Variation Of G With Altitude

Question

A weather balloon rises to an altitude equal to half the radius of the Earth. What fraction of its ground-level weight does the instrument package retain at that height?

RankGuru · Accepted Answer

NCERT Class 11, Chapter 8 (Gravitation) gives the variation of gravity with altitude as $g_h = g\left(\dfrac{R}{R+h}\right)^2$. 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 $h = \dfrac{R}{2}$, so the distance from the centre is $R + h = \dfrac{3R}{2}$. Substituting, $\dfrac{g_h}{g} = \left(\dfrac{R}{3R/2}\right)^2 = \left(\dfrac{2}{3}\right)^2 = \dfrac{4}{9}$, so the package weighs $\dfrac{4}{9}$ 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 $\dfrac{4}{9} \approx 0.44$ is a reasonable intermediate value.

Variation Of G With Altitude

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