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To solve this problem correctly, let's go through each step in setting up the differential equation and the conditions, then check if the given solution \( u(t) = 0.2607 e^{-4t} \cos(6.235t + 4.018) \) matches what we would expect.

### 1. **Setting up the problem**:
   - **Mass and Weight**: The mass weighs \( 16 \, \text{lb} \), which we can use to find the mass in slugs since:
     \[
     \text{mass (slugs)} = \frac{\text{weight}}{g} = \frac{16}{32} = 0.5 \, \text{slugs}
     \]
   - **Spring Constant (k)**: The weight stretches the spring 7 inches, or \( \frac{7}{12} \) feet. Since \( mg = k \cdot \text{displacement} \), we have:
     \[
     k = \frac{16}{7/12} = \frac{16 \cdot 12}{7} = \frac{192}{7} \approx 27.43 \, \text{lb/ft}
     \]
   - **Damping Constant (c)**: Given as \( 4 \, \text{lb} \cdot \text{s/ft} \).

### 2. **Formulating the Differential Equation**:
   Using Newton’s second law with damping and spring force, we have:
   \[
   m \frac{d^2u}{dt^2} + c \frac{du}{dt} + k u = 0
   \]
   Plugging in the values:
   \[
   0.5 \frac{d^2u}{dt^2} + 4 \frac{du}{dt} + \frac{192}{7} u = 0
   \]
   Simplifying:
   \[
   \frac{d^2u}{dt^2} + 8 \frac{du}{dt} + \frac{384}{7} u = 0
   \]

### 3. **Solving the Characteristic Equation**:
   The characteristic equation is:
   \[
   r^2 + 8r + \frac{384}{7} = 0
   \]
   Solving this quadratic equation will provide the roots, which indicate whether the solution is underdamped, overdamped, or critically damped.

Since the answer provided is an underdamped oscillation (indicated by the cosine term), we should verify if this aligns with our calculations.

### Verifying the Given Answer

The general solution for an underdamped system is:
\[
u(t) = e^{-\alpha t} (C_1 \cos(\omega t) + C_2 \sin(\omega t))
\]
where:
- \( \alpha = 4 \) is the damping coefficient (from \( e^{-4t} \)).
- The angular frequency \( \omega = 6.235 \) rad/s and phase \( 4.018 \) rad are implied in the cosine term.

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If you'd like, I can further verify the solution details, including solving for constants based on initial conditions. Let me know!

Would you like more details or have any questions?

### Related Questions:
1. How do we find the spring constant in such problems?
2. Why is damping important in oscillatory motion?
3. What happens in critically damped versus underdamped systems?
4. How can we verify the initial conditions in this context?
5. What is the physical interpretation of the phase shift in this solution?

### Tip:
Always double-check units for consistency, especially when working across different unit systems (like inches and feet).

A mass weighing 16 lb stretches a spring 7 in. The mass is attached to a viscous damper with a damping constant of 4 lb·s/ft. The mass is pushed upward, contracting the spring a distance of 2 in, and then set into motion with a downward velocity of 7 in/s. Determine the position u of the mass at any time t. Use 32 ft/s² as the acceleration due to gravity. Pay close attention to the units.

Explore how to solve for the position of a mass in a spring-mass-damper system subjected to damping and an initial displacement and velocity. This problem covers differential equations and key concepts in oscillatory motion with damping, suitable for university-level students.

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