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Consider moving outward from the centre of the Earth up to an altitude equal to half the Earth's radius. At which of the following locations is the acceleration due to gravity g the greatest?

1. centre of the earth
2. At a depth half the radius of the earth
3. At the surface of the earth
4. At an altitude equal to half the radius of the earth.

Correct answer: At the surface of the earth

Solution

Inside a uniform Earth, g(r) = g_surface * (r/R), so it increases linearly from 0 at the centre to maximum at the surface (r = R). Above the surface, g = g_surface * (R/(R+h))² decreases. Among centre (g=0), depth R/2 (g = gₛ/2), surface (g = gₛ, maximum), and altitude R/2 (g = gₛ*(R/(1.5R))² = gₛ*4/9 ~ 0.44 gₛ), the largest value is at the surface.

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