To solve this problem, let's analyze the information given:

1. Mass and Weight: The mass weighs $3 \, \text{lb}$. Since weight $W = mg$, where $g = 32 \, \text{ft/s}^2$, we can determine the mass $m$ in slugs. $m = \frac{W}{g} = \frac{3 \, \text{lb}}{32 \, \text{ft/s}^2} = \frac{3}{32} \, \text{slugs}$

2. Spring Constant: The spring constant $k = 8 \, \text{lb/in}$. We should convert this to consistent units, such as pounds per foot. Since there are 12 inches in a foot: $k = 8 \times 12 = 96 \, \text{lb/ft}$

3. External Force: The system experiences an external force $F(t) = 8 \cos(5t)$ lb.

4. Equation of Motion: For an undamped spring-mass system under external forcing, the equation of motion is: $m \frac{d^2 u}{dt^2} + k u = F(t)$ Substituting $m = \frac{3}{32}$ slugs, $k = 96 \, \text{lb/ft}$, and $F(t) = 8 \cos(5t)$: $\frac{3}{32} \frac{d^2 u}{dt^2} + 96 u = 8 \cos(5t)$

5. Simplifying the Equation: Multiply through by $32/3$ to clear the fraction: $\frac{d^2 u}{dt^2} + 1024 u = \frac{256}{3} \cos(5t)$

Now, we solve this non-homogeneous differential equation with the form: $\frac{d^2 u}{dt^2} + 1024 u = \frac{256}{3} \cos(5t)$

Solution Steps:

1. Solve the Homogeneous Equation: $\frac{d^2 u}{dt^2} + 1024 u = 0$ The characteristic equation is: $r^2 + 1024 = 0$ $r = \pm 32i$ Thus, the solution to the homogeneous equation is: $u_h(t) = C_1 \cos(32t) + C_2 \sin(32t)$

2. Particular Solution: For the particular solution, assume $u_p(t) = A \cos(5t) + B \sin(5t)$ and substitute into the differential equation.

3. Combine General Solution: The general solution will be: $u(t) = u_h(t) + u_p(t)$

Would you like me to continue with the detailed calculation of the particular solution and the application of initial conditions?

Here are related questions to deepen understanding:

1. How do we find the particular solution of a non-homogeneous differential equation?
2. What are the steps to convert units for consistency in a physical problem?
3. Why is it necessary to transform the spring constant from lb/in to lb/ft?
4. What role does the characteristic equation play in solving homogeneous differential equations?
5. How can initial conditions be used to solve for arbitrary constants in the general solution?

Tip: When working with physical units, ensure all units are consistent to avoid errors in the solution.