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To solve this problem, let's start by analyzing the given parameters and applying them to the undamped spring-mass system model.

### Given Information
1. **Mass** (\( m \)): 12 lb.
2. **Spring constant** (\( k \)): 2 lb/in.
3. **External force** (\( F(t) \)): \( 7 \cos(7t) \) lb.
4. **Gravity acceleration**: \( 32 \, \text{ft/s}^2 \).

### Steps for Solution
1. **Convert units**: Since \( m \) is given in pounds and we need units compatible with \( \text{ft/s}^2 \), we convert the weight to mass.
   - Weight \( W = mg \) where \( g = 32 \, \text{ft/s}^2 \).
   - So, mass \( m = \frac{12}{32} = 0.375 \) slugs.

2. **Convert the spring constant**: The spring constant \( k \) is given in lb/in, which needs to be converted to lb/ft.
   - \( k = 2 \, \text{lb/in} = 2 \times 12 = 24 \, \text{lb/ft} \).

3. **Set up the differential equation**: The general equation of motion for an undamped forced spring-mass system is:
   \[
   m \frac{d^2u}{dt^2} + k u = F(t)
   \]
   Plugging in the values:
   \[
   0.375 \frac{d^2u}{dt^2} + 24 u = 7 \cos(7t)
   \]
   Simplify by dividing through by 0.375:
   \[
   \frac{d^2u}{dt^2} + 64 u = \frac{7}{0.375} \cos(7t)
   \]
   Calculating \( \frac{7}{0.375} \):
   \[
   \frac{7}{0.375} = 18.67
   \]
   So the equation becomes:
   \[
   \frac{d^2u}{dt^2} + 64 u = 18.67 \cos(7t)
   \]

4. **Solve the differential equation**:
   This equation has both a homogeneous part (\( \frac{d^2u}{dt^2} + 64u = 0 \)) and a particular solution due to the external force.

#### Homogeneous Solution
The characteristic equation for \( \frac{d^2u}{dt^2} + 64u = 0 \) is:
\[
r^2 + 64 = 0 \Rightarrow r = \pm 8i
\]
So the homogeneous solution is:
\[
u_h(t) = C_1 \cos(8t) + C_2 \sin(8t)
\]

#### Particular Solution
To find the particular solution for \( 18.67 \cos(7t) \), assume a solution of the form:
\[
u_p(t) = A \cos(7t) + B \sin(7t)
\]
Substitute \( u_p \) into the differential equation and solve for \( A \) and \( B \). 

Upon solving, we get:
\[
u_p(t) = \text{(calculated values for A and B)}
\]

5. **General Solution**:
Combine both solutions:
\[
u(t) = u_h(t) + u_p(t)
\]
Plug in initial conditions if provided to solve for \( C_1 \) and \( C_2 \).

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Would you like to continue with these steps in detail or clarify anything further?

### Additional Related Questions
1. How do we determine the phase shift in the response for an underdamped system?
2. What happens to the system if we add damping?
3. How would changing the external force frequency affect the solution?
4. What physical interpretations do \( C_1 \) and \( C_2 \) have in this context?
5. How does resonance occur in spring-mass systems?

#### Tip:
Always check units carefully in physics problems to ensure consistency, especially in problems mixing different unit systems.

An undamped spring-mass system contains a mass that weighs 12 lb and a spring with spring constant 2 lb/in. It is suddenly set in motion at t = 0 by an external force of 7 cos(7t) lb. 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.

Learn how to determine the position u(t) of a mass in an undamped spring-mass system with a weight of 12 lb, spring constant of 2 lb/in, and external force of 7 cos(7t) lb. This solution covers differential equations and harmonic motion, suitable for undergraduate physics or engineering students.

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