The reason why a handrail doesn't move after an ice skater pushes off from it is primarily due to B. balanced force.
When the ice skater pushes against the handrail, they exert an action force on it, but the handrail (being fixed in place) exerts an equal and opposite reaction force back on the skater, according to Newton's third law of motion. The handrail does not move because the forces acting on it (the push from the skater and the structural forces holding it in place) are balanced.
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