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To solve this problem, we need to model the motion of the mass-spring system, which is under the influence of an external force. Here's the step-by-step solution process:

### Step 1: Setting Up the Differential Equation
1. **Weight of the Mass:**  
   Since the weight of the mass is \( 11 \, \text{lb} \), we can determine its mass \( m \) using the relation:
   \[
   W = mg \Rightarrow m = \frac{W}{g} = \frac{11}{32} \, \text{slugs}
   \]
   where \( g = 32 \, \text{ft/s}^2 \) is the gravitational acceleration.

2. **Spring Constant \( k \):**  
   The mass stretches the spring by 8 inches (or \( \frac{8}{12} = \frac{2}{3} \, \text{ft} \)) under its weight of \( 11 \, \text{lb} \). The spring constant \( k \) can be calculated using Hooke's Law:
   \[
   kx = W \Rightarrow k = \frac{W}{x} = \frac{11}{\frac{2}{3}} = 16.5 \, \text{lb/ft}
   \]

3. **Formulating the Differential Equation:**  
   The general equation for a forced, damped mass-spring system is:
   \[
   m \frac{d^2 u}{dt^2} + c \frac{du}{dt} + ku = F(t)
   \]
   where:
   - \( m = \frac{11}{32} \),
   - \( c = 0 \) (no damping mentioned),
   - \( k = 16.5 \),
   - \( F(t) = 9 \sin(4 \sqrt{3} t) \).

   Substituting these values, the equation becomes:
   \[
   \frac{11}{32} \frac{d^2 u}{dt^2} + 16.5 u = 9 \sin(4 \sqrt{3} t)
   \]
   Multiplying through by \( \frac{32}{11} \) to simplify, we get:
   \[
   \frac{d^2 u}{dt^2} + \frac{528}{11} u = \frac{288}{11} \sin(4 \sqrt{3} t)
   \]
   Simplifying the coefficient of \( u \):
   \[
   \frac{d^2 u}{dt^2} + 48 u = \frac{288}{11} \sin(4 \sqrt{3} t)
   \]

### Step 2: Solving the Homogeneous Equation
The homogeneous equation associated with this system is:
\[
\frac{d^2 u}{dt^2} + 48 u = 0
\]
The characteristic equation is:
\[
r^2 + 48 = 0 \Rightarrow r = \pm i \sqrt{48} = \pm 4 \sqrt{3} i
\]
So, the solution to the homogeneous equation is:
\[
u_h(t) = C_1 \cos(4 \sqrt{3} t) + C_2 \sin(4 \sqrt{3} t)
\]

### Step 3: Solving the Particular Solution
Since the forcing function is \( F(t) = \frac{288}{11} \sin(4 \sqrt{3} t) \), we can use the method of undetermined coefficients. We assume a particular solution of the form:
\[
u_p(t) = A t \cos(4 \sqrt{3} t) + B t \sin(4 \sqrt{3} t)
\]
Differentiating and substituting back into the differential equation will yield values for \( A \) and \( B \).

### Step 4: Initial Conditions
The mass is pulled down 4 inches (or \( \frac{1}{3} \, \text{ft} \)) and then released, which gives the initial conditions:
\[
u(0) = -\frac{1}{3}, \quad u'(0) = 0
\]
Using these conditions, we can solve for \( C_1 \) and \( C_2 \) in the general solution:
\[
u(t) = u_h(t) + u_p(t)
\]

Would you like further details on solving for \( u_p(t) \) and applying the initial conditions, or do you have any specific questions?

### Relative Questions
1. How do we apply initial conditions to determine constants in differential equations?
2. What method can be used to solve for particular solutions in forced oscillatory systems?
3. Why do we use the method of undetermined coefficients in this context?
4. How do we interpret the physical significance of each term in the differential equation?
5. What are other common external forces in oscillatory systems?

#### Tip:
Always double-check units when working with physical constants in differential equations to ensure consistency across terms.

A mass that weighs 11 lb stretches a spring 8 in. The system is acted on by an external force 9 sin(4√3 t) lb. If the mass is pulled down 4 in and then released, determine the position of the mass at any time t. Use 32 ft/s^2 as the acceleration due to gravity.

Learn how to solve a differential equation for a mass-spring system acted on by an external force. This problem involves calculating spring constant, setting up the differential equation, solving the homogeneous and particular solutions, and applying initial conditions.

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