Final answer:
To calculate g for a spherical shell planet, apply the shell theorem for r greater than the outer radius and note the gravitational acceleration is zero within the inner radius of the shell.
For r within the shell, the equation g = (4/3)πGρr applies, and for travel within the shell, there is no gravitational acceleration.
Step-by-step explanation:
To find the value of gravitational acceleration (g) as a function of radius r from the center of a spherical shell planet with constant density ρ and inner and outer radii Rin and Rout, we need to apply the shell theorem for the regions r > Rout and Rin < r < Rout, and use the concept of no gravitational force within a hollow shell for r < Rin.
For r > Rout, the gravitational pull is as if all the mass were concentrated at the center of the planet, so we use the standard gravitation equation:
g = G*M/r2, where M is the total mass of the shell.
For Rin < r < Rout, the shell theorem states that the gravitational pull from the mass of the shell outside your radius cancels out, so only the mass at a radius less than r contributes to g:
g = (4/3)πGρr (correct for a solid sphere of uniform density, not a shell)
For r < Rin, inside a spherical shell, the gravitational force (and thus acceleration) is zero. This is because all the mass elements of the shell provide forces in different directions that cancel each other out.
Therefore, when considering travel inside the spherical shell planet, there would be no gravitational acceleration, making such travel quite unusual and challenging as there would be no ‘down’ direction.