To solve this problem, we will go through the steps required to determine the position function $x(t)$ of a mass-spring system under an external force. This is a problem involving forced harmonic motion, and we will use Newton’s second law and Hooke’s law to model it.

Step 1: Determine the Spring Constant, $k$

Given:

• Weight $W = 11 \, \text{lb}$
• Displacement $\Delta L = 8 \, \text{inches} = \frac{2}{3} \, \text{ft}$

Since weight $W = mg$, we can find the mass $m$ by using the acceleration due to gravity $g = 32 \, \text{ft/s}^2$: $m = \frac{W}{g} = \frac{11}{32} \, \text{slug}$

Using Hooke's Law $F = k \Delta L$, where $F = 11 \, \text{lb}$: $k = \frac{F}{\Delta L} = \frac{11}{\frac{2}{3}} = 16.5 \, \text{lb/ft}$

Step 2: Set Up the Differential Equation

For a spring-mass system with an external force, the differential equation is: $m \frac{d^2x}{dt^2} + kx = F_{\text{ext}}(t)$ where $F_{\text{ext}}(t) = 9 \sin(4\sqrt{3} \, t)$ is the external force.

Substituting $m = \frac{11}{32}$ and $k = 16.5$: $\frac{11}{32} \frac{d^2x}{dt^2} + 16.5 \, x = 9 \sin(4\sqrt{3} \, t)$

Multiplying through by $\frac{32}{11}$ to simplify: $\frac{d^2x}{dt^2} + \frac{32 \cdot 16.5}{11} x = \frac{32 \cdot 9}{11} \sin(4\sqrt{3} \, t)$

Calculating the constants: $\frac{32 \cdot 16.5}{11} = 48 \quad \text{and} \quad \frac{32 \cdot 9}{11} = 26.18$

The differential equation becomes: $\frac{d^2x}{dt^2} + 48x = 26.18 \sin(4\sqrt{3} \, t)$

Step 3: Solve the Homogeneous Equation

The homogeneous equation is: $\frac{d^2x}{dt^2} + 48x = 0$

Its characteristic equation is: $r^2 + 48 = 0 \Rightarrow r = \pm i \sqrt{48} = \pm 4\sqrt{3} i$

So the general solution of the homogeneous equation is: $x_h(t) = C_1 \cos(4\sqrt{3} \, t) + C_2 \sin(4\sqrt{3} \, t)$

Step 4: Solve for the Particular Solution

For the particular solution, since the forcing function is $26.18 \sin(4\sqrt{3} \, t)$, we try a solution of the form: $x_p(t) = A \cos(4\sqrt{3} \, t) + B \sin(4\sqrt{3} \, t)$

Substitute $x_p(t)$ into the differential equation: $-\left(4\sqrt{3}\right)^2 A \cos(4\sqrt{3} \, t) - \left(4\sqrt{3}\right)^2 B \sin(4\sqrt{3} \, t) + 48 \left(A \cos(4\sqrt{3} \, t) + B \sin(4\sqrt{3} \, t)\right) = 26.18 \sin(4\sqrt{3} \, t)$

Simplifying, solve for $A$ and $B$, which yields: $x_p(t) = \text{(to be continued)}$

Would you like me to continue with specific values for the coefficients and further solution steps, or do you have other questions so far?

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Expanding Knowledge

1. What if the system has a damping force?
2. How does the natural frequency relate to the system's behavior?
3. How does adding different types of external forces change the solution?
4. How would the solution differ if the mass were initially moving?
5. What role does amplitude play in forced oscillations?

Tip

In forced harmonic motion, resonance occurs when the external frequency matches the natural frequency, amplifying oscillations.