Animated Solution for Physics - System of Particles: If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that

When we analyze a system of particles—whether it's a rigid block of wood, a swirling cloud of interstellar gas, or a complex machine—we must distinguish between two types of forces: internal forces and external forces.

Internal forces are the pushes and pulls that the particles within the system exert on each other. According to Newton's Third Law, these forces always occur in equal and opposite pairs. Therefore, when we sum up all the forces acting on every particle in the system, the internal forces perfectly cancel each other out.

This brings us to the master equation of system dynamics, derived directly from Newton's Second Law:

Here, $\sum {\vec{F}}_{\text{ext}}$ is the vector sum of all external forces acting on the system, and $\vec{P}$ is the total linear momentum of the system.

The problem explicitly states a powerful condition: the resultant of all external forces acting on the system is zero.

By substituting this into our master equation, we get a profound mathematical statement:

In calculus, if the time derivative of a quantity is zero, it means that the quantity is not changing with time. It is a constant. Therefore, we can definitively conclude that the total linear momentum of the system, $\vec{P}$, remains constant. This is the celebrated Law of Conservation of Linear Momentum.

It is equally important to understand why we cannot guarantee the conservation of the other quantities listed in the options.

Kinetic and Potential Energy: While internal forces cancel out when calculating the net force, they do not necessarily cancel out when calculating work. Imagine a firecracker floating in deep space. There are no external forces, so its linear momentum is conserved. However, when it explodes, the internal chemical forces do a massive amount of work, converting chemical potential energy into kinetic energy. Thus, kinetic and potential energy can change drastically even when the net external force is zero.

Angular Momentum: Can the angular momentum change if the net external force is zero? Absolutely. Imagine a steering wheel. If you pull down on the left side with a force of $10\text{ N}$ and pull up on the right side with a force of $10\text{ N}$, the net force is zero. However, these forces create a 'couple' which produces a net torque. According to the rotational analog of Newton's Second Law ($\sum {\vec{\tau }}_{\text{ext}}=\frac{d\vec{L}}{dt}$), this net torque will cause the angular momentum $\vec{L}$ to change.

Therefore, the only quantity we can surely say is conserved under these conditions is the linear momentum.