To find the acceleration of the center of mass of the system of three particles, we will follow these steps:
### Step 1: Identify the masses and positions of the particles
We have three particles with the following properties:
- Particle 1: Mass \( m_1 = 8 \, \text{kg} \) at position \( (4, 1) \)
- Particle 2: Mass \( m_2 = 4 \, \text{kg} \) at position \( (-2, 2) \)
- Particle 3: Mass \( m_3 = 4 \, \text{kg} \) at position \( (1, -3) \)
### Step 2: Identify the forces acting on each particle
The external forces acting on the particles are:
- Force on Particle 1: \( \vec{F}_1 = 6 \hat{j} \, \text{N} \)
- Force on Particle 2: \( \vec{F}_2 = -6 \hat{i} \, \text{N} \)
- Force on Particle 3: \( \vec{F}_3 = 14 \hat{i} \, \text{N} \)
### Step 3: Calculate the acceleration of each particle
Using Newton's second law, the acceleration \( \vec{a} \) of each particle can be calculated as:
\[
\vec{a}_i = \frac{\vec{F}_i}{m_i}
\]
- For Particle 1:
\[
\vec{a}_1 = \frac{6 \hat{j}}{8} = \frac{3}{4} \hat{j} \, \text{m/s}^2
\]
- For Particle 2:
\[
\vec{a}_2 = \frac{-6 \hat{i}}{4} = -\frac{3}{2} \hat{i} \, \text{m/s}^2
\]
- For Particle 3:
\[
\vec{a}_3 = \frac{14 \hat{i}}{4} = \frac{7}{2} \hat{i} \, \text{m/s}^2
\]
### Step 4: Calculate the total mass of the system
The total mass \( M \) of the system is:
\[
M = m_1 + m_2 + m_3 = 8 + 4 + 4 = 16 \, \text{kg}
\]
### Step 5: Calculate the acceleration of the center of mass
The acceleration of the center of mass \( \vec{a}_{cm} \) is given by:
\[
\vec{a}_{cm} = \frac{m_1 \vec{a}_1 + m_2 \vec{a}_2 + m_3 \vec{a}_3}{M}
\]
Substituting the values:
\[
\vec{a}_{cm} = \frac{8 \left(\frac{3}{4} \hat{j}\right) + 4 \left(-\frac{3}{2} \hat{i}\right) + 4 \left(\frac{7}{2} \hat{i}\right)}{16}
\]
Calculating each term:
- For Particle 1:
\[
8 \cdot \frac{3}{4} \hat{j} = 6 \hat{j}
\]
- For Particle 2:
\[
4 \cdot -\frac{3}{2} \hat{i} = -6 \hat{i}
\]
- For Particle 3:
\[
4 \cdot \frac{7}{2} \hat{i} = 14 \hat{i}
\]
Now, summing these:
\[
\vec{a}_{cm} = \frac{(-6 \hat{i} + 14 \hat{i} + 6 \hat{j})}{16} = \frac{(8 \hat{i} + 6 \hat{j})}{16}
\]
This simplifies to:
\[
\vec{a}_{cm} = \frac{1}{2} \hat{i} + \frac{3}{8} \hat{j} \, \text{m/s}^2
\]
### Step 6: Calculate the magnitude of the acceleration of the center of mass
The magnitude \( |\vec{a}_{cm}| \) is calculated as:
\[
|\vec{a}_{cm}| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{3}{8}\right)^2}
\]
Calculating each term:
\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}, \quad \left(\frac{3}{8}\right)^2 = \frac{9}{64}
\]
Finding a common denominator (64):
\[
\frac{1}{4} = \frac{16}{64}
\]
Thus,
\[
|\vec{a}_{cm}| = \sqrt{\frac{16}{64} + \frac{9}{64}} = \sqrt{\frac{25}{64}} = \frac{5}{8} \, \text{m/s}^2
\]
### Final Answer
The acceleration of the center of mass of the system is:
\[
\frac{5}{8} \, \text{m/s}^2 \quad \text{or} \quad 0.625 \, \text{m/s}^2
\]
