Conservation of Linear Momentum for Zero Net External Force

To determine which physical quantity remains conserved, let us analyze the laws of mechanics for a system of particles from an inertial reference frame:

1. Linear Momentum:
   According to Newton's second law for a system of particles, the rate of change of total linear momentum $\vec{P}$ is equal to the net external force $\vec{F}_{\text{ext}}$ acting on the system:
   $\vec{F}_{\text{ext}} = \frac{d\vec{P}}{dt}$
   Given that the resultant of all external forces is zero, i.e., $\vec{F}_{\text{ext}} = \vec{0}$, we have:
   $\frac{d\vec{P}}{dt} = \vec{0} \implies \vec{P} = \text{constant}$
   Therefore, the linear momentum of the system does not change in time.

2. Kinetic Energy and Potential Energy:
   The work done by internal forces within the system can be non-zero even if no external forces act on it. For example, in an internal explosion or an oscillating spring-mass system with no external forces, internal forces do work:
   $W_{\text{int}} = \Delta K = -\Delta U$
   Consequently, the kinetic energy $K$ and potential energy $U$ of the system can both change with time.

3. Angular Momentum:
   The rate of change of total angular momentum $\vec{L}$ about a fixed point is equal to the net external torque $\vec{\tau}_{\text{ext}}$ about that point:
   $\vec{\tau}_{\text{ext}} = \frac{d\vec{L}}{dt}$
   Even if the resultant external force is zero ($\sum \vec{F}_i = \vec{0}$), the forces may constitute a couple (i.e., equal and opposite forces acting along different lines of action). In such a case, the net external torque is non-zero:
   $\vec{\tau}_{\text{ext}} = \sum \vec{r}_i \times \vec{F}_i \neq \vec{0}$
   This results in a change in the total angular momentum of the system over time.

Hence, one can surely say only that the linear momentum of the system does not change in time.

Correct Answer: A