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This solution outlines the setup and analysis of a forced, undamped harmonic oscillator. Let’s break down the key parts of the process:

1. **Establishing the Standard Form**: The problem is set up using the differential equation \( m u'' + \gamma u' + k u = F(t) \), where:
   - \( m \) is the mass,
   - \( \gamma \) is the damping coefficient,
   - \( k \) is the spring constant, and
   - \( F(t) \) is the external forcing function.

2. **Given Data and Unit Conversions**:
   - The mass \( mg = 13 \) lb, which is converted to slugs as \( m = \frac{13}{32} = 0.40625 \) slugs (using \( g = 32 \, \text{ft/s}^2 \)).
   - The spring constant \( k \) is calculated using \( k s = 13 \) and the stretch \( s = \frac{8}{12} = \frac{2}{3} \) ft. This gives \( k = \frac{13}{(2/3)} = 19.5 \, \text{lb/ft} \).

3. **Damping and Forcing**:
   - Since there’s no damping, \( \gamma = 0 \).
   - The external force is given as \( F(t) = 7 \sin(4 \sqrt{3} \, t) \, \text{lb} \).

4. **Initial Conditions**:
   - The mass is pulled down 3 inches, giving an initial displacement of \( u(0) = \frac{3}{4} = 0.75 \, \text{ft} \).
   - The initial velocity is zero, so \( u'(0) = 0 \).

5. **Simplifying the Differential Equation**:
   - The differential equation becomes \( 0.40625 u'' + 19.5 u = 7 \sin(4 \sqrt{3} \, t) \).
   - Dividing through by \( 0.40625 \) simplifies it to \( u'' + 48 u = \frac{224}{13} \sin(4 \sqrt{3} \, t) \).

6. **Characteristic Equation**:
   - The characteristic equation for the homogeneous system \( u'' + 48 u = 0 \) is \( \lambda^2 + 48 = 0 \), leading to solutions in terms of imaginary roots, indicating oscillatory motion.

Would you like a more detailed analysis on a specific part of this solution or explanations of any steps?

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Here are 5 related questions that could deepen your understanding:

1. How would the solution change if there was a damping coefficient \( \gamma \neq 0 \)?
2. What are the general solutions for the homogeneous equation \( u'' + 48u = 0 \)?
3. How does the external forcing function \( F(t) = 7 \sin(4 \sqrt{3} \, t) \) affect the system's steady-state behavior?
4. Can we determine the amplitude of the resulting oscillation from this solution?
5. What role does the initial displacement \( u(0) = 0.75 \) ft play in the solution?

**Tip:** When solving differential equations with sinusoidal forcing functions, look out for resonance conditions, which occur when the forcing frequency matches the natural frequency of the system.

Using standard form to solve the differential equation of a forced, undamped harmonic oscillator

This solution demonstrates how to set up and solve the differential equation for a forced, undamped harmonic oscillator. It covers key steps such as calculating the mass in slugs, determining the spring constant, setting initial conditions, and simplifying the differential equation. Suitable for college-level students studying physics or engineering.

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