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To determine the position \( u(t) \) of the mass at any time \( t \), we'll follow the steps below, analyzing the motion of a damped harmonic oscillator.

### Step 1: Establish Equations and Parameters

Given:
- Mass \( m = 16 \, \text{lb} \) (note that \( 1 \, \text{lb} \) is the force unit, so mass in slugs is \( m/32 \, \text{ft/s}^2 \)).
- Damping constant \( c = 4 \, \text{lb} \cdot \text{s/ft} \).
- Spring stretches \( 7 \, \text{in} \) under the weight, which means \( k \) can be determined from this.
- The mass is displaced \( 2 \, \text{in} \) upward from the equilibrium position.
- Initial velocity \( v(0) = 7 \, \text{in/s} \) downward.

### Step 2: Convert Units

Let's convert the given information to consistent units (feet):
- Displacement \( 7 \, \text{in} = \frac{7}{12} \, \text{ft} \).
- Initial displacement \( 2 \, \text{in} = \frac{2}{12} \, \text{ft} \).
- Initial velocity \( v(0) = 7 \, \text{in/s} = \frac{7}{12} \, \text{ft/s} \).

### Step 3: Determine Spring Constant \( k \)

Since the mass \( m = \frac{16}{32} = 0.5 \, \text{slugs} \), we use the equilibrium condition \( mg = kx \), where:
- \( g = 32 \, \text{ft/s}^2 \) (acceleration due to gravity),
- \( x = \frac{7}{12} \, \text{ft} \) (displacement).

Thus:
\[
k = \frac{mg}{x} = \frac{16}{\frac{7}{12}} = \frac{16 \cdot 12}{7} = \frac{192}{7} \approx 27.4286 \, \text{lb/ft}
\]

### Step 4: Write the Differential Equation

The standard differential equation for a damped harmonic oscillator is:
\[
m \frac{d^2u}{dt^2} + c \frac{du}{dt} + ku = 0
\]

Plugging in the values:
\[
0.5 \frac{d^2u}{dt^2} + 4 \frac{du}{dt} + \frac{192}{7} u = 0
\]

Simplifying:
\[
\frac{d^2u}{dt^2} + 8 \frac{du}{dt} + \frac{384}{7} u = 0
\]

### Step 5: Solve the Characteristic Equation

The characteristic equation is:
\[
r^2 + 8r + \frac{384}{7} = 0
\]

Solving for \( r \) using the quadratic formula:
\[
r = \frac{-8 \pm \sqrt{64 - \frac{1536}{7}}}{2}
\]

\[
r = \frac{-8 \pm \sqrt{\frac{448}{7} - \frac{1536}{7}}}{2}
\]

\[
r = \frac{-8 \pm \sqrt{\frac{-1088}{7}}}{2}
\]

\[
r = -4 \pm i\sqrt{\frac{1088}{7}} \Rightarrow r = -4 \pm i\sqrt{155.4286}
\]

### Step 6: General Solution

The general solution for underdamped motion is:
\[
u(t) = e^{-4t} \left( A \cos(\omega t) + B \sin(\omega t) \right)
\]
where \( \omega = \sqrt{155.4286} / 2 \).

Using the given initial conditions \( u(0) = -\frac{2}{12} \, \text{ft} \) and \( v(0) = \frac{7}{12} \, \text{ft/s} \), solve for \( A \) and \( B \).

This will give:
### Result: 
\[
u(t) = e^{-4t} \left( -\frac{1}{6} \cos(\omega t) + \frac{7}{12} \frac{\sin(\omega t)}{\omega} \right)
\]

Do you need details or want clarifications? Here are some relative questions:
1. Why did we convert measurements from inches to feet?
2. How to solve for constants \( A \) and \( B \)?
3. What if damping was doubled?
4. Can energy conservation apply in such damping conditions?
5. Explain steps on damping solutions classification.

**Tip**: Always remember that units must be consistent!

A mass weighing 16 lb stretches a spring 7 in. The mass is attached to a viscous damper with damping constant 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/s^2 as the acceleration due to gravity. Pay close attention to the units.

Solve a problem involving a 16 lb mass attached to a spring with damping, where the spring is stretched and then set into motion. The solution explores differential equations and concepts like damped harmonic oscillators, requiring unit conversion for consistent results. Ideal for college-level physics or engineering students.

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