Secondly I think you got mixed up in your question 🙂 . Gravity does affect us on Earth it is what keeps us standing on the floor and not floating off all the time. Gravity is a force which a planet exerts on things around it, pulling things towards its centre. The bigger and heavier the planet the bigger the gravitational force. However as an object moves away from the centre of a planet the gravitational force acting on it reduces.
It is not actually true that there is no gravity on the moon. For a start the moon orbits the earth because it is feeling the pull of the earth’s gravitational force. If it was not affected by the earth’s gravity it would fly off into space and we wouldn’t see it again 🙁 .
However the moon also has it’s own gravitational force because although it is smaller than earth it is still pretty big and pretty heavy 🙂 . So when an astronaut is standing on the surface of the moon he is being affected by the moon’s gravitational force. It just that, that force is a fifth of the force you feel on Earth so they bounce around a bit and find it hard to stay on the ground. If there was no gravity when they bounced up they would never come back down!
This is a great question that naomi has covered really well. Just wanted to add that this is why you see astronauts floating upside down in their shuttles because they are usually travelling too far away from a planet for it to have enough of an effect on them. When astronauts are in space they have to do a lot of physical activity because our body reacts to being used. If we don’t use parts of or bodies the muscles get weaker and the strengths of our bones reduce. when astronauts are in space, because of the reduced gravity, the body thinks its not being used and starts to get weaker so they take gym equipment with them to keep their bodies strong.
Something a little bit different from what Naomi and Rachel have said…
Have you heard of or seen an anti-gravity treadmill?
They are used in rehab by people who want to carry on running whilst letting their body heal, and without causing any further injury. It basically removes the impact of running from the legs! Running puts a lot of forces through our ankles and knees, and can cause them to wear. This lower impact method means that the knees and ankles remain healthy!
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