Variation Of G With Altitude

NCERT Class 11, Chapter 8 (Gravitation) gives the variation of gravity with altitude as $g_h=g{\left(\frac{R}{R+h}\right)}^2$. This full inverse-square form must be used whenever the height is comparable to the Earth's radius, because the simple linear approximation $g_h\approx g(1-2h/R)$ holds only for heights very small compared with $R$. Weight is directly proportional to the local value of gravity, so the ratio of the package's weight to its ground weight equals this same factor exactly. Here $h=\frac{R}{2}$, so the distance from the centre is $R+h=\frac{3R}{2}$. Substituting, $\frac{g_h}{g}={\left(\frac{R}{3R/2}\right)}^2={\left(\frac{2}{3}\right)}^2=\frac{4}{9}$, so the package weighs $\frac{4}{9}$ of its ground value. The option $2/3$ forgets to square the distance ratio. The option $1/2$ incorrectly assumes the weight halves because the altitude is half a radius, treating the fall-off as linear. The option $1/4$ uses $R+h=2R$, which corresponds to $h=R$, not $h=R/2$. As a plausibility check, since $h$ is a significant fraction of $R$, the weight should drop noticeably but not vanish, and $\frac{4}{9}\approx 0.44$ is a reasonable intermediate value.