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?
Answer (3)
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.)
The force of gravity on the astronaut will be 1000 N on the new planet where the planet's mass is half and its radius is half that of Earth's.
To determine the force of gravity on the astronaut on the new planet, we use Newton's law of universal gravitation. The formula for gravitational force is:
F = r 2 G ∗ m 1 ∗ m 2
where G is the gravitational constant, m 1 and m 2 are the masses of the two objects, and r is the distance between their centers.
Given that the astronaut weighs 500 N on Earth, we first find his mass:
Weight = Mass * Acceleration due to gravity
500 N = Mass * 9.8 s 2 m
Mass of the astronaut, m = 9.8 s 2 m 500 N ≈ 51 k g
For the new planet, the mass is half that of Earth's (0.5M) and the radius is half (0.5R). Using the gravitational formula:
New gravitational force, F ′ = ( 0.5 R ) 2 G ∗ m ∗ 0.5 M
Simplifying the equation:
F ′ = 0.25 R 2 G ∗ m ∗ 0.5 M = 4 ∗ R 2 G ∗ m ∗ 0.5 M = 2 ∗ R 2 G ∗ m ∗ M
Since R 2 G ∗ m ∗ M is the gravitational force on Earth (500 N):
F' = 2 * (500 N) = 1000 N
Therefore, the force of gravity on the astronaut on the new planet will be 1000 N.
The astronaut's weight on the new planet will be approximately 1000 N. This is due to the combination of the planet's mass being half of Earth's and its radius also being half, which effectively doubles the gravitational pull compared to what he would experience on Earth. Therefore, the astronaut will feel heavier on this new planet, despite the decreased mass and radius.
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