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!