining the Position of the Mass** Find the position \( x \) of the mass at the specified times: \( t = \pi/12, \pi/8, \pi/6, \pi/4, \) and \( 9\pi/32 \) seconds. (Use \( g = 32 \, \text{ft/s}^2 \) for the acceleration due to gravity.) - \( x(\pi/12) = -0.125 \, \text{ft} \) ✔️ - \( x(\pi/8) = -0.25 \, \text{ft} \) ✔️ - \( x(\pi/6) = -0.125 \, \text{ft} \) ✔️ - \( x(\pi/4) = 0.25 \, \text{ft} \) ✔️ - \( x(9\pi/32) = \frac{1}{4\sqrt{2}} \, \text{ft} \) ✔️ **(b) Velocity of the Mass** - What is the velocity of the mass when \( t = 3\pi/16 \, \text{s}? \) \[\_\_\_\_\_\_\_\_\_ \, \text{ft/s}\] - In which direction is the mass heading at this instant? - [ ] downward - [ ] upward **(c) Equilibrium Position** At what times does the mass pass through the equilibrium position? \[ n = 0, 1, 2, \ldots \] --- **Explanation:** This problem involves analyzing the simple harmonic motion of a mass-spring system. Each part of the task aims to deduce different aspects of this motion, including the position and velocity at specific times, as well as moments when the system passes through equilibrium.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Title: Oscillatory Motion of a Spring-Mass System**
A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 3 inches below the equilibrium position.
**(a) Determining the Position of the Mass**
Find the position \( x \) of the mass at the specified times: \( t = \pi/12, \pi/8, \pi/6, \pi/4, \) and \( 9\pi/32 \) seconds. (Use \( g = 32 \, \text{ft/s}^2 \) for the acceleration due to gravity.)
- \( x(\pi/12) = -0.125 \, \text{ft} \) ✔️
- \( x(\pi/8) = -0.25 \, \text{ft} \) ✔️
- \( x(\pi/6) = -0.125 \, \text{ft} \) ✔️
- \( x(\pi/4) = 0.25 \, \text{ft} \) ✔️
- \( x(9\pi/32) = \frac{1}{4\sqrt{2}} \, \text{ft} \) ✔️
**(b) Velocity of the Mass**
- What is the velocity of the mass when \( t = 3\pi/16 \, \text{s}? \)
\[\_\_\_\_\_\_\_\_\_ \, \text{ft/s}\]
- In which direction is the mass heading at this instant?
- [ ] downward
- [ ] upward
**(c) Equilibrium Position**
At what times does the mass pass through the equilibrium position?
\[ n = 0, 1, 2, \ldots \]
---
**Explanation:**
This problem involves analyzing the simple harmonic motion of a mass-spring system. Each part of the task aims to deduce different aspects of this motion, including the position and velocity at specific times, as well as moments when the system passes through equilibrium.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa103160f-6870-471c-8ca1-5357643d2f88%2F47b87abf-7e74-46ef-8446-4fd9d74e350b%2Ftjqyecp_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Oscillatory Motion of a Spring-Mass System**
A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 3 inches below the equilibrium position.
**(a) Determining the Position of the Mass**
Find the position \( x \) of the mass at the specified times: \( t = \pi/12, \pi/8, \pi/6, \pi/4, \) and \( 9\pi/32 \) seconds. (Use \( g = 32 \, \text{ft/s}^2 \) for the acceleration due to gravity.)
- \( x(\pi/12) = -0.125 \, \text{ft} \) ✔️
- \( x(\pi/8) = -0.25 \, \text{ft} \) ✔️
- \( x(\pi/6) = -0.125 \, \text{ft} \) ✔️
- \( x(\pi/4) = 0.25 \, \text{ft} \) ✔️
- \( x(9\pi/32) = \frac{1}{4\sqrt{2}} \, \text{ft} \) ✔️
**(b) Velocity of the Mass**
- What is the velocity of the mass when \( t = 3\pi/16 \, \text{s}? \)
\[\_\_\_\_\_\_\_\_\_ \, \text{ft/s}\]
- In which direction is the mass heading at this instant?
- [ ] downward
- [ ] upward
**(c) Equilibrium Position**
At what times does the mass pass through the equilibrium position?
\[ n = 0, 1, 2, \ldots \]
---
**Explanation:**
This problem involves analyzing the simple harmonic motion of a mass-spring system. Each part of the task aims to deduce different aspects of this motion, including the position and velocity at specific times, as well as moments when the system passes through equilibrium.
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