An astronaut lands on a new planet. The new planet’s mass is half that of the
Earth and the new planet’s radius is half that of the Earth. If he weighs 500 N on
Earth, what will the force of gravity be on him on the new planet?

Ah hah !  Very nice problem.-- Since the mass is 1/2 of the Earth's mass, the gravitational forces between him and the new planet would be 1/2 of what they are on Earth.-- Since the distance between him and the center of the new planet is 1/2 of the distance between him and the Earth's center, the gravitational forces between him and the new planet would be (2)² = 4 times what they are on Earth.-- These conditions combine to make his weight  (1/2) x (4) = 2 times his earth weight.-- He weighs 1,000 N (about 224.8 pounds) on the new planet.(Maybe that's why they selected an astronaut for this mission who only weighs about 112.5 pounds on Earth.)