Answer to Question #86969 in Mechanics | Relativity for Maria

Question #86969
You are explaining to friends why an astronaut feels weightless orbiting in the space shuttle, and they respond that they thought gravity was just a lot weaker up there.
Convince them that it isn't so by calculating how much weaker (in %) gravity is 440 km above the Earth's surface.
Express your answer using three significant figures.
1
Expert's answer
2019-03-25T09:10:06-0400

We can find the acceleration of gravity at any height above the Earth’s surface from the formula:


"g_E = G \\dfrac{M_E}{(R_E + h)^2},"


here, "G" is the gravitational constant, "M_E=5.98 \\cdot 10^{24} kg" is the mass of the Earth, "R_E=6.38 \\cdot 10^6 m" is the radius of the Earth and "h" is the height above the Earth’s surface.

Let’s calculate the acceleration of gravity at "440 km" above the Earth’s surface:


"g_{440 km} = 6.67 \\cdot 10^{-11} \\dfrac{N \\cdot m^2}{kg^2} \\cdot \\dfrac{5.98 \\cdot 10^{24} kg}{(6.38 \\cdot 10^6 m + 4.4 \\cdot 10^5 m)^2} = 8.57 \\dfrac{m}{s^2}."

Let’s compare (in %) the acceleration of gravity at "440 km" above the Earth’s surface to the acceleration of gravity at the Earth’s surface:


"\\dfrac{g_{440 km}}{g_{Earth's surface}} = \\dfrac{8.57 \\dfrac{m}{s^2}}{9.8 \\dfrac{m}{s^2}} \\cdot 100 \\% = 87.4 \\%."

Answer:

Therefore, the gravity is 87.4% as strong as at the Earth’s surface.


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