Question

If acceleration due to gravity $g$ at height $h \ll R$ (where $R$ is radius of earth) is $g_h = g_0(1+\frac{h}{R})^{-2}$, then using binomial theorem which is correct?

• A. $g_h = g_0$
• B. $g_h = g_0(1-\frac{2h}{R})$
• C. $g_h = g_0(1+\frac{2h}{R})$
• D. $g_h = g_0(1-\frac{h}{2R})$

Answer 1

Answer: (B)

$g_h = g_0 \left(1 - \frac{2h}{R}\right)$

Answer 2

The acceleration due to gravity at height $h$ is given by $g_h = g_0(1+\frac{h}{R})^{-2}$. Since $h \ll R$, we can use the binomial approximation $(1+x)^n \approx 1+nx$ for $x \ll 1$. Here, $x = \frac{h}{R}$ and $n = -2$. Substituting these values, we get $(1+\frac{h}{R})^{-2} \approx 1 - 2\frac{h}{R}$. Therefore, $g_h = g_0(1 - \frac{2h}{R})$.