The Moon, in the absence of the Earth, would move in a straight line. Earth's pull at a given instant of time pulls the moon closer to Earth, the new direction of Moon now being the resultant of it's old motion and Earth's inward pull. The moon would then like to move in a straight line, but along this new direction, on account of its inertia as enunciated by Newton's first law, but the next instant, the Earth pulls again the moon a bit more. If you keep doing this exercise, you will realize that the Moon describes an ellipse. So, the answer to your question is yes and no, yes because Earth makes the Moon move around it in a particular way due to gravity, but the cause of the initial motion of the Moon till Earth got hold of it is probably is buried in the formation of the Solar System!
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The gravity on the Moon is about 1/6th that of Earth's gravity, due to its smaller mass and size. This means objects on the Moon weigh much less compared to Earth. Astronauts can jump higher and move more easily on the Moon due to the weaker gravitational force.
Gravity on Earth is about 6 times stronger than on the moon. This means that objects weigh less on the moon compared to Earth due to its weaker gravitational pull.
It takes the moon 27.322 days to orbit the Earth. While the moon rotates, the moon also rotates. Due to this rotation, we always see the same side of the moon.
Anything, including an apple, weighs heavier on earth due to earth's gravity. The gravity on the moon is much less, so objects are lighter.
The golf ball would travel much farther than on Earth because of the moon's lower gravity and lack of atmosphere to slow it down. It would also move in a straighter path due to the absence of air resistance and gravitational pull.