Newton’s third law states that for every action, there is an equal and opposite reaction. Thus, when the Earth exerts a gravitational force on the Moon, the Moon exerts an equal and opposite gravitational force on the Earth.
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The force of attraction of the Earth provides the centripetal force that keeps the moon in orbit around it.
The gravitational force of the Earth attracts a falling apple towards it.
The acceleration due to the Earth’s mass is much larger than the acceleration due to the apple’s mass, making the motion of the Earth practically imperceptible compared to the motion of the falling apple.
The centripetal force acts towards the center of the circular path, keeping the stone moving along the path.