Introduction:
When a system of particles experiences mutually interacting forces, it is important to analyze the effect of these forces on the system's total linear momentum. Linear momentum is a vector quantity that represents the motion of an object and is defined as the product of its mass and velocity. In order to show that mutually interacting forces have no effect on the total linear momentum of a system, we need to consider the principles of Newton's third law of motion and the conservation of linear momentum.

Newton's Third Law of Motion:
Newton's third law states that for every action, there is an equal and opposite reaction. This means that when two particles interact, the forces they exert on each other are equal in magnitude and opposite in direction. These forces act on different particles and therefore do not cancel each other out.

Conservation of Linear Momentum:
The principle of conservation of linear momentum states that the total linear momentum of an isolated system remains constant if no external forces act on it. This means that the total linear momentum before an interaction is equal to the total linear momentum after the interaction.

Explanation:
1. Consider a system of particles, each with their own mass and velocity.
2. When these particles interact with each other, they exert forces on each other according to Newton's third law.
3. These mutually interacting forces may cause the individual particles to accelerate or change their velocities.
4. However, the total linear momentum of the system remains constant before and after the interaction.
5. This is because the forces acting on different particles in the system are equal and opposite, resulting in a cancellation of their contributions to the total linear momentum.
6. The change in momentum of one particle due to the interaction is exactly balanced by the change in momentum of another particle, resulting in no net change in the total linear momentum of the system.
7. Therefore, the mutually interacting forces on a system of particles have no effect on its total linear momentum.

Conclusion:
Mutually interacting forces on a system of particles do not affect its total linear momentum. This is due to the equal and opposite nature of these forces as described by Newton's third law of motion. The principle of conservation of linear momentum further confirms that the total linear momentum of an isolated system remains constant when no external forces act on it. Understanding these principles allows us to analyze the motion of systems of particles and predict their behavior.