How is the acceleration of the center of mass of a system related to external forces?

How is the acceleration of the center of mass of a system related to external forces?

• A. a_cm = F_ext / m
• B. a_cm = F_ext / M
• C. a_cm = sum of internal forces / M
• D. a_cm is independent of external forces

Answer: (B)

The acceleration of the center of mass depends on the net external force acting on the system, not on the internal interactions inside it. If you apply Newton’s second law to every particle in the system and add up all those equations, the internal forces cancel in pairs because action and reaction forces occur between parts of the system. What remains is the net external force, F_ext, driving the entire system with total mass M (the sum of all the individual masses). This gives the relationship F_ext = M a_cm, so a_cm = F_ext / M. The net external force sets the direction and magnitude of the center-of-mass acceleration. Why the other ideas don’t fit: using a single particle’s mass would not describe a system’s motion, internal forces don’t change how the center of mass accelerates, and saying the center of mass is independent of external forces contradicts Newton’s laws.

The acceleration of the center of mass depends on the net external force acting on the system, not on the internal interactions inside it. If you apply Newton’s second law to every particle in the system and add up all those equations, the internal forces cancel in pairs because action and reaction forces occur between parts of the system. What remains is the net external force, F_ext, driving the entire system with total mass M (the sum of all the individual masses). This gives the relationship F_ext = M a_cm, so a_cm = F_ext / M. The net external force sets the direction and magnitude of the center-of-mass acceleration.

Why the other ideas don’t fit: using a single particle’s mass would not describe a system’s motion, internal forces don’t change how the center of mass accelerates, and saying the center of mass is independent of external forces contradicts Newton’s laws.