A mass weighting 56 lbs stretches a spring 3 inches. The mass is in a medium that exerts a viscous resistance of 207 lbs when the mass has a velocity of 6 ft/sec. Suppose the object is displaced an additional 3 inches and released. Find an equation for the object's displacement, u(t), in feet after t seconds.
A mass weighting 56 lbs stretches a spring 3 inches. The mass is in a medium that exerts a viscous resistance of 207 lbs when the mass has a velocity of 6 ft/sec. Suppose the object is displaced an additional 3 inches and released. Find an equation for the object's displacement, u(t), in feet after t seconds.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Problem Statement:**
A mass weighing 56 lbs stretches a spring 3 inches. The mass is in a medium that exerts a viscous resistance of 207 lbs when the mass has a velocity of 6 ft/sec.
Suppose the object is displaced an additional 3 inches and released.
Find an equation for the object's displacement, \( u(t) \), in feet after \( t \) seconds.
\[ u(t) = \quad \text{\_\_\_\_\_\_\_\_\_\_\_\_} \]
**Explanation:**
This problem involves an object connected to a spring within a medium that provides viscous resistance. This forms a classic physics problem often modeled by differential equations.
- **Given Data:**
- Weight of the mass: 56 lbs
- Stretch caused by mass: 3 inches
- Viscous resistance: 207 lbs at 6 ft/sec
- **Assumptions:**
- The spring follows Hooke's Law.
- The medium provides linear viscous resistance.
**Steps to Solve:**
1. **Convert Units:**
- Convert the 3 inches stretch to feet (3 inches = 0.25 feet).
2. **Determine Parameters:**
- Compute the spring constant \( k \) using Hooke’s Law: \( F = kx \)
\( 56 = k \times 0.25 \)
\( k = 224 \) lbs/ft
- Compute the damping coefficient \( c \) using the given viscous resistance:
\( c \times 6 = 207 \)
\( c = 34.5 \) lbs-sec/ft
3. **Determine the Displacement Equation:**
- The differential equation modeling the system is:
\( m \frac{d^2 u}{dt^2} + c \frac{du}{dt} + k u = 0 \)
Here, \( m = \frac{56}{32} = 1.75 \) slugs (using \( g = 32 \) ft/sec\(^2\))
- Plug in the values:
\( 1.75 \frac{d^2 u}{dt^2} + 34.5 \frac{du}{dt} + 224 u = 0 \)
- Solve the differential equation to find \( u(t) \).
**Note](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc700bc18-61e4-4e04-a31d-52f10b21c2d0%2F12a269b0-ee48-481d-94c0-66491406d4de%2Fg1ayjp8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A mass weighing 56 lbs stretches a spring 3 inches. The mass is in a medium that exerts a viscous resistance of 207 lbs when the mass has a velocity of 6 ft/sec.
Suppose the object is displaced an additional 3 inches and released.
Find an equation for the object's displacement, \( u(t) \), in feet after \( t \) seconds.
\[ u(t) = \quad \text{\_\_\_\_\_\_\_\_\_\_\_\_} \]
**Explanation:**
This problem involves an object connected to a spring within a medium that provides viscous resistance. This forms a classic physics problem often modeled by differential equations.
- **Given Data:**
- Weight of the mass: 56 lbs
- Stretch caused by mass: 3 inches
- Viscous resistance: 207 lbs at 6 ft/sec
- **Assumptions:**
- The spring follows Hooke's Law.
- The medium provides linear viscous resistance.
**Steps to Solve:**
1. **Convert Units:**
- Convert the 3 inches stretch to feet (3 inches = 0.25 feet).
2. **Determine Parameters:**
- Compute the spring constant \( k \) using Hooke’s Law: \( F = kx \)
\( 56 = k \times 0.25 \)
\( k = 224 \) lbs/ft
- Compute the damping coefficient \( c \) using the given viscous resistance:
\( c \times 6 = 207 \)
\( c = 34.5 \) lbs-sec/ft
3. **Determine the Displacement Equation:**
- The differential equation modeling the system is:
\( m \frac{d^2 u}{dt^2} + c \frac{du}{dt} + k u = 0 \)
Here, \( m = \frac{56}{32} = 1.75 \) slugs (using \( g = 32 \) ft/sec\(^2\))
- Plug in the values:
\( 1.75 \frac{d^2 u}{dt^2} + 34.5 \frac{du}{dt} + 224 u = 0 \)
- Solve the differential equation to find \( u(t) \).
**Note
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