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This problem involves analyzing the motion of a mass-spring system with an external force. Let's break it down step by step to determine the equation of motion.

### Step 1: Understanding the mass-spring system

We are given:
- A mass \( m = 15 \, \text{kg} \) elongates the spring by \( \frac{5}{32} \) meters.
- At \( t = 0 \), the mass is released from a point 2 meters above the equilibrium position.
- Initial velocity is \( v_0 = 4 \, \text{m/s} \), directed upward (positive direction).
- There is an external force \( f(t) = -945 \cos(t) \).
- Gravitational acceleration is \( g = 10 \, \text{m/s}^2 \).

### Step 2: Determine the spring constant \( k \)

The mass elongates the spring by \( \Delta L = \frac{5}{32} \, \text{m} \). This means the spring force balances the gravitational force at equilibrium:
\[
mg = k \Delta L
\]
Solving for \( k \):
\[
k = \frac{mg}{\Delta L} = \frac{15 \times 10}{\frac{5}{32}} = \frac{150}{\frac{5}{32}} = 960 \, \text{N/m}
\]

### Step 3: Set up the equation of motion

The general equation of motion for a damped spring-mass system with an external force is:
\[
m \frac{d^2x}{dt^2} + kx = f(t)
\]
Substitute the known values:
\[
15 \frac{d^2x}{dt^2} + 960x = -945 \cos(t)
\]
Simplify the equation:
\[
\frac{d^2x}{dt^2} + \frac{960}{15} x = \frac{-945}{15} \cos(t)
\]
\[
\frac{d^2x}{dt^2} + 64x = -63 \cos(t)
\]

This is the second-order differential equation governing the motion of the mass.

### Step 4: Initial conditions

At \( t = 0 \):
- \( x(0) = 2 \, \text{m} \) (2 meters above equilibrium).
- \( \frac{dx}{dt}(0) = 4 \, \text{m/s} \) (initial velocity).

### Step 5: Solve the equation of motion

The general solution to the equation of motion \( \frac{d^2x}{dt^2} + 64x = -63 \cos(t) \) consists of two parts:
1. **The homogeneous solution** (solution to \( \frac{d^2x}{dt^2} + 64x = 0 \)):
   \[
   x_h(t) = A \cos(8t) + B \sin(8t)
   \]
2. **The particular solution** (solution to \( \frac{d^2x}{dt^2} + 64x = -63 \cos(t) \)):
   We assume a particular solution of the form \( x_p(t) = C \cos(t) \). Substituting into the differential equation:
   \[
   -C \cos(t) + 64C \cos(t) = -63 \cos(t)
   \]
   \[
   C(64 - 1) = -63 \quad \Rightarrow \quad C = -\frac{63}{63} = -1
   \]
   So, the particular solution is:
   \[
   x_p(t) = -\cos(t)
   \]

The general solution is:
\[
x(t) = A \cos(8t) + B \sin(8t) - \cos(t)
\]

### Step 6: Apply initial conditions

Using the initial conditions \( x(0) = 2 \) and \( \frac{dx}{dt}(0) = 4 \):

1. At \( t = 0 \):
   \[
   x(0) = A - 1 = 2 \quad \Rightarrow \quad A = 3
   \]

2. Differentiate \( x(t) \) to find \( \frac{dx}{dt}(t) \):
   \[
   \frac{dx}{dt} = -8A \sin(8t) + 8B \cos(8t) + \sin(t)
   \]
   At \( t = 0 \):
   \[
   4 = 8B + 0 \quad \Rightarrow \quad B = \frac{1}{2}
   \]

### Final Equation of Motion:

The final equation of motion is:
\[
x(t) = 3 \cos(8t) + \frac{1}{2} \sin(8t) - \cos(t)
\]

This describes the motion of the mass under the influence of the spring and external force.

### Graphical Representation

You can represent the function \( x(t) = 3 \cos(8t) + \frac{1}{2} \sin(8t) - \cos(t) \) using an app like SLOPES to visualize the oscillations and external force interaction.

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Would you like further details or clarifications on any part of this problem?

Here are 5 related questions to expand the discussion:
1. How does the external force affect the oscillation of the mass?
2. What would happen if the external force had a different frequency?
3. How would the equation change if the spring had some damping?
4. What are the effects of initial velocity on the motion of the system?
5. How can resonance occur in a mass-spring system?

**Tip**: When solving second-order differential equations with external forces, always check the form of the external force to determine the particular solution accurately.

Una masa de 15kg alarga 5/32 metros un resorte. En t=0 se libera la masa desde un punto que está a 2 metros arriba de la posición de equilibrio con una velocidad ascendente de 4m/s. Esta masa tiene asociado una fuerza externa f(t) = −945cos(t). Determine la ecuación del movimiento (g=10m/s). Represente gráficamente, use SLOPES (app).

Determine the equation of motion for a mass-spring system where a 15kg mass elongates the spring by 5/32 meters. The system has an external force f(t) = −945 cos(t). This solution includes step-by-step analysis and graphical representation.

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