Final answer:
The center of mass for two astronauts in space, connected by a cable and at rest, will not move unless acted upon by external forces, due to the conservation of momentum.
Step-by-step explanation:
The question asks about the center of mass movement for a system of two astronauts in space connected by a lightweight cable. When two space walkers, one with a mass of 120 kg and the other with a mass of 85 kg, are at rest and hold on to the ends of a 12m long cable, the system's center of mass will not move unless an external force acts on the system. This is due to the conservation of momentum, which holds that if there are no external forces, the center of mass of an isolated system will remain stationary.
If we wish to calculate the actual position of the center of mass within the system, we'd use the formula:
Center of mass = (m1 * x1 + m2 * x2) / (m1 + m2),
where m1 and m2 are the masses of the astronauts and x1 and x2 are their respective positions from a chosen reference point. Because no external forces act on the system, there will be no net movement of the center of mass.