Two spheres of mass M and 2M float in space in the absence of external gravitational forces, as shown in the figure. Which of the following predictions is correct about the motion of the center of mass of the two-sphere system?

The correct prediction about the motion of the center of masses is ;

( A ) The center of mass remains at rest.

Since the two spheres are floating without the presence an an external force ( external gravitational force ) their momentum will be conserved because we assume they are both at rest .

Given that momentum is conserved in both sphere their centers of mass will also remain at rest just like the spheres are assumed to be at rest in the absence of an external force.  

Hence In conclusion the center of mass of both spheres will remain at rest

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-- If the two spheres are both at rest when we arrive and start watching them, then their total momentum is zero.  

-- If their center of mass were to move, then that would mean that some momentum had somehow magically appeared.

-- But we know that momentum is conserved so this can't happen.

-- The center of mass must remain at rest. (A)

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b. lifting dumbbell weight at 20-degree angle from the body after a 45-degree angle