Physics

Physics

Problem

Which one of the following expressions gives the acceleration due to gravity gdg_d at a depth dd below Earth’s surface (where d≪REd\ll R_{\mathrm E}) in terms of the surface value gg? a) gd=g(1−2dRE)g_d = g\left(1-\dfrac{2d}{R_{\mathrm E}}\right) b) gd=g(1−dRE)g_d = g\left(1-\dfrac{d}{R_{\mathrm E}}\right) c) gd=g(1−(dRE)2)g_d = g\left(1-\left(\dfrac{d}{R_{\mathrm E}}\right)^{2}\right) d) gd=g(RERE−d)2g_d = g\left(\dfrac{R_{\mathrm E}}{R_{\mathrm E}-d}\right)^{2}
Intuitive explanation

The key insight is that only the mass inside a spherical shell contributes to gravity—the outer layers cancel out. As you go deeper, you're excluding more mass above you, so the effective gravitating mass shrinks. Since d≪REd \ll R_E, you can treat Earth's density as roughly constant and use a linear approximation: the gravitating mass scales with the remaining radius, making gdg_d proportional to (RE−d)(R_E - d) rather than RE2R_E^2.

Intuitive explanation

The key insight is that only the mass inside a spherical shell contributes to gravity—the outer layers cancel out. As you go deeper, you're excluding more mass above you, so the effective gravitating mass shrinks. Since d≪REd \ll R_E, you can treat Earth's density as roughly constant and use a linear approximation: the gravitating mass scales with the remaining radius, making gdg_d proportional to (RE−d)(R_E - d) rather than RE2R_E^2.

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