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We are given a spring-mass system with the following dataWe are given a spring-mass system with the following 

- **Mass:** \( m = 15 \, \text{kg} \)
- **Spring stretch:** \( \frac{5}{32} \, \text{m} \)
- **At \( t = 0 \):**
  - **Initial position:** 2 meters above the equilibrium position
  - **Initial velocity:** 4 m/s (upward)
- **External force:** \( f(t) = -945 \cos(t) \)
- **Gravitational acceleration:** \( g = 10 \, \text{m/s}^2 \)

We need to find the equation of motion.

### Step 1: Find the spring constant \( k \)

The force due to the spring is \( F = kx \), where \( x \) is the displacement from equilibrium. From Hooke's law, the weight of the mass is balanced by the spring's restoring force:
\[
mg = kx
\]
Given \( m = 15 \, \text{kg} \), \( g = 10 \, \text{m/s}^2 \), and \( x = \frac{5}{32} \, \text{m} \), we can solve for \( k \):
\[
15 \times 10 = k \times \frac{5}{32}
\]
\[
k = \frac{150 \times 32}{5} = 960 \, \text{N/m}
\]

### Step 2: Set up the differential equation

The equation of motion for a forced spring-mass system is given by:
\[
m \ddot{x} + kx = f(t)
\]
Substituting the known values of \( m = 15 \), \( k = 960 \), and \( f(t) = -945 \cos(t) \):
\[
15 \ddot{x} + 960x = -945 \cos(t)
\]
Dividing the equation by 15:
\[
\ddot{x} + \frac{960}{15}x = \frac{-945}{15} \cos(t)
\]
\[
\ddot{x} + 64x = -63 \cos(t)
\]

### Step 3: Solve the equation

This is a second-order non-homogeneous differential equation. The general solution is the sum of the homogeneous and particular solutions.

1. **Homogeneous equation:**
   \[
   \ddot{x} + 64x = 0
   \]
   The characteristic equation is:
   \[
   r^2 + 64 = 0 \quad \Rightarrow \quad r = \pm 8i
   \]
   So, the general solution of the homogeneous part is:
   \[
   x_h(t) = A \cos(8t) + B \sin(8t)
   \]

2. **Particular solution:**
   Since the non-homogeneous term is \( -63 \cos(t) \), we propose a particular solution of the form:
   \[
   x_p(t) = C \cos(t)
   \]
   Substituting into the differential equation:
   \[
   -C \cos(t) + 64C \cos(t) = -63 \cos(t)
   \]
   Simplifying:
   \[
   C(-1 + 64) = -63 \quad \Rightarrow \quad C = \frac{-63}{63} = -1
   \]
   So, the particular solution is:
   \[
   x_p(t) = -\cos(t)
   \]

### Step 4: General solution

The general solution is the sum of the homogeneous and particular solutions:
\[
x(t) = A \cos(8t) + B \sin(8t) - \cos(t)
\]

### Step 5: Apply initial conditions

- At \( t = 0 \), \( x(0) = 2 \):
  \[
  x(0) = A \cos(0) + B \sin(0) - \cos(0) = A - 1 = 2 \quad \Rightarrow \quad A = 3
  \]

- At \( t = 0 \), \( \dot{x}(0) = 4 \, \text{m/s} \) (upward):
  \[
  \dot{x}(t) = -8A \sin(8t) + 8B \cos(8t) + \sin(t)
  \]
  At \( t = 0 \), \( \dot{x}(0) = 4 \):
  \[
  \dot{x}(0) = 8B + 0 + 0 = 4 \quad \Rightarrow \quad B = \frac{4}{8} = \frac{1}{2}
  \]

### Final Equation of Motion:

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

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Do you want me to explain the steps in more detail or answer any specific questions?

### Related questions:
1. How does the external force \( f(t) = -945 \cos(t) \) influence the motion?
2. What are the physical meanings of the constants \( A \) and \( B \) in the general solution?
3. How do you graphically represent the equation of motion \( x(t) \)?
4. What would happen to the motion if the external force was removed?
5. How can energy conservation be applied to this spring-mass system?

#### Tip:
When solving second-order differential equations, always verify if the system is underdamped, overdamped, or critically damped to understand its behavior.

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) = -945 cos(t). Determine la ecuación del movimiento (g=10m/s).

Solve a spring-mass system where a 15kg mass stretches a spring by 5/32 meters. The system is influenced by an external force f(t) = -945 cos(t). Learn how to derive the equation of motion using second-order differential equations and harmonic motion principles.

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