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The problem shows three particles on which external forces act. Particles 1 and 2 are experiencing forces (5 N to the left for particle 1 and 3 N to the right for particle 2). We are asked to determine the magnitude and direction of the force on particle 3 under three different conditions for the center of mass of the system:

### Given:
- Particle 1: Force of 5 N to the left (negative x-direction).
- Particle 2: Force of 3 N to the right (positive x-direction).
- Particle 3: Force is unknown.

We need to consider the system’s center of mass in three cases: 
(a) Stationary (no net force).
(b) Moving with constant velocity (no net force again due to Newton's first law).
(c) Accelerating rightward (net force must accelerate the system to the right).

### Solution Strategy:
1. **Total Force**: The sum of forces on all particles must equal the net force acting on the system's center of mass.
2. **Conditions**:
    - In case (a) and (b), there’s no acceleration, so the net force must be zero.
    - In case (c), there is acceleration to the right, meaning there must be a net positive force acting in the positive x-direction.

### (a) Center of Mass is Stationary:
- For the system to be stationary, the net external force must be zero:
  \[
  F_1 + F_2 + F_3 = 0
  \]
  Using the given forces:
  \[
  -5 \, \text{N} + 3 \, \text{N} + F_3 = 0
  \]
  Solving for \( F_3 \):
  \[
  F_3 = 2 \, \text{N}
  \]
  So, the force on particle 3 is \( 2 \, \text{N} \) to the right.

### (b) Moving with Constant Velocity Rightward:
- Since the system is moving with constant velocity, the net force is again zero (Newton's First Law):
  \[
  F_1 + F_2 + F_3 = 0
  \]
  The same equation as in (a) applies, so the force on particle 3 is still \( 2 \, \text{N} \) to the right.

### (c) Accelerating Rightward:
- In this case, the center of mass is accelerating to the right, so there must be a net positive force in the x-direction.
  Let's assume the net force needed for the acceleration is \( F_\text{net} \). Then:
  \[
  -5 \, \text{N} + 3 \, \text{N} + F_3 = F_\text{net}
  \]
  For acceleration to the right, \( F_\text{net} \) must be a positive number. Therefore, \( F_3 \) must be greater than \( 2 \, \text{N} \). The exact value depends on the specified acceleration (which isn't provided directly in the problem).

Would you like more details on how to determine the exact force in case (c), or have any questions? Here are some related questions:

1. How do we calculate net force from given accelerations?
2. Why does a constant velocity imply no net force on a system?
3. How would forces differ if the particles had different masses?
4. How do we calculate the force on particle 3 in the case of upward acceleration?
5. How does Newton’s Third Law affect force calculations in multiple particle systems?

**Tip:** In systems with multiple particles, always start by analyzing the net external force to get a clear picture of how forces interact.

The figure shows an overhead view of three identical particles on which external forces act. The magnitudes and directions of the forces on two of the particles are indicated. What are the magnitude and direction of the force acting on the third particle if the center of mass of the three-particle system is (a) stationary, (b) moving at a constant velocity rightward, and (c) accelerating rightward?

Explore how to find the magnitude and direction of forces acting on a particle in a three-particle system, using Newton's Laws of Motion. This problem involves calculating the net force in cases where the system is stationary, moving with constant velocity, or accelerating. Suitable for high school physics students.

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