The four systems slide on a smooth horizontal surface and...

The four systems slide on a smooth horizontal surface and have the same mass $m$. The configurations of mass in the two pairs are identical. What can be said about the acceleration of the mass center for each system? Explain any difference in the accelerations of the members.

Key Concepts

Center of Mass Motion

This concept states that the overall motion of a system of particles is determined solely by the net external forces acting on the system. In a situation where external forces are balanced or uniformly applied, the acceleration of the center of mass is predictable regardless of the configuration of the system's individual parts.

Newton’s Second Law

Newton’s second law connects the acceleration of an object or a system's center of mass with the net external forces acting upon it by the relation F = ma. This principle ensures that if the external force remains the same, the acceleration of the center of mass will be identical for systems having the same total mass.

Internal vs External Forces

In any multi-body system, internal forces (forces between the system’s parts) occur in equal and opposite pairs as per Newton’s third law and cancel out when considering the net force on the whole system. Therefore, only external forces affect the acceleration of the system’s center of mass.

Relative Acceleration

Although the acceleration of the center of mass is governed by external forces, the individual masses in a system can have different accelerations relative to the center of mass due to internal constraints or differences in configuration. This relative motion between parts does not alter the center of mass acceleration, which remains defined by the overall external influence.

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